arXiv:math/0201175v3 [math.AG] 22 Jul 2002 · Faltings’ “almost purity theorem”, which is the...

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ALMOST RING THEORY

OFER GABBER AND LORENZO RAMERO

sixth (and final) release

Ofer GabberI.H.E.S.Le Bois-Marie35, route de ChartresF-91440 Bures-sur-Yvettee-mail address:[email protected]

Lorenzo RameroUniversite de Bordeaux ILaboratoire de Mathematiques351, cours de la LiberationF-33405 Talence Cedexe-mail address:[email protected] page:http://www.math.u-bordeaux.fr/∼ramero

1

http://arXiv.org/abs/math/0201175v3

http://www.math.u-bordeaux.fr/~ramero

2 OFER GABBER AND LORENZO RAMERO

CONTENTS

1. Introduction 32. Homological theory 72.1. Some ring-theoretic preliminaries 72.2. Categories of almost modules and algebras 112.3. Uniform spaces of almost modules 162.4. Almost homological algebra 222.5. Almost homotopical algebra 293. Almost ring theory 383.1. Flat, unramified and etale morphisms 393.2. Nilpotent deformations of almost algebras and modules 413.3. Nilpotent deformations of torsors 473.4. Descent 553.5. Behaviour of etale morphisms under Frobenius 644. Fine study of almost projective modules 724.1. Almost traces 724.2. Endomorphisms ofGm. 784.3. Modules of almost finite rank 834.4. Localisation in the flat site 884.5. Construction of quotients by flat equivalence relations 965. Henselization and completion of almost algebras 1025.1. Henselian pairs 1035.2. Criteria for unramified morphisms 1065.3. Topological algebras and modules 1115.4. Henselian approximation of structures over adically complete rings 1165.5. Lifting theorems for henselian pairs 1275.6. Smooth locus of an affine almost scheme 1345.7. Quasi-projective almost schemes 1415.8. Lifting and descent of torsors 1466. Valuation theory 1516.1. Ordered groups and valuations 1516.2. Basic ramification theory 1596.3. Algebraic extensions 1626.4. Logarithmic differentials 1706.5. Transcendental extensions 1736.6. Deeply ramified extensions 1807. Analytic geometry 1867.1. Derived completion functor 1877.2. Cotangent complex for formal schemes and adic spaces 1927.3. Deformations of formal schemes and adic spaces 2007.4. Analytic geometry over a deeply ramified base 2097.5. Semicontinuity of the discriminant 2138. Appendix 2198.1. Simplicial almost algebras 2198.2. Fundamental group of an almost algebra 223References 230Index 232

ALMOST RING THEORY 3

Le bruit des vagues etait encore plus paresseux, plusetale qu’a midi. C’etait le meme soleil, la memelumiere sur le meme sable qui se prolongeait ici.

A. Camus -L’ etranger

1. INTRODUCTION

From a pragmatic standpoint, one can describe the theory of almost rings as a useful tool forperforming calculations of Galois cohomology groups. Indeed, this is the main application ofFaltings’ “almost purity theorem”, which is the technical heart of [34].

Though almost ring theory is developed here as an independent branch of mathematics,stretching somewhere in between commutative algebra and category theory, the original ap-plications to Galois cohomology still provide the main motivation and influence largely theevolution of the subject.

It is therefore fitting to introduce the present work by reviewing briefly the main ideas behindthese calculations. Let us consider first a complete discretely valued fieldK of zero character-istic, with perfect residue field of characteristicp > 0, and uniformizerπ; we denote byK+ thering of integers ofK. The valuationv of K extends uniquely to any algebraic extension, andwe want to normalize the value group in such a way thatv(p) = 1 in every such extension.

Let E be a finite Galois extension ofK, with Galois groupG. Typically, one is given adiscreteE+[G]-moduleM (such that theG-action onM is semilinear, that is, compatible withtheG-action onE+), and is interested in studying the (modified) Tate cohomology H i :=

H i(G,M) (for i ∈ Z). (Recall thatH i(M) agrees with Galois cohomologyRiΓGM for i > 0,with Galois homology fori < −1, and fori = 0 it equalsMG/TrE/K(M), theG-invariantsdivided by the image of the trace map).

In such a situation, the scalar multiplication mapE+⊗ZM → M induces natural cup productpairingsH i(G,E+) ⊗Z H

j → H i+j. Especially, the action of(E+)G = K+ on H i factorsthroughK+/TrE/K(E+); in other words, the image ofE+ under the trace map annihilates themodified Tate cohomology.

If now the extensionE is tamely ramifiedoverK, thenTrE/K(E+) = K+, so theH i vanishfor all i ∈ Z. Even sharper results can be achieved when the extension isunramified. Indeed, insuch caseE+ is aG-torsor for the etale topology ofK+, hence, some basic descent theory tellsus that the natural map

E+ ⊗K+ RΓGM →M [0]

is an isomorphism in the derived category of the category ofE+[G]-modules (where we havedenoted byM [0] the complex consisting ofM placed in degree zero).

In Tate’s paper [64] there occurs a variant of the above situation : instead of the finite ex-tensionE one considers the algebraic closureKa of K, so thatG is the absolute Galois groupof K, and the discreteG-moduleM is replaced by thetopologicalmoduleC(χ), whereCis thep-adic completion ofKa, whose naturalG-action we “twist” by a continuous characterχ : G → K×. Then the relevantH• is thecontinuousGalois cohomologyH•cont(G,C(χ)),which is defined in general as the homology of a complex of continuous cochains. Under thepresent assumptions,H i can be computed by the formula:

H icont(G,C(χ)) := (lim

←−n

H i(G,Ka+(χ)⊗Z Z/pnZ))⊗Z Q.

Let nowK∞ be a totally ramified Galois extension with Galois groupH isomorphic toZp. Taterealized that, for cohomological purposes, the extensionK∞ plays the role of a maximal totallyramified Galois extension ofK. More precisely, letL be any finite extension ofK, and setLn := L ·Kn, whereKn is the subfield ofK∞ fixed byHpn ≃ pn ·Zp. The extensionKn ⊂ Ln


is unramified if and only if the different idealDn := DL+n /K

+n

equalsL+n . In case this fails,

the valuationv(δn) of a generatorδn of Dn will be a strictly positive rational number, giving aquantitative measure for the ramification of the extension.With this notation, [64,§3.2, Prop.9]reads

limn→∞

v(δn) = 0(1.0.1)

(indeed,v(δn) approaches zero about as fast asp−n). In this sense, one can say that the extensionK∞ ⊂ L∞ := L ·K∞ is almost unramified. One immediate consequence is that the maximalidealm ofK+

∞ is contained inTrL∞/K∞(L+∞). If, additionally,L is a Galois extension ofK, we

can consider the subgroup

G∞ := Gal(L∞/K∞) ⊂ Gal(L/K)

and the foregoing implies thatm annihilatesH icont(G∞,M), for everyi > 0, and every topolog-

icalL+∞[G∞]-moduleM . More precisely, the homology of the cone of the natural morphism

L+∞ ⊗K+

∞RΓG∞M →M [0](1.0.2)

is annihilated bym in all degrees,i.e. it is almost zero. Equivalently, one says that the maps onhomology induced by (1.0.2) arealmost isomorphismsin all degrees.

Tate goes on to apply these cohomological vanishings to the study of p-divisible groups;in turns, this study enables him to establish a comparison between the etale and the Hodgecohomology of an abelian scheme overK+, which has become the prototype for all subsequentinvestigation ofp-adic Hodge theory.

A first generalization of (1.0.1) can be found in the work [37]by Fresnel and Matignon; oneinteresting aspect of this work is that it does away with any consideration of local class fieldtheory (which was used to get the main estimates in [64]); instead, Fresnel and Matignon writea general extensionL as a tower of monogenic subextensions, whose structure is sufficientlywell understood to allow a direct and very explicit analysis. The main tool in [37] is a notionof different idealDE+/K+ for a possibly infinite algebraic field extensionK ⊂ E; then theextensionK∞ considered in [64] is replaced by any extensionE ofK such thatDE+/K+ = (0),and (1.0.1) is generalized by the claim thatDF+/E+ = F+, for every finite extensionE ⊂ F .

In some sense, the arguments of [37] anticipate those used byFaltings in the first few para-graphs of his fundamental article [33]. There we find, first ofall, a further extension of (1.0.1):the residue field ofK is now not necessarily perfect, instead one assumes only that it admits afinite p-basis; then the relevantK∞ is an extension whose residue field is perfect, and whosevalue group isp-divisible. This generalization paves the way to the almostpurity theorem, ofwhich it represents the one-dimensional case. In order to state and prove the higher dimen-sional case, Faltings invents the method of “almost etale extensions”, and indeed sketches in afew pages a whole program of “almost commutative algebra”, with the aim of transposing tothe almost context as much as possible of the classical theory. So, for instance, ifA is a givenK+∞-algebra, andM is anA-module, one says thatM is almost flatif, for everyA-moduleN ,

the natural map of complexes

ML

⊗A N →M ⊗A N [0]

induces almost isomorphisms on homology in all degrees. Similarly, M is almost projectiveifthe same holds for the map of complexesHomA(M,N)[0]→ RHomA(M,N). Then, accordingto [33], a mapA → B of K+

∞-algebras is calledalmostetale if B is almost projective as anA-module and as aB ⊗A B-module (moreover,B is required to bealmost finitely generated:the discussion of finiteness conditions in almost ring theory is a rather subtle business, and wededicate the better part of section 2.3 to its clarification).

With this new language, the almost purity theorem should be better described as an almostversion of Abhyankar’s lemma, valid for morphismsA → B of K+-algebras that are etale


in characteristic zero and possibly wildly ramified on the locus of positive characteristic. Theactual statement goes as follows. Suppose thatA admits global etale coordinates, that is, thereexists an etale mapK+[T±1

1 , ..., T±1d ]→ A; whereas in the tamely ramified case a finite ramified

base changeK+ → K+[π1/n] (with (p, n) = 1) suffices to kill all ramification, the infiniteextensionA→ A∞ := A[T

±1/p∞

1 , ..., T±1/p∞

d ]⊗K+ K+∞ is required in the wildly ramified case,

to kill almost allramification, which means that the normalizationB∞ of A∞ ⊗A B is almostetale overA∞.

Faltings has proposed two distinct strategies for the proofof his theorem : the first one, pre-sented in [33], consists in adapting Grothendieck’s proof of Zariski-Nagata’s purity1; a more re-cent one ([34]) uses the action of Frobenius on some local cohomology modules, and is actuallyvalid under more general assumptions (one does not require the existence of etale coordinates,but only a weaker semi-stable reduction hypothesis on the special fibre).

As a corollary, one deduces cohomological vanishings generalizing the foregoing : indeed,suppose that the extension of fraction fieldsFrac(A) ⊂ Frac(B) is Galois with groupG; then,granting almost purity,B∞ is an “almostG-torsor” overA∞, therefore, for anyB∞[G]-moduleM , the natural map of complexesB∞ ⊗A∞ RΓGM → M [0] induces almost isomorphismson homology. Finally, these results can be used (together with a lot of hard work) to deducecomparison theorems betweenp-adic etale cohomology and deRham (or other kinds of) coho-mology, for arbitrary smooth projective varieties overK. This method can even be extendedto treat cohomology with not necessarily constant coefficients (see [34]), thereby providing themost comprehensive approach top-adic Hodge theory found so far.

The purpose of our text is to fully work out the foundations of“almost commutative algebra”outlined by Faltings; in the process we generalize and simplify considerably the theory, and alsoextend it in directions that were not explored in [33], [34].

It turns out that most of almost ring theory can be built up satisfactorily from a very slimand general set of assumptions: our basic setup, introducedin section 2.1, consists of a ringVand an idealm ⊂ V such thatm = m2; starting from (2.5.14) we also assume thatm ⊗V m isa flatV -module : simple considerations show this to be a natural hypothesis, often verified inpractice.

TheV -modules killed bym are the objects of a (full) Serre subcategoryΣ of the categoryV -Mod of all V -modules, and the quotientV a-Mod := V -Mod/Σ is an abelian categorywhich we call the category ofalmostV -modules. It is easy to check that the usual tensor productof V -modules descends to a bifunctor⊗ on almostV -modules, so thatV a-Mod is a monoidalabelian category in a natural way. Then analmost ringis just an almostV -moduleA endowedwith a “multiplication” morphismA⊗A→ A satisfying certain natural axioms. Together withthe obvious morphisms, these gadgets form a categoryV a-Alg. Given any almostV -algebraA, one can then define the notion ofA-module andA-algebra, just like for usual rings. Thepurpose of the game is to reconstruct in this new framework asmuch as possible (and useful) ofclassical linear and commutative algebra. Essentially, this is the same as the ideology informingDeligne’s paper [23], which sets out to develop algebraic geometry in the context of abstracttannakian categories. We could also claim an even earlier ancestry, in that some of the leadingmotifs resonating throughout our text, can be traced as far back as Gabriel’s memoir [38] “Descategories abeliennes”.

In evoking Deligne’s and Gabriel’s works, we have unveiled another source of motivationwhose influence has steadily grown throughout the long gestation of our paper. Namely, wehave come to view almost ring theory as a contribution to thatexpanding body of research ofstill uncertain range and shifting boundaries, that we could call “abstract algebraic geometry”.We would like to encompass under this label several heterogeneous developments: notably, it

1At the time of writing, there are still some obscure points inthis proof


should include various versions of non-commutative geometry that have been proposed in thelast twenty years, but also the relative schemes of [43], as well as Deligne’s ideas for algebraicgeometry over symmetric monoidal categories.

The common thread loosely unifying these works is the realization that ”geometric spaces”do not necessarily consist of set-theoretical points, and -perhaps more importantly - functionson such ”spaces” do not necessarily form (sheaves of) commutative rings. Much effort hasbeen devoted to extending the reach of geometric intuition to non-commutative algebras; al-ternatively, one can retain commutativity, but allow “structure sheaves” which take values intensor categories other than the category of rings. As a casein point, to any given almost ringA one can attach itsspectrumSpecA, which is justA viewed as an object of the opposite ofthe categoryV a-Alg. SpecA has even a naturalflat topology, which allows to define moregeneralalmost schemesby gluing (i.e. taking colimits of) diagrams of affine spectra; all this isexplained in section 5.7, where we also introducequasi-projectivealmost schemes and investi-gate some basic properties of thesmooth locusof a quasi-projective almost scheme. By way ofillustration, these generalities are applied in section 5.8 in order to solve a deformation problemfor torsors over affine almost group schemes; let us stress that the problem in question is statedpurely in terms of affine objects (i.e. almost rings and “almost Hopf algebras”), but the solutionrequires the introduction of certain auxiliary almost schemes that are not affine.

Having soared into the thin air of abstract geometry, we comeback to earth in the last twochapters, which deal with applications to valuation theoryand top-adic analytic geometry :especially the reader will find there our own contributions to almost purity. This general pre-sentation has thus come full circle, and we defer to the introductory remarks at the beginning ofthe respective chapters for a more detailed description of our results.

AcknowledgementsThe second author is very much indebted to Gerd Faltings for many patient explanations onthe method of almost etale extensions. Next he would like toacknowledge several interesting discussions withIoannis Emmanouil. He is also much obliged to Pierre Deligne, for a useful list of critical remarks. Finally, heowes a special thank to Roberto Ferretti, who has read the first tentative versions of this work, has corrected manyslips and has made several valuable suggestions.

This project began in 1997 while the second author was supported the IHES, and has been brought to completionwhile the second author was a guest of the Laboratoire de Theorie des Nombres of the University of Paris VI.


2. HOMOLOGICAL THEORY

As explained in the introduction, in order to define a category of almost modules one requiresa pair(V,m) consisting of a ringV and an idealm ⊂ V such thatm = m2. In section 2.1we collect a few useful ring-theoretic preliminaries concerning such pairs. In section 2.2 weintroduce the categoryV a-Mod of almost modules: it is a quotientV -Mod/Σ of the categoryof V -modules, whereΣ is the thick subcategory of theV -modules killed bym. V a-Mod is anabelian tensor category and its commutative unitary monoids, calledalmost algebras, are thechief objects of study in this work. The first useful observation is that the localization functorV -Mod→ V a-Mod admits both left and right adjoints. Taken together, these functors exhibitthe kind of exactness properties that one associates to openembeddings of topoi, perhaps a hintof some deeper geometrical structure, still to be unearthed.

After these generalities, we treat in section 2.3 the question of finiteness conditions for almostmodules. LetA denote an almost algebra, fixed for the rest of this introduction. It is certainlypossible to define as usual a notion of finitely generatedA-module, however this turns out tobe too restrictive a class for applications. The main idea here is to define a uniform structureon the set of equivalence classes ofA-modules ; then we will say that anA-module isalmostfinitely generatedif its isomorphism class lies in the topological closure of the subspace offinitely generatedA-modules. Similarly we definealmost finitely presentedA-modules. Theuniform structure also comes handy when we want to constructoperators on almost modules :if one can show that the operator in question is uniformly continuous on a classC of almostmodules, then its definition extends right away by continuity to the topological closureC of C .This is exemplified by the construction of the (almost) Fitting ideals forA-modules, at the endof section 2.3.

In section 2.4 we introduce the basic toolkit of homologicalalgebra, beginning with thenotion of flat almost module, which poses no problem, since wedo have a tensor product in ourcategory. The notion of projectivity is more subtle : it turns out that the category ofA-modulesusually doesnot have enough projectives. The useful notion isalmost projectivity: simply oneuses the standard definition, except that the role of theHom functor is played by the internalalHom functor. The scarcity of projectives should not be regardedas surprising or pathological:it is quite analogous to the lack of enough projective objects in the category of quasi-coherentOX-modules on a non-affine schemeX.

Section 2.5 introduces the cotangent complex of a morphism of almost rings, and establishesits usual properties, such as transitivity and Tor-independent base change theorems. These foun-dations will be put to use in chapter 3, to study infinitesimaldeformations of almost algebras.

2.1. Some ring-theoretic preliminaries. Unless otherwise stated, every ring is commutativewith unit. This section collects some results of general nature that will be used throughout thiswork.

2.1.1. Ourbasic setupconsists of a fixed base ringV containing an idealm such thatm2 = m.Starting from (2.5.14), we will also assume thatm := m⊗V m is a flatV -module.

Example 2.1.2. (i) The main example is given by a non-discrete valuation ring (V, | · |) of rankone; in this casem will be the maximal ideal.

(ii) Takem := V . This is the “classical limit”. In this case almost ring theory reduces to usualring theory. Thus, all the discussion that follows specialises to, and sometimes gives alternativeproofs for, statements about rings and their modules.

2.1.3. LetM be a givenV -module. We say thatM is almost zeroif mM = 0. A mapφ ofV -modules is analmost isomorphismif both Kerφ andCoker φ are almost zeroV -modules.


Remark 2.1.4. (i) It is easy to check that aV -moduleM is almost zero if and only ifm⊗VM =0. Similarly, a mapM → N of V -modules is an almost isomorphism if and only if the inducedmapm⊗V M → m⊗V N is an isomorphism. Notice also that, ifm is flat, thenm ≃ m.

(ii) Let V → W be a ring homomorphism. For aV -moduleM setMW := W ⊗V M . Wehave an exact sequence

0→ K → mW → mW → 0(2.1.5)

whereK := TorV1 (V/m,W ) is an almost zeroW -module. By (i) it follows thatm ⊗V K ≃(mW )⊗W K ≃ 0. Then, applyingmW ⊗W − and−⊗W (mW ) to (2.1.5) we derive

mW ⊗W mW ≃ mW ⊗W (mW ) ≃ (mW )⊗W (mW )

i.e. mW ≃ (mW )∼. In particular, ifm is a flatV -module, thenmW is a flatW -module. Thismeans that our basic assumptions on the pair(V,m) are stable under arbitrary base extension.Notice that the flatness ofm does not imply the flatness ofmW . This partly explains why weinsist thatm, rather thanm, be flat.

2.1.6. Before moving on, we want to analyze in some detail howour basic assumptions relateto certain other natural conditions that can be postulated on the pair(V,m). Indeed, let usconsider the following two hypotheses :

(A) m = m2 andm is a filtered union of principal ideals.

(B) m = m2 and, for all integersk > 1, thek-th powers of elements ofm generatem.

Clearly (A) implies (B). Less obvious is the following result.

Proposition 2.1.7. (i) (A) implies thatm is flat.

(ii) If m is flat then(B) holds.

Proof. Suppose that (A) holds, so thatm = colimα∈I

V xα, whereI is a directed set parametrizing

elementsxα ∈ m (andα ≤ β ⇔ V xα ⊂ V xβ). For anyα ∈ I we have natural isomorphisms

V xα ≃ V/AnnV (xα) ≃ (V xα)⊗V (V xα).(2.1.8)

Forα ≤ β, let jαβ : V xα → V xβ be the imbedding; we have a commutative diagram

Vµz2 //

πα

V

πβ

(V xα)⊗V (V xα)

jαβ⊗jαβ // (V xβ)⊗V (V xβ)

wherez ∈ V is such thatxα = z ·xβ, µz2 is multiplication byz2 andπα is the projection inducedby (2.1.8) (and similarly forπβ). Sincem = m2, for all α ∈ I we can findβ such thatxα is amultiple ofx2

β . Sayxα = y ·x2β; then we can takez := y ·xβ, soz2 is a multiple ofxα and in the

above diagramKerπα ⊂ Kerµz2. Hence one can define a mapλαβ : (V xα) ⊗V (V xα) → Vsuch thatπβ λαβ = jαβ ⊗ jαβ andλαβ πα = µz2. It now follows that for everyV -moduleN , the induced morphismTorV1 (N, (V xα) ⊗V (V xα)) → TorV1 (N, (V xβ) ⊗V (V xβ)) is thezero map. Taking the colimit we derive thatm is flat. This shows (i). In order to show (ii) weconsider, for any prime numberp, the following condition

(∗p) m/p ·m is generated (as aV -module) by thep-th powers of its elements.

Clearly (B) implies (∗p) for all p. In fact we have :

Claim 2.1.9. (B) holds if and only if (∗p) holds for every primep.


Proof of the claim:Suppose that (∗p) holds for every primep. The polarization identity

k! · x1 · x2 · ... · xk =∑

I⊂1,2,...,k

(−1)k−|I| ·(∑

i∈I

xi

)k

shows that ifN :=∑

x∈mV xk thenk! · m ⊂ N . To prove thatN = m it then suffices to show

that for every primep dividing k! we havem = p · m + N . Let φ : V/pV → V/pV be theFrobenius (x 7→ xp); we can denote by(V/pV )φ the ringV/pV seen as aV/pV -algebra via thehomomorphismφ. Also setφ∗M := M ⊗V/pV (V/pV )φ for aV/pV -moduleM . Then the mapφ∗(m/p · m) → (m/p · m) (defined by raising top-th power) is surjective by (∗p). Hence so is(φr)∗(m/p · m) → (m/p · m) for everyr > 0, which says thatm = p · m + N whenk = pr,hence for everyk.

Next recall (see [6, Exp. XVII 5.5.2]) that, ifM is aV -module, the module of symmetrictensorsTSk(M) is defined as(⊗kVM)Sk , the invariants under the natural action of the symmetricgroupSk on⊗kVM . We have a natural mapΓk(M)→ TSk(M) that is an isomorphism whenMis flat (seeloc. cit. 5.5.2.5; hereΓk denotes thek-th graded piece of the divided power algebra).

Claim2.1.10. The groupSk acts trivially on⊗kV m and the mapm ⊗V m → m (x ⊗ y ⊗ z 7→x⊗ yz) is an isomorphism.

Proof of the claim:The first statement is reduced to the case of transpositions and tok = 2.There we can compute :x ⊗ yz = xy ⊗ z = y ⊗ xz = yz ⊗ x. For the second statement notethat the imbeddingm → V is an almost isomorphism, and apply remark 2.1.4(i).

Suppose now thatm is flat and pick a primep. ThenSp acts trivially on⊗pV m. Hence

Γp(m) ≃ ⊗pV m ≃ m.(2.1.11)

But Γp(m) is spanned as aV -module by the productsγi1(x1) · ... · γik(xk) (wherexi ∈ m and∑j ij = p). Under the isomorphism (2.1.11) these elements map to

(p

i1,...,ik

)· xi11 · ... · xikk ; but

such an element vanishes inm/p·m unlessik = p for somek. Thereforem/p·m is generated byp-th powers, so the same is true form/p ·m, and by the above, (B) holds, which shows (ii).

Theorem 2.1.12.Let (εi | i ∈ I) be a family of generators ofm and, for every subsetS ⊂ I,denote bymS ⊂ m the subideal generated by(εi | i ∈ S). Then we have:

(i) Every countable subsetS ⊂ I is contained in another countableS ′ ⊂ I such that:(a) m2

S′ = mS′.(b) If m is a flatV -module, the same holds formS′ ⊗V mS′ .

(ii) Suppose thatm is countably generated as aV -module. Then :(a) m is countably presented as aV -module.(b) If m is a flatV -module, then it is of homological dimension≤ 1.

Proof. For everyi ∈ I, we can writeεi =∑

j xijεj, for certainxij ∈ m. For anyi, j ∈ I suchthatxij 6= 0, let us writexij =

∑k xijkεk for somexijk ∈ V . We say that a subsetS ⊂ I is

saturatedif the following holds: wheneveri ∈ S andxijk 6= 0, we havej, k ∈ S.

Claim2.1.13. Every countable subsetS ⊂ I is contained in a countable saturated subsetS∞ ⊂I.

Proof of the claim:We let S0 := S and define recursivelySn for everyn > 0 as follows.SupposeSn−1 has already been given; then we set

Sn := Sn−1 ∪ i ∈ I | there existsa ∈ Sn−1, b ∈ I such that eitherxaib 6= 0 or xabi 6= 0.Notice that, for everyi ∈ I we havexij = 0 for all but finitely manyj ∈ I, hencexijk = 0 forall but finitely manyj, k ∈ I. It follows easily thatS∞ :=

⋃n∈N Sn will do.


(i.a) is now straightforward, when one remarks thatmS = m2S for every saturated subsetS.

Next, letS ⊂ I be any saturated subset. Clearly the family(εi ⊗ εj | i, j ∈ S) generatesmS := mS ⊗V mS andεi · εj · (εk ⊗ εl) = εk · εl · (εi⊗ εj) for all i, j, k, l ∈ S. LetF (S) be theV -module defined by generators(eij)i,j∈S, subject to the relations:

εi · εj · ekl = εk · εl · eij eik =∑

j

xijejk for all i, j, k, l ∈ S.

We get an epimorphismπS : F (S) → mS by eij 7→ εi ⊗ εj. The relations imply that, ifx :=

∑k,l yklekl ∈ KerπS, thenεi · εj · x = 0, somS ·KerπS = 0. WhencemS ⊗V Ker πS = 0

and1mS⊗V πS is an isomorphism. We consider the diagram

mS ⊗V F (S)∼ //

φ

mS ⊗V mS

ψ

F (S)

πS // mS

whereφ andψ are induced by scalar multiplication. We already know thatψ is an isomorphism,and sinceF (S) = mS · F (S), we see thatφ is an epimorphism, soπS is an isomorphism. IfnowI is countable, this shows that (ii.a) holds. Now (ii.b) follows from (ii.a) and the followinglemma 2.1.16. In order to show (i.b) we will use the followingwell known criterion.

Claim 2.1.14. ([51, Ch.I, Th.1.2]). LetR be a ring,M anR-module. ThenM is R-flat ifand only if, for every finitely presentedR-moduleN , every morphismN → M factors as acompositionN → L→M whereL is a freeR-module of finite rank.

In order to apply claim 2.1.14 we show:

Claim 2.1.15. Let S ⊂ I be a countable saturated subset. Then there exists a countable satu-rated subsetσ(S) ⊂ I containingS, with the following property. For every finitely presentedV -moduleN and every morphismf : N → mS, we can find a commutative diagram:

Nf //

mS

j

L // mσ(S)

whereL is freeV -module of finite rank andj is the natural map.

Proof of the claim:In view of (ii.a), we can writemS ≃ colimα∈A

Mα, whereA is some filtered

countable set and everyMα is a finitely presentedV -module. Givenf as in the claim, we canfind α ∈ A such thatf factors thorugh the natural mapια : Mα → mS, so we are reduced toprove the claim forN = Mα andf = ια. However, by assumptionm is flat, so by claim 2.1.14the compositionMα

ια→ mS → m factors through some mapLα → m, with Lα free of finiterank overV . Furthermore, thanks to claim 2.1.13 we havem = colim

JmJ , whereJ runs over

the family of all countable saturated subsets ofI. It follows that, for some countable saturatedSα ⊃ S, the mapLα → m factors throughmSα . Clearlyσ(S) :=

⋃α∈A Sα will do.

Finally, for any countable subsetS ⊂ I, let us setS ′ :=⋃n∈N σ

n(S∞), whereS∞ is thesaturation ofS as in claim 2.1.13. One verifies easily that (i.b) holds for this choice ofS ′.

The proof of following well known lemma is due to D.Lazard ([51, Ch.I, Th.3.2]), up to someslight imprecisions which were corrected in [55, pp.49-50]. For the convenience of the readerwe reproduce the argument.


Lemma 2.1.16.Let R be any ring. A flat countably presentedR-module has homologicaldimension≤ 1.

Proof. Let M be flat and countable presented overR, and choose a presentationF1φ→ F0 →

M → 0 with F0 andF1 freeR-modules of (infinite) countable rank. Let(ej | j ∈ N) bea basis ofF0 and (fi | i ∈ N) a basis ofF1. Say thatφ(fi) =

∑i xijej with xij ∈ R, for

every i ∈ N. For everyi ∈ N we define the finite setSi := j ∈ N | xij 6= 0; also, letS ′n :=

⋃i≤n(i ∪ Si). DefineR-modulesGn := ⊕i≤nfiR, Hn := ⊕j∈S′

nejR; the restriction

of φ induces mapsφn : Gn → Hn and we haveM ≃ colimn∈N

Coker φn. By claim 2.1.14, the

natural mapCoker φn → M factors as a compositionCoker φnαn→ Ln

ψn→ M , whereLn isa freeR-module of finite rank. Thenψn factors through a mapψ′n : Ln → Cokerφt(n) forsomet(n) ∈ N. We further defineψ′′n : Ln → Coker φk(n) as the composition ofψ′n withthe natural transition mapCoker φt(n) → Coker φk(n), wherek(n) is suitably chosen, so thatk(n) > max(n, t(n)) and the compositionψ′′n αn : Coker φn → Coker φk(n) is the naturaltransition map. We define by induction onn ∈ N a direct system of mapsτn : Lhn → Lhn+1 , asfollows. Seth0 := 0; next, suppose thathn ∈ N has already been given up to somen ∈ N; welet hn+1 := k(hn) andτn := αhn+1 ψ′′hn

. ClearlyM ≃ colimn∈N

Lhn ; setL := ⊕n∈NLhn and let

τ : L→ L be the map given by the ruleτ(xn | n ∈ N) := (xn − τn(xn−1) | n ∈ N). We derivea short exact sequence:0→ L

τ→ L→ M → 0, whence the claim.

2.2. Categories of almost modules and algebras.If C is a category, andX, Y two objectsof C , we will usually denote byHomC (X, Y ) the set of morphisms inC fromX to Y and by1X the identity morphism ofX. Moreover we denote byC o the opposite category ofC andby s.C the category of simplicial objects overC , that is, functors∆o → C , where∆ is thecategory whose objects are the ordered sets[n] := 0, ..., n for each integern ≥ 0 and where amorphismφ : [p]→ [q] is a non-decreasing map. A morphismf : X → Y in s.C is a sequenceof morphismsf[n] : X[n] → Y [n], n ≥ 0 such that the obvious diagrams commute. We canimbedC in s.C by sending each objectX to the “constant” objects.X such thats.X[n] = Xfor all n ≥ 0 ands.X[φ] = 1X for all morphismsφ in ∆.

2.2.1. IfC is an abelian category,D(C ) will denote the derived category ofC . As usual wehave also the full subcategoriesD+(C ),D−(C ) of complexes of objects ofC that are exact forsufficiently large negative (resp. positive) degree. IfR is a ring, the category ofR-modules(resp.R-algebras) will be denoted byR-Mod (resp.R-Alg). Most of the times we will writeHomR(M,N) instead ofHomR-Mod(M,N).

We denote bySet the category of sets. The symbolN denotes the set of non-negative inte-gers; in particular0 ∈ N.

2.2.2. The full subcategoryΣ of V -Mod consisting of allV -modules that are almost isomor-phic to0 is clearly a Serre subcategory and hence we can form the quotient categoryV -Mod/Σ.There is a localization functor

V -Mod→ V -Mod/Σ M 7→Ma

that takes aV -moduleM to the same module, seen as an object ofV -Mod/Σ. In particular,we have the objectV a associated toV ; it seems therefore natural to use the notationV a-Mod

for the categoryV -Mod/Σ, and an object ofV a-Mod will be indifferently referred to as “aV a-module” or “an almostV -module”. In case we need to stress the dependance on the idealm, we can write(V,m)a-Mod.

Since the almost isomorphisms form a multiplicative system(seee.g. [67, Exerc.10.3.2]), itis possible to describe the morphisms inV a-Mod via a calculus of fractions, as follows. LetV -al.Iso be the category that has the same objects asV -Mod, but such thatHomV -al.Iso(M,N)


consists of all almost isomorphismsM → N . If M is any object ofV -al.Iso we write(V -al.Iso/M) for the category of objects ofV -al.Iso overM (i.e. morphismsφ : X → M).If φi : Xi → M (i = 1, 2) are two objects of(V -al.Iso/M) thenHom(V -al.Iso/M)(φ1, φ2)consists of all morphismsψ : X1 → X2 in V -al.Iso such thatφ1 = φ2 ψ. For any twoV -modulesM,N we define a functorFN : (V -al.Iso/M)o → V -Mod by associating to anobjectφ : P → M the V -moduleHomV (P,N) and to a morphismα : P → Q the mapHomV (Q,N)→ HomV (P,N) : β 7→ β α. Then we have

HomV a-Mod(Ma, Na) = colim(V -al.Iso/M)o

FN .(2.2.3)

However, formula (2.2.3) can be simplified considerably by remarking that for anyV -moduleM , the natural morphismm ⊗V M → M is an initial object of(V -al.Iso/M). Indeed, letφ : N →M be an almost isomorphism; the diagram

m⊗V N ∼ //

m⊗V M

N

φ // M

(cp. remark 2.1.4(i)) allows one to define a morphismψ : m ⊗V M → N overM . Weneed to show thatψ is unique. But ifψ1, ψ2 : m ⊗V M → N are two maps overM , thenIm(ψ1−ψ2) ⊂ Ker(φ) is almost zero, henceIm(ψ1−ψ2) = 0, sincem⊗V M = m ·(m⊗V M).Consequently, (2.2.3) boils down to

HomV a-Mod(Ma, Na) = HomV (m⊗V M,N).(2.2.4)

In particularHomV a-Mod(M,N) has a natural structure ofV -module for any twoV a-modulesM,N , i.e. HomV a-Mod(−,−) is a bifunctor that takes values in the categoryV -Mod.

2.2.5. One checks easily (for instance using (2.2.4)) that the usual tensor product induces abifunctor− ⊗V − on almostV -modules, which, in the jargon of [24] makes ofV a-Mod anabelian tensor category. Then analmostV -algebra is just a commutative unitary monoid inthe tensor categoryV a-Mod. Let us recall what this means. Quite generally, let(C ,⊗, U) beany abelian tensor category, so that⊗ : C × C → C is a biadditive functor,U is the identityobject ofC (see [24, p.105]) and for any two objectsM andN in C we have a “commutativityconstraint” (i.e. a functorial isomorphismθM |N : M ⊗ N → N ⊗M that “switches the twofactors”) and a functorial isomorphismνM : U ⊗M → M . Then aC -monoidA is an objectof C endowed with a morphismµA : A ⊗ A → A (the “multiplication” ofA) satisfying theassociativity condition

µA (1A ⊗ µA) = µA (µA ⊗ 1A).

We say thatA is unitary if additionallyA is endowed with a “unit morphism”1A : U → Asatisfying the (left and right) unit property :

µA (1A ⊗ 1A) = νA µA (1A ⊗ 1A) θA|U = µA (1A ⊗ 1A).

Finally A is commutativeif µA = µA θA|A (to be rigorous, in all of the above one shouldindicate the associativity constraints, which we have omitted : see [24]). A commutative unitarymonoid will also be simply called analgebra. With the morphisms defined in the obvious way,theC -monoids form a category; furthermore, given aC -monoidA, a leftA-moduleis an objectM of C endowed with a morphismσM/A : A ⊗M → M such thatσM/A (1A ⊗ σM/A) =σM/A (µA ⊗ 1M). Similarly one defines rightA-modules andA-bimodules. In the case ofbimodules we have left and right morphismsσM,l : A⊗M →M , σM,r : M ⊗A→M and oneimposes that they “commute”,i.e. that

σM,r (σM,l ⊗ 1A) = σM,l (1A ⊗ σM,r).


Clearly the (left resp. right)A-modules (and theA-bimodules) form an additive category withA-linear morphismsdefined as one expects. One defines the notion of a submodule asanequivalence class of monomorphismsN →M such that the compositionA⊗N → A⊗M →M factors throughN . Especially, atwo-sided idealof A is anA-sub-bimoduleI → A. Forgiven submodulesI, J ofA one denotesIJ := Im(I⊗J → A⊗A µA→ A). For anA-moduleM ,theannihilator AnnA(M) of M is the largest idealj : I → A of A such that the composition

I ⊗M j⊗1M−→ A⊗M σM/A−→ M is the zero morphism.

2.2.6. Iff : M → N is a morphism of leftA-modules, thenKer(f) exists in the underlyingabelian categoryC and one checks easily that it has a unique structure of leftA-module whichmakes it a submodule ofM . If moreover⊗ is right exactwhen either argument is fixed, thenalsoCoker f has a uniqueA-module structure for whichN → Coker f is A-linear. In thiscase the category of leftA-modules is abelian. Similarly, ifA is a unitaryC -monoid, then onedefines the notion ofunitary left A-module by requiring thatσM/A (1A⊗1M ) = νM and theseform an abelian category when⊗ is right exact.

2.2.7. Specialising to our case we obtain the categoryV a-Alg of almostV -algebras and,for every almostV -algebraA, the categoryA-Mod of unitary leftA-modules. Clearly thelocalization functor restricts to a functorV -Alg → V a-Alg and for anyV -algebraR we havea localization functorR-Mod→ Ra-Mod.

Next, if A is an almostV -algebra, we can define the categoryA-Alg of A-algebras. Itconsists of all the morphismsA→ B of almostV -algebras.

2.2.8. Let again(C ,⊗, U) be any abelian tensor category. By [24, p.119], the endomorphismring EndC (U) of U is commutative. For any objectM of C , denoteM∗ = HomC (U,M);thenM 7→ M∗ defines a functorC → EndC (U)-Mod. Moreover, ifA is a C -monoid,A∗is an associativeEndC (U)-algebra, with multiplication given as follows. Fora, b ∈ A∗ leta · b := µA (a ⊗ b) ν−1

U . Similarly, if M is anA-module,M∗ is anA∗-module in a naturalway, and in this way we obtain a functor fromA-modules andA-linear morphisms toA∗-modules andA∗-linear maps. Using [24, Prop.1.3], one can also check thatEndC (U) = U∗ asEndC (U)-algebras, whereU is viewed as aC -monoid usingνU .

2.2.9. All this applies especially to our categories of almost modules and almost algebras : inthis case we callM 7→ M∗ the functor of almost elements. So, ifM is an almost module, analmost element ofM is just an honest element ofM∗. Using (2.2.4) one can show easily thatfor everyV -moduleM the natural mapM → (Ma)∗ is an almost isomorphism.

2.2.10. LetA be aV a-algebra. For any twoA-modulesM,N , the setHomA-Mod(M,N) hasa natural structure ofA∗-module and we obtain an internal Hom functor by letting

alHomA(M,N) = HomA-Mod(M,N)a.

This is the functor ofalmost homomorphismsfromM toN .

2.2.11. For anyA-moduleM we have also a functor of tensor productM⊗A− onA-moduleswhich, in view of the following proposition 2.2.13 can be shown to be a left adjoint to thefunctoralHomA(M,−). It can be defined asM ⊗A N := (M∗ ⊗A∗ N∗)

a.With this tensor product,A-Mod is an abelian tensor category as well, andA-Alg could

also be described as the category of (A-Mod)-algebras. Under this equivalence, a morphismφ : A→ B of almostV -algebras becomes the unit morphism1B : A→ B of the correspondingmonoid. We will sometimes drop the subscript and write simply 1.


Remark 2.2.12. Let V → W be a map of base rings,W taken with the extended idealmW .ThenW a is an almostV -algebra so we have defined the categoryW a-Mod using base ringVand the category(W,mW )a-Mod using baseW . One shows easily that they are equivalent:we have an obvious functor(W,mW )a-Mod → W a-Mod and a quasi-inverse is provided byM 7→M∗. Similar base comparison statements hold for the categories of almost algebras.

Proposition 2.2.13.LetA be aV a-algebra,R a V -algebra.

(i) There is a natural isomorphismA ≃ Aa∗ of almostV -algebras.(ii) The functorM 7→ M∗ fromRa-Mod to R-Mod (resp. fromRa-Alg toR-Alg) is right

adjoint to the localization functorR-Mod→ Ra-Mod (resp.R-Alg→ Ra-Alg).(iii) The counit of the adjunctionMa

∗ → M is a natural isomorphism from the composition ofthe two functors to the identity functor1Ra-Mod (resp.1Ra-Alg).

Proof. (i) has already been remarked. We show (ii). In light of remark 2.2.12 (applied withW = R) we can assume thatV = R. LetM be aV -module andN an almostV -module; wehave natural bijections

HomV a-Mod(Ma, N)≃ HomV a-Mod(Ma, (N∗)a) ≃ HomV (m⊗V M,N∗)

≃ HomV (M,HomV (m, N∗)) ≃ HomV (M,HomV a-Mod(V, (N∗)a))

≃ HomV (M,N∗)

which proves (ii). Now (iii) follows by inspecting the proofof (ii), or by [38, Ch.III Prop.3].

Remark 2.2.14. (i) Let M1,M2 be twoA-modules. By proposition 2.2.13(iii) it is clear thata morphismφ : M1 → M2 of A-modules is uniquely determined by the induced morphismM1∗ → M2∗. On this basis, we will very often define morphisms ofA-modules (orA-algebras)by saying how they act on almost elements.

(ii) It is a bit tricky to deal with preimages of almost elements under morphisms: for instance,if φ : M1 → M2 is an epimorphism (by which we mean thatCoker φ ≃ 0) andm2 ∈M2∗, thenit is not true in general that we can find an almost elementm1 ∈ M1∗ such thatφ∗(m1) = m2.What remains true is that for arbitraryε ∈ m we can findm1 such thatφ∗(m1) = ε ·m2.

(iii) The existence of the right adjointM 7→ M∗ follows also directly from [38, Chap.III§3Cor.1 or Chap.V§2].

Corollary 2.2.15. The categoriesA-Mod andA-Alg are both complete and cocomplete.

Proof. We recall that the categoriesA∗-Mod andA∗-Alg are both complete and cocomplete.Now let I be any small indexing category andM : I → A-Mod be any functor. Denote byM∗ : I → A∗-Mod the composed functori 7→ M(i)∗. We claim thatcolim

IM = (colim

IM∗)

a.

The proof is an easy application of proposition 2.2.13(iii). A similar argument also works forlimits and for the categoryA-Alg.

2.2.16. For anyV a-algebraA, The abelian categoryA-Mod satisfies axiom (AB5) (seee.g.[67, §A.4]) and it has a generator, namely the objectA itself. It then follows by a general resultthatA-Mod has enough injectives.

Corollary 2.2.17. The functorM 7→ M∗ fromRa-Mod to R-Mod sends injectives to injec-tives and injective envelopes to injective envelopes.

Proof. The functorM 7→ M∗ is right adjoint to an exact functor, hence it preserves injectives.Now, let J be an injective envelope ofM ; to show thatJ∗ is an injective envelope ofM∗, itsuffices to show thatJ∗ is an essential extension ofM∗. However, ifN ⊂ J∗ andN ∩M∗ = 0,thenNa ∩M = 0, hencemN = 0, butJ∗ does not containm-torsion, thusN = 0.


2.2.18. Note that the essential image ofM 7→ M∗ is closed under limits. Next recall that theforgetful functorA∗-Alg → Set (resp.A∗-Mod → Set) has a left adjointA∗[−] : Set →A∗-Alg (resp.A(−) : Set→ A∗-Mod) that assigns to a setS the freeA∗-algebraA∗[S] (resp.the freeA∗-moduleA(S)

∗ ) generated byS. If S is any set, it is natural to writeA[S] (resp.A(S))for theA-algebra(A∗[S])a (resp. for theA-module(A

(S)∗ )a. This yields a left adjoint, called

thefreeA-algebrafunctorSet→ A-Alg (resp. thefreeA-modulefunctorSet→ A-Mod) tothe “forgetful” functorA-Alg→ Set (resp.A-Mod→ Set) B 7→ B∗.

2.2.19. Now letR be anyV -algebra; we want to construct a left adjoint to the localisationfunctorR-Mod→ Ra-Mod. For a givenRa-moduleM , let

M! := m⊗V (M∗).(2.2.20)

We have the natural map (unit of adjunction)R→ Ra∗, so that we can viewM! as anR-module.

Proposition 2.2.21.LetR be aV -algebra.

(i) The functorRa-Mod→ R-Mod defined by(2.2.20)is left adjoint to localisation.(ii) The unit of the adjunctionM → Ma

! is a natural isomorphism from the identity functor1Ra-Mod to the composition of the two functors.

Proof. (i) follows easily from (2.2.4) and (ii) follows easily from(i).

Corollary 2.2.22. Suppose thatm is a flatV -module. Then we have :

(i) the functorM 7→M! is exact;(ii) the localisation functorR-Mod→ Ra-Mod sends injectives to injectives.

Proof. By proposition 2.2.21 it follows thatM 7→ M! is right exact. To show that it is also leftexact whenm is a flatV -module, it suffices to remark thatM 7→ M∗ is left exact. Now, by (i),the functorM 7→ Ma is right adjoint to an exact functor, so (ii) is clear.

2.2.23. LetB be anyA-algebra. The multiplication onB∗ is inherited byB!, which is there-fore a non-unital ring in a natural way. We endow theV -moduleV ⊕B! with the ring structuredetermined by the rule:(v, b) · (v′, b′) := (v · v′, v · b′ + v′ · b + b · b′) for all v, v′ ∈ V andb, b′ ∈ B!. ThenV ⊕ B! is a (unital) ring. We notice that theV -submodule generated by all theelements of the form(x · y,−x ⊗ y ⊗ 1) (for arbitraryx, y ∈ m) forms an idealI of V ⊕ B!.SetB!! := (V ⊕ B!)/I. Thus we have a sequence ofV -modules

0→ m→ V ⊕ B! → B!! → 0(2.2.24)

which in general is only right exact.

Definition 2.2.25. We say thatB is anexactV a-algebraif the sequence (2.2.24) is exact.

Remark 2.2.26. Notice that ifm∼→ m (e.g. whenm is flat), then allV a-algebras are exact.

In the general case, ifB is anyA-algebra, thenV a × B is always exact. Indeed, we have(V a × B)∗ ≃ V a

∗ × B∗ and, by remark 2.1.4(i),m⊗V V a∗ ≃ m.

Clearly we have a natural isomorphismB ≃ Ba!!.

Proposition 2.2.27.The functorB 7→ B!! is left adjoint to the localisation functorA!!-Alg →A-Alg.

Proof. Let B be anA-algebra,C anA!!-algebra andφ : B → Ca a morphism ofA-algebras.By proposition 2.2.21 we obtain a naturalA∗-linear morphismB! → C. Together with thestructure morphismV → C this yields a mapφ : V ⊕ B! → C which is easily seen to be aring homomorphism. It is equally clear that the idealI defined above is mapped to zero byφ,


hence the latter factors through a map ofA!!-algebrasB!! → C. Conversely, such a map inducesa morphism ofA-algebrasB → Ca just by taking localisation. It is easy to check that the twoprocedures are inverse to each other, which shows the assertion.

Remark 2.2.28. (i) The functor of almost elements commutes with arbitrary limits, becauseall right adjoints do. It does not in general commute with colimits, not even with arbitraryinfinite direct sums. Dually, the functorsM 7→ M! andB 7→ B!! commute with all colimits. Inparticular, the latter commutes with tensor products.

(ii) Resume the notation of remark 2.2.12. The change of setup functor(W,mW )a-Mod→W a-Mod commutes with the operationsM 7→ M∗ andM 7→ M!. The corresponding func-tor F : (W,mW )a-Alg → W a-Alg satisfies the identity:F (B)!! ⊗W a

!!W = B!! for every

(W,mW )a-algebraB.

2.3. Uniform spaces of almost modules.LetA be aV a-algebra. For any cardinal numberc,we letMc(A) be the set of isomorphism classes ofA-modules which admit a set of generatorsof cardinality≤ c. In the following we fix some (very) large infinite cardinality ω, and supposethat the isomorphism classes of all ourA-modules lie inMω(A). The choice ofω is required toavoid set-theoretical inconsistencies, but it is immaterial for our purposes, so we will henceforthjust writeM (A) instead ofMω(A).

Definition 2.3.1. LetA be aV a-algebra andM anA-module.(i) We define a uniform structure on the setIA(M) of A-submodules ofM , as follows.

For every finitely generated idealm0 ⊂ m, the subset ofIA(M) × IA(M) given byEM(m0) := (M0,M1) | m0M0 ⊂ M1 andm0M1 ⊂ M0 is an entourage for the uniformstructure, and the subsets of this kind form a fundamental system of entourages.

(ii) We define a uniform structure onM (A) as follows. For every finitely generated idealm0 ⊂ m and every integern ≥ 0 we define the entourageEM (m0) ⊂ M (A) ×M (A),which consists of all pairs ofA-modules(M,M ′) such that there exist a third moduleN and morphismsφ : N → M , ψ : N → M ′, such thatm0 annihilates the kerneland cokernel ofφ andψ. We declare that theEM (m0) form a fundamental system ofentourages for the uniform structure ofM (A).

Remark 2.3.2. Notice that the entourageEM (m0) can be defined equivalently by all the pairsof A-modules(M,M ′) such that there exists a third moduleL and morphismsφ′ : M → L,ψ : M ′ → L such thatm0 annihilates the kernel and cokernel ofφ andφ. Indeed, given apair (M,M ′) ∈ EM (m0), and a datum(N, φ, ψ) as in definition 2.3.1(ii), a datum(L, φ′, ψ′)satisfying the above condition is obtained from the push outdiagram

Nφ //

ψ

M

φ′

M ′

ψ′

// L.

(2.3.3)

Conversely, given a datum(L, φ′, ψ′), one obtains another diagram as (2.3.3), by lettingN bethe fibred product ofM andM ′ overL.

2.3.4. We will also need occasionally a notion of “Cauchy product” : let∏∞

n=0 In be a formalinfinite product of idealsIn ⊂ A. We say that the formal productsatisfies the Cauchy condition(or briefly : is a Cauchy product) if, for every neighborhoodU of A in IA(A) there existsn0 ≥ 0 such that

∏n+pm=n Im ∈ U for all n ≥ n0 and allp ≥ 0.

Lemma 2.3.5. LetM be anA-module.

(i) IA(M) with the uniform structure of definition2.3.1is complete and separated.


(ii) The following maps are uniformly continuous :(a) IA(M)×IA(M)→ IA(M) : (M ′,M ′′) 7→M ′ ∩M ′′.(b) IA(M)×IA(M)→ IA(M) : (M ′,M ′′) 7→M ′ +M ′′.(c) IA(A)×IA(A)→ IA(A) : (I, J) 7→ IJ .

(iii) For anyA-linear morphismφ : M → N , the following maps are uniformly continuous:(a) IA(M)→ IA(N) : M ′ 7→ φ(M ′).(b) IA(N)→ IA(M) : N ′ 7→ φ−1(N ′).

Proof. (i) : The separation property is easily verified. We show thatIA(M) is complete.Therefore, suppose thatF is some Cauchy filter ofIA(M). Concretely, this means thatfor every finitely generatedm0 ⊂ m, there existsF (m0) ∈ F such thatm0I ⊂ J for ev-ery I, J ∈ F (m0). Let L :=

⋃F∈F (

⋂I∈F I). We claim thatL is the limit of our fil-

ter. Indeed, for a given finitely generatedm0 ⊂ m, we havem0I ⊂⋂J∈F (m0)

J , for everyI ∈ F (m0), whencem0I ⊂ L. On the other hand, ifI ∈ F ⊂ F (m0), we can write:m0L =

⋃F ′⊂F m0(

⋂J∈F ′ J) ⊂ ⋃

F ′⊂F (⋂J∈F ′ m0J) ⊂ ⋃

F ′⊂F I = I (whereF ′ runs overall the subsetsF ′ ∈ F such thatF ′ ⊂ F ). This shows that(L, I) ∈ EM (m0) wheneverI ∈ F (m0), which implies the claim. (ii) and (iii) are easy and will be left to the reader.

Remark 2.3.6. In general, the uniform spaceM (A) is not separated. In view of proposition3.2.30, a counterexample is provided by remark 3.2.29.

Lemma 2.3.7. Let φ : M → N be anA-linear morphism,B an A-algebra. The followingmaps are uniformly continuous :

(i) M (A)→ IA(A) : M 7→ AnnA(M).(ii) IA(M)×IA(N)→M (A) : (M ′, N ′) 7→ (φ(M ′) +N ′)/φ(M ′).(iii) M (A)×M (A)→M (A) : (M ′,M ′′) 7→ alHomA(M ′,M ′′).(iv) M (A)×M (A)→M (A) : (M ′,M ′′) 7→ M ′ ⊗AM ′′.(v) M (A)→M (B) : M 7→ B ⊗AM .(vi) M (A)→M (A) : M 7→ Λr

AM for anyr ≥ 0, provided(B) holds.

Proof. We show (iv) and leave the others to the reader. By symmetry, we reduce to verifyingthat, if (M ′,M ′′) ∈ EM (m0) andN is an arbitraryA-module, then(N ⊗A M ′, N ⊗A M ′′) ∈EM (m2

0). Then we can further assume that there is a morphismφ : M ′ → M ′′ with m0 ·Kerφ = m0 · Coker φ = 0. We factorφ! as an epimorphism followed by a monomorphism

M ′!φ1→ Imφ!

φ2→ M ′′! , and then we reduce to checking that the kernels and cokernels of both1N ⊗A φ1 and1N ⊗A φ2 are killed bym0. This is clear for1N ⊗A φ1, and it follows easily for1N ⊗A φ2 as well, by using the Tor sequences.

Definition 2.3.8. For a subsetS of a topological spaceT , letS denote the adherence ofS in T .LetM be anA-module.

(i) M is said to befinitely generatedif its isomorphism class lies in⋃n∈N Mn(A).

(ii) M is said to bealmost finitely generatedif its isomorphism class lies in⋃n∈N Mn(A).

(iii) M is said to beuniformly almost finitely generatedif there exists an integern ≥ 0 suchthat the isomorphism class ofM lies in Un(A) := Mn(A). Then we will say thatn is auniform boundfor M .

(iv) M is said to befinitely presentedif it is isomorphic to the cokernel of a morphism offree finitely generatedA-modules. We denote byFP(A) ⊂ M (A) the subset of theisomorphism classes of finitely presentedA-modules.

(v) M is almost finitely presentedif its isomorphism class lies inFP(A).


Remark 2.3.9. Under condition (A), anA-moduleM lies in Un(A) if and only if, for everyε ∈ m there exists anA-linear morphismAn → M whose cokernel is killed byε.

Proposition 2.3.10.LetM be anA-module.

(i) M is almost finitely generated if and only if for every finitely generated idealm0 ⊂ mthere exists a finitely generated submoduleM0 ⊂M such thatm0M ⊂M0.

(ii) The following conditions are equivalent:(a) M is almost finitely presented.(b) for arbitrary ε, δ ∈ m there exist positive integersn = n(ε), m = m(ε) and a three

term complexAmψε→ An

φε→ M with ε · Coker(φε) = 0 andδ ·Kerφε ⊂ Imψε.

(c) For every finitely generated idealm0 ⊂ m there is a complexAmψ→ An

φ→ M withm0 · Coker φ = 0 andm0 ·Kerφ ⊂ Imψ.

Proof. (i): Let M be an almost finitely generatedA-module, andm0 ⊂ m a finitely generatedsubideal. Choose a finitely generated subidealm1 ⊂ m such thatm0 ⊂ m3

1; by hypothesis, thereexistA-modulesM ′ andM ′′, whereM ′′ is finitely generated, and morphismsf : M ′ →M , g :M ′ →M ′′ whose kernels and cokernels are annihilated bym1. We get morphismsm1⊗VM ′′ →Im(g) andm1 ⊗V Im(g)→ M ′, hence a composed morphismφ : m1 ⊗V m1 ⊗V M ′′ → M ′; itis easy to check thatCoker(f φ) is annihilated bym3

1, henceM0 := Im(f φ) will do.To show (ii) we will need the following :

Claim 2.3.11. Let F1 be a finitely generatedA-module and suppose that we are givena, b ∈ Vand a (not necessarily commutative) diagram

F1p //

φ

M

F2

ψ

OO

q

>>

such thatq φ = a · p, p ψ = b · q. Let I ⊂ V be an ideal such thatKer q has a finitelygenerated submodule containingI · Ker q. Then Ker p has a finitely generated submodulecontainingab · I ·Ker p.

Proof of the claim:LetR be the submodule ofKer q given by the assumption. We haveIm(ψ φ − ab · 1F1) ⊂ Ker p andψ(R) ⊂ Ker p. We takeR1 := Im(ψ φ − ab · 1F1) + ψ(R).Clearlyφ(Ker p) ⊂ Ker q, soI · φ(Ker p) ⊂ R, henceI · ψ φ(Ker p) ⊂ ψ(R) and finallyab · I ·Ker p ⊂ R1.

Claim 2.3.12. If M satisfies condition (b) of the proposition, andφ : F → M is a morphismwith F ≃ An, then for every finitely generated idealm1 ⊂ m ·AnnV (Coker φ) there is a finitelygenerated submodule ofKerφ containingm1 ·Kerφ.

Proof of the claim:Now, letδ ∈ AnnV (Coker φ) andε1, ε2, ε3, ε4 ∈ m. By assumption there is

a complexArt→ As

q→ M with ε1 ·Coker q = 0, ε2 ·Ker q ⊂ Im t. LettingF1 := F , F2 := As,a := ε1 ·ε3, b := ε4 ·δ, one checks easily thatψ andφ can be given such that all the assumptionsof claim 2.3.11 are fulfilled. So, withI := ε2 · V we see thatε1 · ε2 · ε3 · ε4 · δ · Kerφ lies ina finitely generated submodule ofKerφ. But m1 is contained in an ideal generated by finitelymany such productsε1 · ε2 · ε3 · ε4 · δ.

Now, it is clear that (c) implies (a) and (b). To show that (b) implies (c), take a finitelygenerated idealm1 ⊂ m such thatm0 ⊂ m ·m1, pick a morphismφ : An → M whose cokernelis annihilated bym1, and apply claim 2.3.12. We show that (a) implies (c). For a given finitelygenerated subidealm0 ⊂ m, pick another finitely generatedm1 ⊂ m such thatm0 ⊂ m3

1; find


morphismsf : M ′ → M andg : M ′ →M ′′ whose kernels and cokernels are annihilated bym1,and such thatM ′′ is finitely presented. Letε := (ε1, ..., εr) be a finite sequence of generators ofm1, and denote byK• := K•(ε) the Koszul complex ofV -modules associated to the sequenceε. Setm′1 := Coker(K2 → K1); we derive a natural surjection∂ : m′1 → m1 and, for everyi = 1, ..., r, mapsei : V → m′1 such that the compositions

m′1∂→ m1 → V

ei→ m′1 Vei→ m′1

∂→ m1 → V

are both scalar multiplication byεi. Hence, for everyV a-moduleN , the kernel and cokernel ofthe natural morphismm′1⊗V N → N are annihilated bym1. Let nowφ be as in the proof of (i);notice that the diagram:

m1 ⊗V m1 ⊗V M ′ //

M ′

m1 ⊗V m1 ⊗V M ′′ //

φ

77ooooooooooooo

M ′′

commutes. It follows that the composed morphism

m′1 ⊗V m′1 ⊗V M ′′ → m1 ⊗V m1 ⊗V M ′′ φ→ M ′ → M

has kernel and cokernel annihilated bym31, so the claim follows.

2.3.13. Suppose thatm =⋃λ∈Λ mλ, where(mλ | λ ∈ Λ) is a filtered family of subideals such

thatm2λ = mλ for everyλ ∈ Λ. LetR be aV -algebra,M anR-module, and denote byRa

λ (resp.Ra) the(V,mλ)

a-algebra (resp.(V,m)-algebra) associated toR; define similarlyMa andMaλ ,

for all λ ∈ Λ.

Lemma 2.3.14.With the notation of(2.3.13), the following conditions are equivalent:

(i) Ma is an almost finitely generated (resp. almost finitely presented)Ra-module.(ii) Ma

λ is an almost finitely generated (resp. almost finitely presented)Raλ-module for all

λ ∈ Λ.

Proof. We only give the proof for the case of almost finitely presented modules, and let thereader spell out the analogous (and easier) argument for almost finitely generated modules.

(i) ⇒ (ii) : for given λ ∈ Λ and a finitely generated subidealm0 ⊂ mλ, pick a complex(Ra)m → (Ra)n → Ma as in proposition 2.3.10(ii.c); after applying termwise the naturalfunctorRa-Mod→ Ra

λ-Mod, we obtain another complex that satisfies again the condition ofproposition 2.3.10(ii.c), so the claim follows.

(ii) ⇒ (i) : for finitely generatedm0 ⊂ m find λ ∈ Λ such thatm0 ⊂ mλ; then pick a complex

(Raλ)m ψ→ (Ra

λ)n φ→ Ma

λ as in the foregoing. SetN := Maλ∗, let Na be the image ofN in

Ra-Mod and notice that(Ma, Na) ∈ EM (m0). Let ε1, ..., εk be a set of generators form0; letα : Rm → (Ra

λ∗)n be the composition of the adjunction mapRm → (Ra

λ∗)m and the mapψ∗.

Let e1, ..., ek be a basis ofV k andf1, ...fm a basis ofRm; we define a mapβ : V k⊗V Rm → Rm

by the rule:ei ⊗ fj 7→ εi · fj . The mapα β lifts to anR-linear mapγ : V k ⊗V Rm → Rn

andφ∗ induces anR-linear mapCoker γ → N whose kernel and cokernel are annihilated bym2

0 ·m. All in all, this shows that(Coker γa,Ma) ∈ EM (m40), so the claim follows.

The following proposition generalises a well-known characterization of finitely presentedmodules over usual rings.

Proposition 2.3.15.LetM be anA-module.(i) M is almost finitely generated if and only if, for every filteredsystem(Nλ, φλµ) (indexed

by a directed setΛ) the natural morphism

M : colimΛ

alHomA(M,Nλ)→ alHomA(M, colimΛ

Nλ)(2.3.16)


is a monomorphism.(ii) M is almost finitely presented if and only if for every filtered inductive sytem as above,

(2.3.16)is an isomorphism.

Proof. The “only if” part in (i) (resp. (ii)) is first checked whenM is finitely generated (resp.finitely presented) and then extended to the general case. Weleave the details to the reader andwe proceed to verify the “if” part. For (i), choose a setI and an epimorphismp : A(I) → M .Let Λ be the directed set of finite subsets ofI, ordered by inclusion. ForS ∈ Λ, let MS :=p(AS). Thencolim

Λ(M/MS) = 0, so the assumption givescolim

ΛalHomA(M,M/MS) = 0, i.e.

colimΛ

HomA(M,M/MS) = 0 is almost zero, so, for everyε ∈ m, the image ofε · 1M in the

above colimit is0, i.e. there existsS ∈ Λ such thatεM ⊂ MS, which proves the contention. For(ii), we presentM as a filtered colimitcolim

ΛMλ, where eachMλ is finitely presented (this can

be donee.g.by taking such a presentation of theA∗-moduleM∗ and applyingN 7→ Na). Theassumption of (ii) gives thatcolim

ΛHomA(M,Mλ)→ HomA(M,M) is an almost isomorphism,

hence, for everyε ∈ m there isλ ∈ Λ andφε : M → Mλ such thatpλ φε = ε · 1M , wherepλ : Mλ → M is the natural morphism to the colimit. If such aφε exists forλ, then it existsfor everyµ ≥ λ. Hence, ifm0 ⊂ m is a finitely generated subideal, saym0 =

∑kj V εj,

then there existλ ∈ Λ andφi : M → Mλ such thatpλ φi = εi · 1M for i = 1, ..., k. HenceIm(φi pλ−εi ·1Mλ

) is contained inKer pλ and containsεi ·Ker pλ. HenceKer pλ has a finitelygenerated submoduleL containingm0 · Ker pλ. Choose a presentationAm → An

π→ Mλ.Then one can liftm0L to a finitely generated submoduleL′ of An. ThenKer(π) + L′ is afinitely generated submodule ofKer(pλ π) containingm2

0 · Ker(pλ π). Since we also havem0 ·Coker(pλ π) = 0 andm0 is arbitrary, the conclusion follows from proposition 2.3.10.

Lemma 2.3.17.Let0→ M ′ → M →M ′′ → 0 be an exact sequence ofA-modules. Then:(i) If M ′,M ′′ are almost finitely generated (resp. presented) then so isM .(ii) If M is almost finitely presented, thenM ′′ is almost finitely presented if and only ifM ′ is

almost finitely generated.

Proof. These facts can be deduced from proposition 2.3.15 and remark 2.4.12(iii), or proveddirectly.

Lemma 2.3.18.Let (Mn ; φn : Mn → Mn+1 | n ∈ N) be a direct system ofA-modules andsuppose there exist sequences(εn | n ∈ N) and(δn | n ∈ N) of ideals ofV such that

(i) limn→∞

εan = V a (for the uniform structure of definition2.3.1) and∏∞

j=0 δj is a Cauchy

product (see(2.3.4);(ii) for all n ∈ N there exist integersN(n) and morphisms ofA-modulesψn : AN(n) → Mn

such thatεn · Cokerψn = 0;(iii) δn · Coker φn = 0 for all n ∈ N.

Thencolimn∈N

Mn is an almost finitely generatedA-module.

Proof. Let M := colimn∈N

Mn. For anyn ∈ N let an =⋂m≥0(

∏n+mj=n δj). Then lim

n→∞an = V .

Form > n setφn,m = φm ... φn+1 φn : Mn → Mm+1 and letφn,∞ : Mn → M be thenatural morphism. An easy induction shows that

∏mj=n δj · Coker φn,m = 0 for all m > n ∈ N.

SinceCoker φn,∞ = colimm∈N

Coker φn,m we obtainan · Coker φn,∞ = 0 for all n ∈ N. Therefore

εn · an · Coker(φn,∞ ψn) = 0 for all n ∈ N. Since limn→∞

εn · an = V , the claim follows.

In the remaining of this sectionwe assume that condition(B) of (2.1.6)holds. We wish todefine the Fitting ideals of an arbitrary uniformly almost finitely generatedA-moduleM . This


will be achieved in two steps: first we will see how to define theFitting ideals of a finitelygenerated module, then we will deal with the general case. Werefer to [51, Ch.XIX] for thedefinition of the Fitting idealsFi(M) of a finitely generated module over an arbitrary ringR.

Lemma 2.3.19.Let R be aV -algebra andM , N two finitely generatedR-modules with anisomorphism ofRa-modulesMa ≃ Na. ThenFi(M)a = Fi(N)a for everyi ≥ 0.

Proof. By the usual arguments, for everyε ∈ m we have morphismsα : M → N , β : N →Mwith kernels and cokernels killed byε2. Then we have:Fi(N) ⊃ Fi(Imα) · F0(Cokerα). IfN is generated byk elements, then the same holds forCokerα, whenceAnnR(Cokerα)k ⊂F0(Cokerα), thereforeε2kR ⊂ F0(Cokerα), and consequentlyFi(N) ⊃ ε2kFi(Imα). SinceIm(α) is a quotient ofM , it is clear thatFi(M) ⊂ Fi(Imα), so finallyε2kFi(M) ⊂ Fi(N).Arguing symmetrically withβ one hasε2kFi(N) ⊂ Fi(M). Since we assume (B), the claimfollows.

2.3.20. LetM be a finitely generatedA-module. In light of lemma 2.3.19, the Fitting idealsFi(M) are well defined as ideals inA.

Lemma 2.3.21.Let m0 ⊂ m be a finitely generated subideal andn ∈ N. Pick ε1, ..., εk ∈ msuch thatm0 ⊂ (ε3n

1 , ..., ε3nk ) and setm1 := (ε1, ..., εk). Then(Fi(M), Fi(M

′)) ∈ EA(m0) forevery(M,M ′) ∈ EM (m1) such thatM andM ′ are generated by at mostn of their almostelements.

Proof. Let M,M ′ be as in the lemma. By hypothesis, there exist anA-moduleN and mor-phismsφ : N →M andψ : N → M ′ such thatm1 annihilates the kernel and cokernel ofφ andψ. By symmetry, it suffices to show thatε3n

i Fi(M) ⊂ Fi(M′) for everyi = 1, ..., k. Now, for

everyi ≤ k, the morphismM → M : x 7→ εi · x factors through a morphismα : M → φ(N),and similarly, scalar multiplication byεi onN factors through a morphismβ : φ(N) → N .Thenη := ψ β α : M → M ′ has kernel and cokernel annihilated byε3

i . Pick finitelygeneratedA∗-modulesL ⊂ M∗, L′ ⊂ M ′∗ such thatLa = M andL′a = M ′. ReplacingL′

by L′ + η∗(L) we can assume thatη∗(L) ⊂ L′. ThenFi(M) = Fi(L)a, Fi(L′)a = Fi(M′)

andFi(L′) ⊃ Fi(L/(L ∩ Ker η∗)) · F0(L′/η∗L). SinceL′/η∗(L) is generated by at most

n elements and is annihilated byε3i · m, we haveε3n

i · m ⊂ F0(L′/(η∗L)). Furthermore

Fi(L/(L ∩Ker η∗)) ⊃ Fi(L), so the claim follows.

Proposition 2.3.22.For everyi, n ≥ 0, the mapFi : Mn(A) → IA(A) is uniformly continu-ous and therefore it extends uniquely to a uniformly continuous mapFi : Un(A)→ IA(A).

Proof. The uniform continuity follows readily from lemma 2.3.21. SinceIA(A) is complete, itfollows thatFi extends to the whole ofUn(A). Finally, the extension is unique becauseIA(A)is separated.

Definition 2.3.23. LetM be a uniformly almost finitely generatedA-module. We callFi(M)thei-th Fitting idealof M .

Proposition 2.3.24. (i) Let0→ M ′φ→ M

ψ→ M ′′ → 0 be a short exact sequence of uniformlyalmost finitely generatedA-modules. Then:

∑j+k=i Fj(M

′)·Fk(M ′′) ⊂ Fi(M) for everyi ≥ 0.

(ii) For every uniformly almost finitely generatedA-moduleM , anyA-algebraB and anyi ≥ 0 we haveFi(B ⊗AM) = Fi(M) · B.

Proof. (i): Let n be uniform bound forM andM ′; by remark 2.3.2 we can find, for every

subidealm0 ⊂ m, A-modulesM0, M ′0, L, L′ and morphismsMα→ L

β← M0,M ′α′

→ L′β′

← M ′0


whose kernels and cokernels are annihilated bym0, and such thatM0 andM ′0 are generated byn almost elements. LetN be defined by the push-out diagram

M ′φ //

α′

Mα // L

γ

L′

γ′ // N.

Furthermore setM ′1 := Im(γ′β ′ : M ′0 → N),M1 := Im((γβ)⊕(γ′β ′) : M0⊕M ′0 → N) andlet M ′′1 be the cokernel of the induced monomorphismM ′1 → M1. We deduce a commutativediagram with short exact rows:

0 // M ′φ //

Mψ //

M ′′

// 0

0 // Im γ′ // N // Coker γ′ // 0

0 // M ′1

OO

// M1

OO

// M ′′1

OO

// 0.

(2.3.25)

One checks easily that the kernels and cokernels of all the vertical arrows in (2.3.25) are an-nihilated bym2

0, i.e. (M,M1), (M′,M ′1), (M

′′,M ′′1 ) ∈ EM (m20). Let x1, ..., xn ∈ M0∗ (resp.

x′1, ..., x′n ∈ M ′0∗) be a set of generators forM0 (resp. forM ′0). For everyi = 1, ..., n, let

zi := γ β(xi) andz′i := γ′ β ′(x′i). Let Q′ ⊂ N∗ (respQ ⊂ N∗) be theA∗-module gen-erated by thez′i (resp. and by thezi). It is clear that the bottom row of (2.3.25) is naturallyisomorphic to the short exact sequence(0 → Q′ → Q → Q/Q′ → 0)a. It is well-knownthat Fi(Q′) · Fj(Q/Q′) ⊂ Fi+j(Q) for every i, j ∈ N; by lemma 2.3.5(ii.c) and proposi-tion 2.3.22 all the operations under considerations are uniformly continuous, so we deduceFi(M

′) · Fj(M/M ′) ⊂ Fi+j(M), which is (i).(ii): since the identity is known for usual finitely generated modules over rings, the claim

follows easily from proposition 2.3.22 and lemma 2.3.7(v).

2.4. Almost homological algebra. In this section we fix an almostV -algebraA and we con-sider various constructions in the category ofA-modules.

2.4.1. By corollary 2.2.15 any inverse system(Mn | n ∈ N) of A-modules has an (inverse)limit lim

n∈NM . As usual, we denote bylim1 the right derived functor of the inverse limit functor.

Notice that [67, Cor. 3.5.4] holds in the almost case since axiom (AB4*) holds inA-Mod (onthe other hand, [67, Lemma 3.5.3] does not hold under (AB4*),(the proof given there useselements : for a counterexample in an exotic abelian category, see [57]).

Lemma 2.4.2. Let (Mn ; φn : Mn → Mn+1 | n ∈ N) (resp.(Nn ; ψn : Nn+1 → Nn | n ∈ N))be a direct (resp. inverse) system ofA-modules and morphisms and(εn | n ∈ N) a sequence ofideals ofV a converging toV a (for the uniform structure of definition2.3.1).

(i) If εn ·Mn = 0 for all n ∈ N thencolimn∈N

Mn ≃ 0.

(ii) If εn ·Nn = 0 for all n ∈ N thenlimn∈N

Nn ≃ 0 ≃ limn∈N

1Nn.

(iii) If εn · Cokerψn = 0 for all n ∈ N and∏∞

j=0 εj is a Cauchy product, thenlimn∈N

1Nn ≃ 0.

Proof. (i) and (ii) : we remark only thatlimn∈N

1Nn ≃ limn∈N

1Nn+p for all p ∈ N and leave the details

to the reader. We prove (iii). From [67, Cor. 3.5.4] it follows easily that(limn∈N

1Nn∗)a ≃ lim

n∈N

1Nn.


It then suffices to show thatlimn∈N

1Nn∗ is almost zero. We haveε2n · Cokerψn∗ = 0 and the

product∏∞

j=0 ε2j is again a Cauchy product. Next letN ′n :=

⋂p≥0 Im(Nn+p∗ → Nn∗). If

Jn :=⋂p≥0(εn · εn+1 · ... · εn+p)

2 thenJnNn∗ ⊂ N ′n and limn→∞

Jan = V a. In view of (ii),

limn∈N

1Nn∗/N′n is almost zero, hence we reduce to showing thatlim

n∈N

1N ′n is almost zero. But

Jn+p+q ·N ′n ⊂ Im(N ′n+p+q → N ′n) ⊂ Im(N ′n+p → N ′n)

for all n, p, q ∈ N. On the other hand, since the idealsJan converge toV a, we get⋃∞q=0 m ·

Jn+p+q = m, hencemN ′n ⊂ Im(N ′n+p → N ′n) and finallymN ′n = m2N ′n ⊂ Im(mN ′n+p →mN ′n) which means thatmN ′n is a surjective inverse system, so itslim1 vanishes and the

result follows.

Example 2.4.3.Let (V,m) be as in example 2.1.2. Then every finitely generated ideal inV isprincipal, so in the situation of the lemma we can writeεj = (xj) for somexj ∈ V . Then thehypothesis in (iii) can be stated by saying that there existsc ∈ N such thatxj 6= 0 for all j ≥ cand the sequencen 7→∏n

j=c |xj | is Cauchy inΓ.

Definition 2.4.4. LetM be anA-module.

(i) We say thatM is flat (resp.faithfully flat) if the functorN 7→M ⊗AN , from the categoryof A-modules to itself is exact (resp. exact and faithful).

(ii) We say thatM is almost projectiveif the functorN 7→ alHomA(M,N) is exact.

For euphonic reasons, we will use the expression ”almost finitely generated projective” to de-note anA-module which is almost projective and almost finitely generated. This conventiondoes not give rise to ambiguities, since we will never consider projective almost modules : in-deed, the following example 2.4.5 explains why the categorical notion of projectivity is uselessin the setting of almost ring theory.

Example 2.4.5.First of all we remark that the functorM 7→ M! preserves (categorical) pro-jectivity, since it is left adjoint to an exact functor. Moreover, ifP! is a projectiveV -modulePis a projectiveV a-module, as one checks easily using the fact the the functorM 7→ M! is rightexact. Hence one has an equivalence from the full subcategory of projectiveV a-modules, tothe full subcategory of projectiveV -modulesP such thatP = m⊗V P . The latter condition isequivalent toP = mP ; indeed, asP is flat, we havemP = m⊗V P .

As an example, suppose thatV is local; then every projectiveV -module is free, so ifm 6= Vthe conditionP = mP impliesP = 0, ergo, there are no non-trivial projectiveV a-modules.

Lemma 2.4.6. LetM be an almost finitely generatedA-module andB a flatA-algebra. ThenAnnB(B ⊗AM) = B ⊗A AnnA(M).

Proof. Using lemma 2.3.7(i),(v) we reduce easily to the case of a finitely generatedA-moduleM . Then, letx1, ..., xk ∈M∗ be a finite set of generators forM ; we have:AnnA(M) = Ker(φ :A→ Mk), whereφ is defined by the rule:a 7→ (a · x1, ..., a · xk) for everya ∈ A∗. SinceB isflat, we haveKer(1B ⊗A φ) ≃ B ⊗A Kerφ, whence the claim.

Lemma 2.4.7. Let P be one of the properties : “flat”, “almost projective”, “almost finitelygenerated”, “almost finitely presented”. IfB is aP A-algebra, andM is aP B-module, thenM is P as anA-module.

Proof. Left to the reader.


2.4.8. LetR be aV -algebra andM a flat (resp. faithfully flat)R-module (in the usual sense,see [54, p.45]). ThenMa is a flat (resp. faithfully flat)Ra-module. Indeed, the functorM ⊗R−preserves the Serre subcategory of almost zero modules, so by general facts it induces an exactfunctor on the localized categories (cp. [38, p.369]). For the faithfullness we have to show thatanR-moduleN is almost zero wheneverM ⊗RN is almost zero. However,M ⊗RN is almostzero⇔M ⊗R (m⊗V N) = 0⇔ m⊗V N = 0⇔ N is almost zero. It is clear thatA-Mod hasenough almost projective (resp. flat) objects.

2.4.9. LetR be aV -algebra. The localisation functor induces a functorG : D(R-Mod) →D(Ra-Mod) and, in view of corollary 2.2.22,M 7→M! induces a functorF : D(Ra-Mod)→D(R-Mod). We have a natural isomorphismG F ≃ 1D(Ra-Mod) and a natural transforma-tion F G → 1D(R-Mod). These satisfy the triangular identities of [53, p.83] soF is a leftadjoint toG. If Σ denotes the multiplicative set of morphisms inD(R-Mod) which inducealmost isomorphisms on the cohomology modules, then the localised categoryΣ−1D(R-Mod)exists (seee.g. [67, Th.10.3.7]) and by the same argument we get an equivalence of categoriesΣ−1D(R-Mod) ≃ D(Ra-Mod).

2.4.10. Given anA-moduleM , we can derive the functorsM ⊗A − (resp.alHomA(M,−),resp. alHomA(−,M)) by taking flat (resp. injective, resp. almost projective) resolutions :one remarks that bounded above exact complexes of flat (resp.almost projective)A-modulesare acyclic for the functorM ⊗A − (resp. alHomA(−,M)) (recall the standard argument: ifF• is a bounded above exact complex of flatA-modules, letΦ• be a flat resolution ofM ; thenTot(Φ•⊗AF•)→M⊗AF• is a quasi-isomorphism since it is so on rows, andTot(Φ•⊗AF•) isacyclic since its columns are; similarly, ifP• is a complex of almost projective objects, one con-siders the double complexalHomA(P•, J

•) whereJ• is an injective resolution ofM ; cp. [67,§2.7]); then one uses the construction detailed in [67, Th.10.5.9]. We denote byTorAi (M,−)(resp.alExtiA(M,−), resp.alExtiA(−,M)) the corresponding derived functors. IfA := Ra forsomeV -algebraR, we obtain easily natural isomorphisms

TorRi (M,N)a ≃ TorAi (Ma, Na)(2.4.11)

for all R-modulesM,N . A similar result holds forExtiR(M,N).

Remark 2.4.12. (i) Clearly, anA-moduleM is flat (resp. almost projective) if and only ifTorAi (M,N) = 0 (resp.alExtiA(M,N) = 0) for all A-modulesN and alli > 0. In particular,an almost projectiveA-module is flat, because for everyε ∈ m the scalar multiplication byε :M →M factors through a free module.

(ii) Let M,N be two flat (resp. almost projective)A-modules. ThenM ⊗A N is a flat (resp.almost projective)A-module and for anyA-algebraB, theB-moduleB ⊗A M is flat (resp.almost projective).

(iii) Resume the notation of proposition 2.3.15. IfM is almost finitely presented, then one hasalso that the natural morphismcolim

ΛalExt1

A(M,Nλ) → alExt1A(M, colim

ΛNλ) is a monomor-

phism. This is deduced from proposition 2.3.15(ii), using the fact that(Nλ) can be injectedinto an inductive system(Jλ) of injectiveA-modules (e.g. Jλ = EHomA(Nλ,E), whereE is aninjective cogenerator forA-Mod), and by applyingalExt sequences.

Lemma 2.4.13.Resume the notation of(2.3.13). The following conditions are equivalent:

(i) Ma is a flat (resp. almost projective)Ra-module.(ii) Ma

λ is a flat (resp. almost projective)Raλ-module for everyλ ∈ Λ.

Proof. It is a straightforward consequence of (2.4.11) and its analogue forExtiR(M,N).

Lemma 2.4.14.LetR be aV -algebra.


(i) There is a natural isomorphism:ExtiR(M!, N) ≃ ExtiRa(M,Na) for everyRa-moduleM ,everyR-moduleN and every integeri ∈ N.

(ii) If P is an almost projectiveRa-module, thenhom.dimRP! ≤ hom.dimVm.

Proof. (herehom.dim denotes homological dimension). (i) is a straightforward consequence ofthe existence of the adjunction(F,G) of (2.4.9). Next we consider, for arbitraryR-modulesMandN , the spectral sequence:

Ep,q2 := ExtpV (m,ExtqR(M,N))⇒ Extp+qR (m⊗V M,N).

(this spectral sequence is constructede.g. from the double complexHomV (Fp,HomR(F ′q, N))whereF• (resp.F ′•) is a projective resolution ofm (resp. ofM)). If now we letM := P!, wededuce from (i) thatExtpV (m,ExtqR(P!, N)) ≃ ExtpV a(V a,ExtqR(P!, N)a) ≃ 0 for everyp ∈ Nand everyq > 0. Sincem⊗V P! ≃ P!, assertion (ii) follows easily.

Lemma 2.4.15.LetM be an almost finitely generatedA-module. ThenM is almost projectiveif and only if, for arbitraryε ∈ m, there existn(ε) ∈ N andA-linear morphisms

Muε−→ An(ε) vε−→M(2.4.16)

such thatvε uε = ε · 1M .

Proof. Let morphisms as in (2.4.16) be given. Pick anyA-moduleN and apply the functoralExtiA(−, N) to (2.4.16) to get morphisms

alExtiA(M,N)→ alExtiA(An(ε), N)→ alExtiA(M,N)

whose composition is again the scalar multiplication byε; henceε · alExtiA(M,N) = 0 forall i > 0. Sinceε is arbitrary, it follows from remark 2.4.12(i) thatM is almost projective.Conversely, suppose thatM is almost projective; by hypothesis, for arbitraryε ∈ m we canfind n := n(ε) and a morphismφε : An → M such thatε · Coker φε = 0. Let Mε be the

image ofφε, so thatφε factors asAn(ε) ψε−→ Mεjε−→ M . Also ε · 1M : M → M factors as

Mγε−→Mε

jε−→M. Since by hypothesisM is almost projective, the natural morphism inducedbyψε :

alHomA(M,An)ψ∗

ε−→ alHomA(M,Mε)

is an epimorphism. Then for arbitraryδ ∈ m the morphismδ · γε is in the image ofψ∗ε , in otherwords, there exists anA-linear morphismuεδ : M → An such thatψε uεδ = δ · γε. If now wetakevεδ := φε, it is clear thatvεδ uεδ = ε · δ · 1M . This proves the claim.

Lemma 2.4.17.LetR be any ring,M anyR-module andC := Coker(φ : Rn → Rm) anyfinitely presented (left)R-module. LetC ′ := Coker(φ∗ : Rm → Rn) be the cokernel of thetranspose of the mapφ. Then there is a natural isomorphism

TorR1 (C ′,M) ≃ HomR(C,M)/Im(HomR(C,R)⊗RM).

Proof. We have a spectral sequence :

E2ij := TorRi (Hj(Cone φ∗),M)⇒ Hi+j(Cone(φ∗)⊗R M).

On the other hand we have also natural isomorphisms

Cone(φ∗)⊗R M ≃ HomR(Cone φ,R)[1]⊗RM ≃ HomR(Coneφ,M)[1].

Hence :E2

10 ≃ E∞10 ≃ H1(Cone(φ∗)⊗RM)/E∞01 ≃H0(HomR(Coneφ,M))/Im(E201)

≃HomR(C,M)/Im(HomR(C,R)⊗R M)

which is the claim.


Proposition 2.4.18.LetA be aV a-algebra.

(i) Every almost finitely generated projectiveA-module is almost finitely presented.(ii) Every almost finitely presented flatA-module is almost projective.

Proof. (ii) : let M be such anA-module. Letε, δ ∈ m and pick a three term complex

Amψ−→ An

φ−→ M

such thatε · Coker φ = δ · Ker(φ)/Im(ψ) = 0. SetP := Cokerψ∗; this is a finitely presentedA∗-module andφ∗ factors through a morphismφ∗ : P → M∗. Let γ ∈ m; from lemma 2.4.17we see thatγ · φ is the image of some element

∑nj=1 φj ⊗ mj ∈ HomA∗(P,A∗) ⊗A∗ M∗.

If we defineL := An∗ and v : P → L, w : L → M∗ by v(x) := (φ1(x), ..., φn(x)) andw(y1, ..., yn) :=

∑nj=1 yj ·mj , then clearlyγ ·φ = w v. LetK := Kerφ∗. Thenδ ·Ka = 0 and

the mapδ ·1P a factors through a morphismσ : (P/K)a → P a. Similarly the mapε ·1M factorsthrough a morphismλ : M → (P/K)a. Letα := vaσλ : M → La andβ := wa : La →M .The reader can check thatβ α = ε · δ · γ · 1M . By lemma 2.4.15 the claim follows.

(i) : let P be such an almost finitely generated projectiveA-module. For any finitely gener-ated idealm0 ⊂ m pick a morphismφ : Ar → P such thatm0 ·Cokerφ = 0. If ε1, ..., εk is a setof generators form0, a standard argument shows that, for anyi ≤ k, εi · 1P lifts to a morphismψi : P → Ar/Kerφ; then, sinceP is almost projective,εjψi lifts to a morphismψij : P → Ar.Now claim 2.3.11 applies withF1 := Ar, F2 := M = P , p := φ, q := 1P andψ := ψijand shows thatKerφ has a finitely generated submoduleMij containingεi · εj · Kerφ. Thenthe span of all suchMij is a finitely generated submodule ofKerφ containingm2

0 · Kerφ. Byproposition 2.3.10(ii), the claim follows.

In general a flat almost finitely generatedA-module is not necessarily almost finitely pre-sented, but one can give the following criterion, which extends [17,§1, Exerc.13].

Proposition 2.4.19. If A→ B is a monomorphism ofV a-algebras andM is an almost finitelygenerated flatA-module such thatB⊗AM is almost finitely presented overB, thenM is almostfinitely presented overA.

Proof. Let m0 be a finitely generated subideal ofm and anA-linear morphismφ : An → Msuch thatm0 · Coker φ = 0. By assumption and claim 2.3.12, we can find a finitely generatedB-submoduleR of Ker(1B ⊗A φ) such that

m20 ·Ker(1B ⊗A φ) ⊂ R.(2.4.20)

By a Tor sequence we havem0 · Coker(1B ⊗A Ker(φ) → Ker(1B ⊗A φ)) = 0, hencem ·m0 ·Coker(1B∗ ⊗A∗ Ker(φ)∗ → Ker(1B ⊗A φ)∗) = 0, therefore there exists a finitely generatedsubmoduleR0 of Kerφ such that

m20R ⊂ B · Im(R0 → Ker(1B ⊗A φ)).(2.4.21)

By lemma 2.4.17, for everyε ∈ m the morphismε · φ : An/R0 → M factors through amorphismψ : An/R0 → F , whereF is a finitely generated freeA-module. SinceF ⊂B⊗A F , we deduce easilyKer(An/R0 → Bn/B ·R0) ⊂ Kerψ; on the other hand, by (2.4.20)and (2.4.21) we derivem4

0 · Kerφ ⊂ Ker(An/R0 → Bn/B · R0). Thusψ factors throughM ′ := An/(R0 + m4

0 · Kerφ). Clearlym40 · Ker(M ′ → M) = m0 · Coker(M ′ → M) = 0;

hence, for everyA-moduleN , the kernel of the induced morphism

alExt1A(M,N)→ alExt1

A(M ′, N)(2.4.22)

is annihilated bym50; however (2.4.22) factors throughalExt1

A(F,N) = 0, thereforem50 ·

alExt1A(M,N) = 0 for everyA-moduleN . This shows thatM is almost projective, which

is equivalent to the conclusion, in view of proposition 2.4.18(i).


Definition 2.4.23. LetM be anA-module,f : A→ M an almost element ofM .(i) ThedualA-moduleof M is theA-moduleM∗ := alHomA(M,A).

(ii) The evaluation morphismis the morphismevM/A : M ⊗AM∗ → A : m⊗ φ 7→ φ(m).(iii) The evaluation idealof M is the idealEM/A := Im evM/A.(iv) Theevaluation idealof f is the idealEM/A(f) := Im(evM/A (f ⊗A 1M∗)).(v) We say thatM is reflexiveif the natural morphism

M → (M∗)∗ m 7→ (f 7→ f(m))(2.4.24)

is an isomorphism ofA-modules.(vi) We say thatM is invertible if M ⊗AM∗ ≃ A.

Remark 2.4.25. Notice that ifB is anA-algebra andM anyB-module, then by “restrictionof scalars”M is also anA-module and the dualA-moduleM∗ has a natural structure ofB-module. This is defined by the rule(b · f)(m) := f(b · m) (b ∈ B∗, m ∈ M∗ andf ∈ M∗∗ ).With respect to this structure (2.4.24) becomes aB-linear morphism. Incidentally, notice thatthe two meanings of “M∗∗ ” coincide,i.e. (M∗)∗ ≃ (M∗)∗.

2.4.26. IfE, F andN areA-modules, there is a natural morphism :

E ⊗A alHomA(F,N)→ alHomA(F,E ⊗A N).(2.4.27)

LetP be anA-module. As a special case of (2.4.27) we have the morphism:

ωP/A : P ⊗A P ∗ → EndA(P )a := alHomA(P, P )

such thatωP/A(p⊗ φ)(q) := p · φ(q) for everyp, q ∈ P∗ andφ : P → A.

Proposition 2.4.28.LetP be an almost projectiveA-module.

(i) For every morphism of algebrasA→ B we haveEB⊗AP/B = EP/A · B.(ii) EP/A = E 2

P/A.(iii) P = 0 if and only ifEP/A = 0.(iv) P is faithfully flat if and only ifEP/A = A.(v) EP/A(f) is the smallest of the idealsJ ⊂ A such thatf ∈ (JP )∗.

Proof. Pick an indexing setI large enough, and an epimorphismφ : F := A(I) → P . Foreveryi ∈ I we have the standard morphismsA

ei→ Fπi→ A such thatπi ej = δij · 1A and∑

i∈I ei πi = 1F . For everyx ∈ m chooseψx ∈ HomA(P, F ) such thatφ ψx = x · 1P . It iseasy to check thatEP/A is generated by the almost elementsπi ψx φ ej (i, j ∈ I, x ∈ m). (i)follows already. For (iii), the “only if” is clear; ifEP/A = 0, thenψxφ = 0 for all x ∈ m, henceψx = 0 and thereforex · 1P = 0, i.e. P = 0. Next, notice that (i) and (iii) implyP/EP/AP = 0,i.e.P = EP/AP , so (ii) follows directly from the definition ofEP/A. SinceP is flat, to show (iv)we have only to verify that the functorM 7→ P ⊗A M is faithful. To this purpose, it sufficesto check thatP ⊗A A/J 6= 0 for every proper idealJ of A. This follows easily from (i) and(iii). Finally, it clear thatEP/A(f) ⊂ J for every idealJ ⊂ A with f ∈ (JP )∗. Conversely, forgivenε ∈ m pick a sequenceP

u→ Anv→ P as in (2.4.16); we haveu(f) = (a1, ..., an) with

a1, ..., an ∈ EP/A(f)∗. (v) follows from the identityεf =∑n

i=1 ai · v(ei) (wheree1, ..., en is thestandard basis of the freeA∗-moduleAn∗ ).

Lemma 2.4.29.LetE, F ,N be threeA-modules.

(i) The morphism(2.4.27)is an isomorphism in the following cases :(a) whenE is flat andF is almost finitely presented;(b) when eitherE or F is almost finitely generated projective;(c) whenF is almost projective andE is almost finitely presented;(d) whenE is almost projective andF is almost finitely generated.


(ii) The morphism(2.4.27)is a monomorphism in the following cases :(a) whenE is flat andF is almost finitely generated;(b) whenE is almost projective.

(iii) The morphism(2.4.27)is an epimorphism whenF is almost projective andE is almostfinitely generated.

Proof. If F ≃ A(I) for some finite setI, thenalHomA(F,N) ≃ N (I) and the claims areobvious. More generally, ifF is almost finitely generated projective, for anyε ∈ m there existsa finite setI := I(ε) and morphisms

Fuε−→ A(I) vε−→ F(2.4.30)

such thatvε uε = ε · 1F . We apply the natural transformation

E ⊗A alHomA(−, N)→ alHomA(−, E ⊗A N)

to (2.4.30) : an easy diagram chase allows then to conclude that the kernel and cokernel of(2.4.27) are killed byε. As ε is arbitrary, it follows that (2.4.27) is an isomorphism in this case.An analogous argument works whenE is almost finitely generated projective, so we get (i.b).If F is only almost projective, then we still have morphisms of the type (2.4.30), but nowI(ε)is no longer necessarily finite. However, the cokernels of the induced morphisms1E ⊗ uε andalHomA(vε, E ⊗A N) are still annihilated byε. Hence, to show (iii) (resp. (i.c)) it sufficesto consider the case whenF is free andE is almost finitely generated (resp. presented). Bypassing to almost elements, we can further reduce to the analogous question for usual ringsand modules, and by the usual juggling we can even replaceE by a finitely generated (resp.presented)A∗-module andF by a freeA∗-module. This case is easily dealt with, and (iii) and(i.c) follow. Case (i.d) (resp. (ii.b)) is similar : one considers almost elements and replacesE∗ by a freeA∗-module (resp. andF∗ by a finitely generatedA∗-module). In case (ii.a) (resp.(i.a)), for every finitely generated submodulem0 of m we can find, by proposition 2.3.10, afinitely generated (resp. presented)A-moduleF0 and a morphismF0 → F whose kernel andcokernel are annihilated bym0. It follows easily that we can replaceF by F0 and suppose thatF is finitely generated (resp. presented). Then the argument in [16, Ch.I§2 Prop.10] can betaken oververbatimto show (ii.a) (resp. (i.a)).

Lemma 2.4.31.LetB be anA-algebra.

(i) Let P be anA-module. If eitherP or B is almost finitely generated projective as anA-module, the natural morphism

B ⊗A alHomA(P,N)→ alHomB(B ⊗A P,B ⊗A N)(2.4.32)

is an isomorphism for allA-modulesN .(ii) Every almost finitely generated projectiveA-module is reflexive.

(iii) If P is an almost finitely generated projectiveB-module, the natural morphism

alHomB(P,B)⊗B alHomA(B,A)→ alHomA(P,A) : φ⊗ ψ 7→ ψ φ(2.4.33)

is an isomorphism ofB-modules.

Proof. (i) is an easy consequence of lemma 2.4.29(i.b). To prove (ii), we apply the natural trans-formation (2.4.24) to (2.4.30) : by diagram chase one sees that the kernel and cokernel of themorphismF → (F ∗)∗ are killed byε. (iii) is analogous : one applies the natural transformation(2.4.33) to (2.4.30).

Lemma 2.4.34.Let (Mn ; φn : Mn → Mn+1 | n ∈ N) be a direct system ofA-modules andsuppose there exist sequences(εn | n ∈ N) and(δn | n ∈ N) of ideals ofV such that

(i) limn→∞

εn = V and∏∞

j=0 δj is a Cauchy product (see(2.3.4));


(ii) εn · alExtiA(Mn, N) = δn · alExtiA(Coker φn, N) = 0 for all A-modulesN , all i > 0 andall n ∈ N;

(iii) δn ·Kerφn = 0 for all n ∈ N.

Thencolimn∈N

Mn is an almost projectiveA-module.

Proof. LetM = colimn∈N

Mn. By remark 2.4.12(i) it suffices to show thatalExtiA(M,N) vanishes

for all i > 0 and allA-modulesN . The mapsφn define a mapφ : ⊕nMn → ⊕nMn such that

we have a short exact sequence0→ ⊕nMn1−φ−→ ⊕nMn −→ M → 0. Applying the long exact

alExt sequence one obtains a short exact sequence (cp. [67, 3.5.10])

0→ limn∈N

1alExti−1A (Mn, N)→ alExtiA(M,N)→ lim

n∈NalExtiA(Mn, N)→ 0.

Then lemma 2.4.2(ii) implies thatalExtiA(M,N) ≃ 0 for all i > 1 and moreoveralExt1A(M,N)

is isomorphic tolimn∈N

1alHomA(Mn, N). Let

φ∗n : alHomA(Mn+1, N)→ alHomA(Mn, N) f 7→ f φnbe the transpose ofφn and writeφn as a compositionMn

pn−→ Im(φn)qn→ Mn+1, so that

φ∗n = q∗np∗n, the composition of the respective transposed morphims. Wehave monomorphisms

Coker p∗n → alHomA(Kerφn, N)Coker q∗n → alExt1

A(Coker φn, N)

for all n ∈ N. Henceδ2n · Cokerφ∗n = 0 for all n ∈ N. Since

∏∞n=0 δ

2n is a Cauchy product,

lemma 2.4.2(iii) shows thatlimn∈N

1alHomA(Mn, N) ≃ 0 and the assertion follows.

Proposition 2.4.35.Suppose thatm is a flatV -module. Then for anyV -algebraR the functorM 7→M! commutes with tensor products and takes flatRa-modules to flatR-modules.

Proof. Let M be a flatRa-module andN → N ′ an injective map ofR-modules. Denote byK the kernel of the induced mapM! ⊗R N → M! ⊗R N ′; we haveKa ≃ 0. We obtain anexact sequence0 → m ⊗V K → m ⊗V M! ⊗R N → m ⊗V M! ⊗R N ′. But one sees easilythat m ⊗V K = 0 andm ⊗V M! ≃ M!, which shows thatM! is a flatR-module. Similarly,letM,N be twoRa-modules. Then the natural mapM∗ ⊗R N∗ → (M ⊗Ra N)∗ is an almostisomorphism and the assertion follows from remark 2.1.4(i).

2.5. Almost homotopical algebra. The formalism of abelian tensor categories provides a min-imal framework wherein the rudiments of deformation theorycan be developed.

2.5.1. Let(C ,⊗, U) be an abelian tensor category; we assume henceforth that⊗ is a rightexact functor. LetA be a givenC -monoid. Then, for any two-sided idealI of A, the quotientA/I in the underlying abelian categoryC has a uniqueC -monoid structure such thatA→ A/Iis a morphism of monoids.A/I is unitary if A is. If I is a two-sided ideal ofA such thatI2 = 0, then, using the right exactness of⊗ one checks thatI has a natural structure of anA/I-bimodule, unitary whenA is.

Definition 2.5.2. A C -extensionof aC -monoidB by aB-bimoduleI is a short exact sequenceof objects ofC

X : 0→ I → Cp→ B → 0(2.5.3)

such thatC is aC -monoid,p is a morphism ofC -monoids,I is a square zero two-sided idealin C and theE/I-bimodule structure onI coincides with the given bimodule structure onI.


TheC -extensions form a categoryExmonC . The morphisms are commutative diagrams withexact rows

X :

0 // I //

f

Ep //

g

B //

h

0

X ′ : 0 // I ′ // E ′p′ // B′ // 0

such thatg andh are morphisms ofC -monoids. We letExmonC (B, I) be the subcategory ofExmonC consisting of allC -extensions ofB by I, where the morphisms are all short exactsequences as above such thatf := 1I andh := 1B.

2.5.4. We have also the variant in which all theC -monoids in (2.5.3) are required to be unitary(resp. to be algebras) andI is a unitaryB-bimodule (resp. whose left and rightB-moduleactions coincide,i.e. are switched by composition with the “commutativity constraints” θB|IandθI|B, see (2.2.5)); we will callExunC (resp.ExalC ) the corresponding category.

2.5.5. For a morphismφ : C → B of C -monoids, and aC -extensionX in ExmonC (B, I),we can pullbackX via φ to obtain an exact sequenceX ∗ φ with a morphismφ∗ : X ∗ φ→ X;one checks easily that there exists a unique structure ofC -extension onX ∗ φ such thatφ∗

is a morphism ofC -extension; thenX ∗ φ is an object inExmonC (C, I). Similarly, given aB-linear morphismψ : I → J , we can push outX and obtain a well defined objectψ ∗ X inExmonC (B, J) with a morphismX → ψ ∗X of ExmonC . In particular, ifI1 andI2 are twoB-bimodules, the functorspi∗ (i = 1, 2) associated to the natural projectionspi : I1 ⊕ I2 → Iiestablish an equivalence of categories

ExmonC (B, I1 ⊕ I2) ∼→ ExmonC (B, I1)×ExmonC (B, I2)(2.5.6)

whose quasi-inverse is given by(E1, E2) 7→ (E1 ⊕ E2) ∗ δ, whereδ : B → B ⊕ B is thediagonal morphism. A similar statement holds forExal andExun. These operations can beused to induce an abelian group structure on the setExmonC (B, I) of isomorphism classes ofobjects ofExmonC (B, I) as follows. For any two objectsX, Y of ExmonC (B, I) we canformX ⊕ Y which is an object ofExmonC (B⊕B, I ⊕ I). Letα : I ⊕ I → I be the additionmorphism ofI. Then we setX+Y := α∗(X⊕Y )∗δ. One can check thatX+Y ≃ Y +X foranyX, Y and that the trivial splitC -extensionB⊕I is a neutral element for+. Moreover everyisomorphism class has an inverse−X. The functorsX 7→ X ∗ φ andX 7→ ψ ∗ X commutewith the operation thus defined, and induce group homomorphisms

∗φ : ExmonC (B, I)→ ExmonC (C, I)ψ∗ : ExmonC (B, I)→ ExmonC (B, J).

2.5.7. We will need the variantExalC (B, I) defined in the same way, starting from the cate-gory ExalC (B, I). For instance, ifA is an almost algebra (resp. a commutative ring), we canconsider the abelian tensor categoryC = A-Mod. In this case theC -extensions will be calledsimplyA-extensions, and we will writeExalA rather thanExalC . In fact the commutative uni-tary case will soon become prominent in our work, and the moregeneral setup is only requiredfor technical reasons, in the proof of proposition 2.5.13 below, which is the abstract version ofa well-known result on the lifting of idempotents over nilpotent ring extensions.

2.5.8. LetA be aC -monoid. We form the biproductA† := U ⊕ A in C . We denote byp1,p2 the associated projections fromA† ontoU and respectivelyA. Also, let i1, i2 be the naturalmonomorphisms fromU , resp.A toA†. A† is equipped with a unitary monoid structure

µ† := i2 µ (p2 ⊗ p2) + i2 ℓ−1A (p1 ⊗ p2) + i2 r−1

A (p2 ⊗ p1) + i1 u−1 (p1 ⊗ p1)

whereℓA, rA are the natural isomorphisms provided by [24, Prop. 1.3] andu : U → U ⊗ U isas in [24,§1]. In terms of the ringA†∗ ≃ U∗ ⊕ A∗ this is the multiplication(u1, b1) · (u2, b2) :=


(u1 · u2, b1 · b2 + b1 · u2 + u1 · b2). Theni2 is a morphism of monoids and one verifies thatthe “restriction of scalars” functori∗2 defines an equivalence from the categoryA†-Uni.Mod ofunitaryA†-modules to the categoryA-Mod of all A-modules; letj denote the inverse functor.A similar discussion applies to bimodules.

2.5.9. Similarly, we derive equivalences of categories

ExunC (A†, j(M))∗i2 //

ExmonC (A,M)(−)†

oo

for all A-bimodulesM .

2.5.10. Next we specialise toA := U : for a givenU-moduleM let eM := σM/U ℓM :M → M ; working out the definitions one finds that the condition that(M,σM/U ) is a modulestructure is equivalent toe2M = eM . Let U × U be the product ofU by itself in the categoryof C -monoids. There is an isomorphism of unitaryC -monoidsζ : U † → U × U given byζ := i1 p1 + i2 p1 + i2 p2. Another isomorphism isτ ζ , whereτ is the flipi1 p2 + i2 p1.Hence we get equivalences of categories

U-Modj //

U †-Uni.Mod(ζ−1)∗

//

i∗2

oo (U × U)-Uni.Mod.(τζ)∗

oo

The compositioni∗2 (ζ−1 τ ζ)∗ j defines a self-equivalence ofU-Mod which associatesto a givenU-moduleM the newU-moduleMflip whose underlying object inC isM and suchthat eMflip = 1M − eM . The same construction applies toU-bimodules and finally we getequivalences

ExmonC (U,M)∼→ ExmonC (U,Mflip) X 7→ Xflip(2.5.11)

for all U-bimodulesM . If X := (0→ M → Eπ→ U → 0) is an extension andXflip := (0 →

Mflip → Eflip → U → 0), then one verifies that there is a natural isomorphismXflip → Xof complexes inC inducing−1M on M , the identity onU and carrying the multiplicationmorphism onEflip to

−µE + ℓ−1E (π ⊗ 1E) + r−1

E (1E ⊗ π) : E ⊗ E → E.

In terms of the associated rings, this corresponds to replacing the given multiplication(x, y) 7→x · y of E∗ by the new operation(x, y) 7→ π∗(x) · y + π∗(y) · x− x · y.

Lemma 2.5.12. If M is aU-bimodule whose left and right actions coincide, then everyexten-sion ofU byM splits uniquely.

Proof. Using the idempotenteM we get aU-linear decompositionM ≃ M1 ⊕ M2 wherethe bimodule structure onM1 is given by the zero morphisms and the bimodule structure onM2 is given byℓ−1

M andr−1M . We have to prove thatExmonC (U,M) is equivalent to a one-

point category. By (2.5.6) we can assume thatM = M1 or M = M2. By (2.5.11) we haveExmonC (U,M2) ≃ ExmonC (U,Mflip

2 ) and onMflip2 the bimodule actions are the zero mor-

phisms. So it is enough to considerM = M1. In this case, ifX := (0→ M → E → U → 0) isany extension,µE : E ⊗E → E factors through a morphismU ⊗U → E and composing withu : U → U ⊗ U we get a right inverse ofE → U , which shows thatX is the split extension.Then it is easy to see thatX does not have any non-trivial automorphisms, which proves theassertion.

Proposition 2.5.13.LetX := (0→ I → Ap→ A′ → 0) be aC -extension.

(i) Let e′ ∈ A′∗ be an idempotent element whose left action on theA′-bimoduleI coincideswith its right action. Then there exists a unique idempotente ∈ A∗ such thatp∗(e) = e′.


(ii) Especially, ifA′ is unitary andI is a unitaryA′-bimodule, then every extension ofA′ by Iis unitary.

Proof. (i) : the hypothesise′2 = e′ implies thate′ : U → A′ is a morphism of (non-unitary)C -monoids. We can then replaceX byX ∗e′ and thereby assume thatA′ = U , p : A→ U andI isa (non-unitary)U-bimodule and the right and left actions onI coincide. The assertion to proveis that1U lifts to a unique idempotente ∈ A∗. However, this follows easily from lemma 2.5.12.To show (ii), we observe that, by (i), the unit1A′ of A′∗ lifts uniquely to an idempotente ∈ A∗.We have to show thate is a unit forA∗. Let us show the left unit property. Viae : U → Awe can view the extensionX as an exact sequence of leftU-modules. We can then splitX asthe direct sumX1 ⊕ X2 whereX1 is a sequence of unitaryU-modules andX2 is a sequenceof U-modules with trivial actions. But by hypothesis, onI and onA theU-module structure isunitary, soX = X1 and this is the left unit property.

2.5.14. So much for the general nonsense; we now return to almost algebras. As alreadyannounced,from here on, we assume throughout thatm is a flatV -module. As an immediateconsequence of proposition 2.5.13 we get natural equivalences of categories

ExalA1(B1,M1)×ExalA2(B2,M2)∼→ ExalA1×A2(B1 × B2,M1 ⊕M2)(2.5.15)

wheneverA1,A2 areV a-algebras,Bi is aAi-algebra andMi is a (unitary)Bi-module,i = 1, 2.

2.5.16. Notice that, ifA := Ra for someV -algebraR, S (resp.J) is aR-algebra (resp. anS-module) andX is any object ofExalR(S, J), then by applying termwise the localisation functorwe get an objectXa of ExalA(Sa, Ja). With this notation we have the following lemma.

Lemma 2.5.17.LetB be anyA-algebra andI aB-module.

(i) The natural functor

ExalA!!(B!!, I∗)→ ExalA(B, I) X 7→ Xa(2.5.18)

is an equivalence of categories.(ii) The equivalence(2.5.18)induces a group isomorphismExalA!!

(B!!, I∗)∼→ ExalA(B, I)

functorial in all arguments.

Proof. Of course (ii) is an immediate consequence of (i). To show (i), let X := (0 → I →E → B → 0) be any object ofExalA(B, I). Using corollary 2.2.22 one sees easily that thesequenceX! := (0→ I! → E!! → B!! → 0) is right exact;X! won’t be exact in general, unlessB (and thereforeE) is an exact algebra. In any case, the kernel ofI! → E!! is almost zero, sowe get an extension ofB!! by a quotient ofI! which maps toI∗. In particular we get by pushoutan extensionX!∗ by I∗, i.e. an object ofExalA!!

(B!!, I∗) and in fact the assignmentX 7→ X!∗ isa quasi-inverse for the functor (2.5.18).

Remark 2.5.19. By inspecting the proof, we see that one can replaceI∗ by I!∗ := Im(I! → I∗)in (i) and (ii) above. WhenB is exact, alsoI! will do.

In [46, II.1.2] it is shown how to associate to any ring homomorphismR → S a naturalsimplicial complex ofS-modules denotedLS/R and called the cotangent complex ofS overR.

Definition 2.5.20. Let A → B be a morphism of almostV -algebras. Thealmost cotangentcomplexof B overA is the simplicialB!!-module

LB/A := B!! ⊗(V a×B)!! L(V a×B)!!/(V a×A)!!.


2.5.21. Usually we will want to viewLB/A as an object of the derived categoryD(s.B!!-Mod)of simplicialB!!-modules. Indeed, the hyperext functors computed in this category relate thecotangent complex to a number of important invariants. Recall that, for any simplicial ringRand any twoR-modulesE,F the hyperext ofE andF is the abelian group defined as

ExtpR(E,F ) := colimn≥−p

HomD(R-Mod)(σnE, σn+pF )

(whereσ is the suspension functor of [46, I.3.2.1.4]).Let us fix an almost algebraA. First we want to establish the relationship with differentials.

Definition 2.5.22. LetB be anyA-algebra,M anyB-module.

(i) An A-derivationof B with values inM is anA-linear morphism∂ : B → M such that∂(b1 · b2) = b1 · ∂(b2)+ b2 · ∂(b1) for b1, b2 ∈ B∗. The set of allM-valuedA-derivations ofB forms aV -moduleDerA(B,M) and the almostV -moduleDerA(B,M)a has a naturalstructure ofB-module.

(ii) We reserve the notationIB/A for the idealKer(µB/A : B ⊗A B → B). Themodule ofrelative differentialsof φ is defined as the (left)B-moduleΩB/A := IB/A/I

2B/A. It is

endowed with a naturalA-derivationδ : B → ΩB/A defined byb 7→ 1⊗ b − b ⊗ 1 for allb ∈ B∗. The assignment(A→ B) 7→ ΩB/A defines a functor

Ω : V a-Alg.Morph→ V a-Alg.Mod

from the category of morphismsA → B of almostV -algebras to the category denotedV a-Alg.Mod, consisting of all pairs(B,M) whereB is an almostV -algebra andMis aB-module. The morphisms inV a-Alg.Morph are the commutative squares; themorphisms(B,M)→ (B′,M ′) in V a-Alg.Mod are all pairs(φ, f) whereφ : B → B′ isa morphism of almostV -algebras andf : B′ ⊗B M →M ′ is a morphism ofB′-modules.

2.5.23. The module of relative differentials enjoys the familiar universal properties that oneexpects. In particularΩB/A represents the functorDerA(B,−), i.e. for any (left)B-moduleMthe morphism

HomB(ΩB/A,M)→ DerA(B,M) f 7→ f δ(2.5.24)

is an isomorphism. As an exercise, the reader can supply the proof for this claim and for thefollowing standard proposition.

Proposition 2.5.25.LetB andC be twoA-algebras.

(i) There is a natural isomorphism:

ΩC⊗AB/C ≃ C ⊗A ΩB/A.

(ii) Suppose thatC is aB-algebra. Then there is a natural exact sequence ofC-modules:

C ⊗B ΩB/A → ΩC/A → ΩC/B → 0.

(iii) LetI be an ideal ofB and letC := B/I be the quotientA-algebra. Then there is a naturalexact sequence:I/I2 → C ⊗B ΩB/A → ΩC/A → 0.

(iv) The functorΩ : V a-Alg.Morph→ V a-Alg.Mod commutes with all colimits.

We supplement these generalities with one more statement which is in the same vein aslemma 2.3.21 and which will be useful in section 6.3 to calculate the Fitting ideals of modulesof differentials.

Lemma 2.5.26.Let φ : B → B′ be a morphism ofA-algebras such thatI · Ker(φ) = I ·Coker(φ) = 0 for an idealI ⊂ A. Let dφ : ΩB/A ⊗B B′ → ΩB′/A be the natural morphism.ThenI · Coker dφ = 0 andI4 ·Ker dφ = 0.


Proof. We will use the standard presentation

H(B/A) : B ⊗A B ⊗A B ∂→ B ⊗A B d→ ΩB/A → 0(2.5.27)

whered is defined by :b1 ⊗ b2 7→ b1 · db2 and∂ is the differential of the Hochschild complex :

b1 ⊗ b2 ⊗ b3 7→ b1b2 ⊗ b3 − b1 ⊗ b2b3 + b1b3 ⊗ b2.By naturality ofH(B), we deduce a morphism of complexes :B′ ⊗B H(B/A) → H(B′/A).Then, by snake lemma, we derive an exact sequence :Ker(1B′⊗Aφ)→ Ker dφ→ X, whereXis a quotient ofB′⊗ACoker(φ⊗Aφ). Using the Tor exact sequences we see thatKer(1B′⊗Aφ) isannihilated byI2. It follows easily thatI4 annihilatesKer dφ. Similarly,Coker dφ is a quotientof Coker(1B′ ⊗A φ), soI · Coker dφ = 0.

Lemma 2.5.28.For anyA-algebraB there is a natural isomorphism ofB!!-modules

(ΩB/A)! ≃ ΩB!!/A!!.

Proof. Using the adjunction (2.5.24) we are reduced to showing thatthe natural map

φM : DerA!!(B!!,M)→ DerA(B,Ma)

is a bijection for allB!!-modulesM . Given∂ : B →Ma we construct∂! : B! →Ma! →M . We

extend∂! to V ⊕ B! by setting it equal to zero onV . Then it is easy to check that the resultingmap descends toB!!, hence giving anA-derivationB!! → M . This procedure yields a rightinverseψM to φM . To show thatφM is injective, suppose that∂ : B!! → M is an almost zeroA-derivation. Composing with the naturalA-linear mapB! → B!! we obtain an almost zeromap∂′ : B! → M . But m · B! = B!, hence∂′ = 0. This implies that in fact∂ = 0, and theassertion follows.

Proposition 2.5.29.LetM be aB-module. There exists a natural isomorphism ofB!!-modules

Ext0B!!(LB/A,M!) ≃ DerA(B,M).

Proof. To ease notation, setA := V a ×A andB := V a ×B. We have natural isomorphisms :

Ext0B!!(LB/A,M!)≃ Ext0

B!!(LB!!/A!!

,M!) by [46, I.3.3.4.4]

≃DerA!!(B!!,M!) by [46, II.1.2.4.2]

≃DerA(B,M) by lemma 2.5.28.

But it is easy to see that the natural mapDerA(B,M)→ DerA(B,M) is an isomorphism.

Theorem 2.5.30.There is a natural isomorphism

ExalA(B,M)∼→ Ext1B!!

(LB/A,M!).(2.5.31)

Proof. With the notation of the proof of proposition 2.5.29 we have natural isomorphisms

Ext1B!!(LB/A,M!)≃ Ext1

B!!(LB!!/A!!

,M!) by [46, I.3.3.4.4]

≃ ExalA!!(B!!,M!) by [46, III.1.2.3]

≃ ExalA(B,M)

where the last isomorphism follows directly from lemma 2.5.17(ii) and the subsequent remark2.5.19. Finally, (2.5.15) shows thatExalA(B,M) ≃ ExalA(B,M), as required.

Moreover we have the following transitivity theorem as in [46, II.2.1.2].


Theorem 2.5.32.LetA → B → C be a sequence of morphisms of almostV -algebras. Thereexists a natural distinguished triangle ofD(s.C!!-Mod)

C!! ⊗B!!LB/A

u→ LC/Av→ LC/B → C!! ⊗B!!

σLB/A

where the morphismsu andv are obtained by functoriality ofL.

Proof. It follows directly from loc. cit.

Proposition 2.5.33.Let (Aλ → Bλ)λ∈I be a system of almostV -algebra morphisms indexedby a small filtered categoryI. Then there is a natural isomorphism inD(s.colim

λ∈IBλ!!-Mod)

colimλ∈I

LBλ/Aλ≃ Lcolim

λ∈IBλ/colim

λ∈IAλ.

Proof. Remark 2.2.28(i) gives an isomorphism :colimλ∈I

Aλ!!∼→ (colim

λ∈IAλ)!! (and likewise for

colimλ∈I

Bλ). Then the claim follows from [46, II.1.2.3.4].

Next we want to prove the almost version of the flat base changetheorem [46, II.2.2.1]. Tothis purpose we need some preparation.

Proposition 2.5.34.LetB andC be twoA-algebras and setTi := TorA!!i (B!!, C!!).

(i) If A,B,C andB⊗AC are all exact, then for everyi > 0 the natural morphismm⊗V Ti →Ti is an isomorphism.

(ii) If, furthermore,TorAi (B,C) ≃ 0 for somei > 0, then the correspondingTi vanishes.

Proof. (i): for any almostV -algebraD we letkD denote the complex ofD!!-modules[m ⊗VD!! → D!!] placed in degrees−1, 0; we have a distiguished triangle

T (D) : m⊗V D!! → D!! → kD → m⊗V D!![1].

By assumption, the natural mapkA → kB is a quasi-isomorphism andm ⊗V B!! ≃ B!. On theother hand, for alli ∈ N we have

TorA!!i (kB, C!!) ≃ TorA!!

i (kA, C!!) ≃ H−i(kA ⊗A!!C!!) = H−i(kC).

In particularTorA!!i (kB, C!!) = 0 for all i > 1. As m is flat overV , we havem ⊗V Ti ≃

TorA!!i (m ⊗V B!!, C!!). Then by the long exact Tor sequence associated toT (B)

L

⊗A!!C!! we

get the assertion for alli > 1. Next we consider the natural map of distinguished triangles

T (A)L

⊗A!!A!! → T (B)

L

⊗A!!C!!; writing down the associated morphism of long exact Tor

sequences, we obtain a diagram with exact rows :

0 // TorA!!1 (kA, A!!)

∂ //

(m⊗V A!!)⊗A!!A!!

i //

A!! ⊗A!!A!!

TorA!!1 (kB, C!!)

∂′ // (m⊗V B!!)⊗A!!C!!

i′ // B!! ⊗A!!C!!.

By the above, the leftmost vertical map is an isomorphism; moreover, the assumption givesKer i ≃ Ker(m → V ) ≃ Ker i′. Then, since∂ is injective, also∂′ must be injective, whichimplies our assertion for the remaining casei = 1. (ii): follows directly from (i).

Theorem 2.5.35.LetB,A′ be twoA-algebras. Suppose that the natural morphismBL

⊗AA′ →B′ := B ⊗A A′ is an isomorphism inD(s.A-Mod). Then the natural morphisms

B′!! ⊗B!!LB/A → LB′/A′

(B′!! ⊗B!!LB/A)⊕ (B′!! ⊗A′

!!LA′/A)→ LB′/A

are quasi-isomorphisms.


Proof. Let us remark that the functorD 7→ V a×D : A-Alg→ (V a×A)-Alg commutes withtensor products; hence the same holds for the functorD 7→ (V a ×D)!! (see remark 2.2.28(i)).Then, in view of proposition 2.5.34(ii), the theorem is reduced immediately to [46, II.2.2.1].

As an application we obtain the vanishing of the almost cotangent complex for a certain classof morphisms.

Theorem 2.5.36.LetR→ S be a morphism of almost algebras such that

TorRi (S, S) ≃ 0 ≃ TorS⊗RSi (S, S) for all i > 0

(for the naturalS ⊗R S-module structure induced byµS/R). ThenLS/R ≃ 0 in D(S!!-Mod).

Proof. SinceTorRi (S, S) = 0 for all i > 0, theorem 2.5.35 applies (withA := R andB :=A′ := S), giving the natural isomorphisms

(S ⊗R S)!! ⊗S!!LS/R ≃ LS⊗RS/S

((S ⊗R S)!! ⊗S!!LS/R)⊕ ((S ⊗R S)!! ⊗S!!

LS/R) ≃ LS⊗RS/R.(2.5.37)

SinceTorS⊗RSi (S, S) = 0, the same theorem also applies withA := S ⊗R S,B := S, A′ := S,

and we notice that in this caseB′ ≃ S; hence we have

LS/S⊗RS ≃ S!! ⊗S!!LS/S⊗RS ≃ LS/S ≃ 0.(2.5.38)

Next we apply transitivity to the sequenceR→ S ⊗R S → S, to obtain (thanks to (2.5.38))

S!! ⊗S⊗RS!!LS⊗RS/R ≃ LS/R.(2.5.39)

Applying S!! ⊗S⊗RS!!− to the second isomorphism (2.5.37) we obtain

LS/R ⊕ LS/R ≃ S!! ⊗S⊗RS!!LS⊗RS/R.(2.5.40)

Finally, composing (2.5.39) and (2.5.40) we derive

LS/R ⊕ LS/R∼→ LS/R.(2.5.41)

However, by inspection, the isomorphism (2.5.41) is the summap. ConsequentlyLS/R ≃ 0, asclaimed.

The following proposition shows thatLB/A is already determined byLaB/A.

Proposition 2.5.42.LetA → B be a morphism of exact almostV -algebras. Then the naturalmapm⊗V LB!!/A!!

→ LB!!/A!!is a quasi-isomorphism.

Proof. By transitivity we may assumeA = V a. LetP• := PV (B!!) be the standard resolution ofB!! (see [46, II.1.2.1]). EachP [n]a containsV as a direct summand, hence it is exact, so that wehave an exact sequence of simplicialV -modules0→ s.m→ s.V ⊕ (P a

• )! → (P a• )!! → 0. The

augmentation(P a• )! → (Ba

!!)! ≃ B! is a quasi-isomorphism and we deduce that(P a• )!! → B!!

is a quasi-isomorphism; hence(P a• )!! → P• is a quasi-isomorphism as well. We haveP [n] ≃

Sym(Fn) for a freeV -moduleFn and the map(P [n]a)!! → P [n] is identified withSym(m ⊗VFn) → Sym(Fn), whenceΩP [n]a!!/V

⊗P [n]a!!P [n] → ΩP [n]/V is identified withm⊗V ΩP [n]/V →

ΩP [n]/V . By [46, II.1.2.6.2] the mapL∆(P a

• )!!/V→ L∆

P•/Vis a quasi-isomorphism. In view of [46,

II.1.2.4.4] we derive thatΩ(P a• )!!/V → ΩP•/V is a quasi-isomorphism,i.e. m⊗V ΩP•/V → ΩP•/V

is a quasi-isomorphism. Sincem is flat andΩP•/V → ΩP•/V ⊗P• B!! = LB!!/V is a quasi-isomorphism, we get the desired conclusion.

Finally we have a fundamental spectral sequence as in [46, III.3.3.2].


Theorem 2.5.43.Let φ : A → B be a morphism of almost algebras such thatB ⊗A B ≃ B(e.g.such thatB is a quotient ofA). Then there is a first quadrant homology spectral sequenceof bigraded almost algebras

E2pq := Hp+q(Symq

B(LaB/A))⇒ TorAp+q(B,B).

Proof. We replaceφ by 1V a × φ and apply the functorB 7→ B!! (which commutes with tensorproducts by remark 2.2.28(i)) thereby reducing the assertion to [46, III.3.3.2].


3. ALMOST RING THEORY

With this chapter we begin in earnest the study of almost commutative algebra: in section 3.1the classes of flat, unramified and etale morphisms are defined, together with some variants. Insection 3.2 we derive the infinitesimal lifting theorems foretale algebras (theorem 3.2.18) andfor almost projective modules (theorem 3.2.28).

In section 3.4 we turn to study some cases of non-flat descent;when we specialize to usualrings, we recover known theorems (of course, standard commutative algebra is a particular caseof almost ring theory). But if the result is not new, the argument is : indeed, we believe thatour treatment, even when specialized to usual rings, improves upon the method found in theliterature.

The last section of chapter 3 calls on stage the Frobenius endomorphism of an almost algebraof positive characteristic. The main results are invariance of etale morphisms under pull-backby Frobenius maps (theorem 3.5.13) and theorem 3.5.28, which can be interpreted as a puritytheorem. Perhaps the most remarkable aspect of the latter result is how cheap it is : in positivecharacteristic, the availability of the Frobenius map allows for a quick and easy proof. Philo-sophically, this proof is not too far removed from the methoddevised by Faltings for his morerecent proof of purity in mixed characteristic.

Taken together, the above-mentioned four sections leave uswith a decent understanding ofthe morphisms “of relative dimension zero” (this expression should be taken with a grain of salt,since we do not try to define the dimension of an almost algebra). On one hand, a good hold onthe case of relative dimension zero is all that is required for the applications currently in sight(especially for the proof of the almost purity theorem, but also for the needs of our chapters6 and 7); on the other hand, having reached this stage, one cannot help wondering what liesahead, for instance whether there is a good notion of smooth morphism of almost algebras. Thefull answer to this question shall be delayed until chapter 5: there we will introduce a classof morphisms that generalize “in higher dimension” the class of weakly etale morphisms, andthat specialize to formally smooth morphisms in the “classical limit” V = m. We will presentevidence that our notion of smoothness is well behaved and worthwhile; however we shall alsosee that smoothness “in higher dimension” exhibits some extra twists that have no analogue instandard commutative algebra, and cannot be easily guessedjust by extrapolating from the caseof etale morphisms ofV a-algebras (which, after all, reproduce very faithfully thebehaviour ofthe classical notion defined in EGA).

Such extra twists are already foreshadowed by our results onthe nilpotent deformation ofalmost projective modules : we show that such deformations exist, but are not unique; however,any two such deformations are “very close” in a precise sense(proposition 3.2.30).

In (usual) algebraic geometry one can also regard projective modules of finite rankn asGLn-torsors (say for the Zariski topology); in almost ring theory this description carries through atleast for the class of almost projective modulesof finite rank(to be defined in section 4.3).From this vantage, one is naturally led to ask how much of the standard deformation theory fortorsors over arbitrary group schemes generalizes to the almost world. We take up this questionin section 3.3, focusing especially on the case ofsmooth affine almost group schemes, as a warmup to the later study of general smooth morphisms. Having committed seriously to almost groupschemes and almost torsors, it is only a short while before one grows impatient at the limitedexpressive range afforded by the purely algebraic terminology introduced thus far, which bythis point starts feeling a little like a linguistic straightjacket. That is why we find ourselvescompelled to introduce a more geometric language : for our purpose anaffine almost schemeisjust an object of the opposite of the category ofV a-algebras; similarly we define quasi-coherentmodules on affine almost schemes, as well as some suggestive notation to go with it, mimickingthe standard usage in algebraic geometry.


Once these preliminaries are in place, the theory proceeds as in [47] : the techniques arerather sophisticated, but all the hard work has already beendone in the previous sections, andwe can just adapt Illusie’s treatise without much difficulty.

3.1. Flat, unramified and etale morphisms. LetA→ B be a morphism of almostV -algebras.Using the natural “multiplication” morphism ofA-algebrasµB/A : B ⊗A B → B we can viewB as aB ⊗A B-algebra.

Definition 3.1.1. Let φ : A→ B be a morphism of almostV -algebras.

(i) We say thatφ is aflat (resp.faithfully flat, resp.almost projective) morphismif B is a flat(resp. faithfully flat, resp. almost projective)A-module.

(ii) We say thatφ is (uniformly) almost finite(resp.finite) if B is a (uniformly) almost finitelygenerated (resp. finitely generated)A-module.

(iii) We say thatφ isweakly unramified(resp.unramified) if B is a flat (resp. almost projective)B ⊗A B-module (via the morphismµB/A defined above).

(iv) φ is weaklyetale(resp.etale) if it is flat and weakly unramified (resp. unramified).

Furthermore, in analogy with definition 2.4.4, we shall write “(uniformly) almost finite projec-tive” to denote a morphismφ which is both (uniformly) almost finite and almost projective.

Lemma 3.1.2. Letφ : A→ B andψ : B → C be morphisms of almostV -algebras.

(i) LetA → A′ be any morphism ofV a-algebras; ifφ is flat (resp. almost projective, resp.faithfully flat, resp. almost finite, resp. weakly unramified, resp. unramified, resp. weaklyetale, resp.etale) then the same holds forφ⊗A 1A′ .

(ii) If both φ andψ are flat (resp. almost projective, resp. faithfully flat, resp. almost finite,resp. weakly unramified, resp. unramified, resp. weaklyetale, resp.etale), then so isψφ.

(iii) If φ is flat andψ φ is faithfully flat, thenφ is faithfully flat.(iv) If φ is weakly unramified andψ φ is flat (resp. weaklyetale), thenψ is flat (resp. weakly

etale).(v) If φ is unramified andψ φ is etale, thenψ is etale.(vi) φ is faithfully flat if and only if it is a monomorphism andB/A is a flatA-module.

(vii) If φ is almost finite and weakly unramified, thenφ is unramified.(viii) If ψ is faithfully flat andψ φ is flat (resp. weakly unramified), thenφ is flat (resp. weakly

unramified).

Proof. For (vi) use the Tor sequences. In view of proposition 2.4.18(ii), to show (vii) it sufficesto know thatB is an almost finitely presentedB ⊗A B-module; but this follows from theexistence of an epimorphism ofB ⊗A B-modules(B ⊗A B) ⊗A B → KerµB/A defined byx ⊗ b 7→ x · (1 ⊗ b − b ⊗ 1). Of the remaining assertions, only (iv) and (v) are not obvious,but the proof is just the “almost version” of a well-known argument. Let us show (v); the sameargument applies to (iv). We remark thatµB/A is an etale morphism, sinceφ is unramified.DefineΓψ := 1C ⊗B µB/A. By (i), Γψ is etale. Define alsop := (ψ φ)⊗A 1B. By (i), p is flat(resp. etale). The claim follows by remarking thatψ = Γψ p and applying (ii).

Remark 3.1.3. (i) Suppose we work in the classical limit case, that is,m := V (cp. example2.1.2(ii)). Then we caution the reader that our notion of “etale morphism” is more generalthan the usual one, as defined in [40]. The relationship between the usual notion and ours isdiscussed in the digression (3.4.44).

(ii) The naive hope that the functorA 7→ A!! might preserve flatness is crushed by the fol-lowing counterexample. Let(V,m) be as in example 2.1.2(i) and letk be the residue fieldof V . Consider the flat mapV × V → V defined as(x, y) 7→ x. We get a flat morphismV a × V a → V a in V a-Alg; applying the left adjoint to localisation yields a mapV ×k V → V


that is not flat. On the other hand, faithful flatnessis preserved. Indeed, letφ : A→ B be a mor-phism of almost algebras. Thenφ is a monomorphism if and only ifφ!! is injective; moreover,B!!/Im(A!!) ≃ B!/A!, which is flat overA!! if and only if B/A is flat overA, by proposition2.4.35.

We will find useful to study certain “almost idempotents”, asin the following proposition.

Proposition 3.1.4. A morphismφ : A → B is unramified if and only if there exists an almostelementeB/A ∈ B ⊗A B∗ such that

(i) e2B/A = eB/A;(ii) µB/A(eB/A) = 1;

(iii) x · eB/A = 0 for all x ∈ IB/A∗.Proof. Suppose thatφ is unramified. We start by showing that for everyε ∈ m there existalmost elementseε of B ⊗A B such that

e2ε = ε · eε µB/A(eε) = ε · 1 IB/A∗ · eε = 0.(3.1.5)

SinceB is an almost projectiveB ⊗A B-module, for everyε ∈ m there exists an “approximatesplitting” for the epimorphismµB/A : B ⊗A B → B, i.e. a B ⊗A B-linear morphismuε :B → B ⊗A B such thatµB/A uε = ε · 1B. Seteε := uε 1 : A → B ⊗A B. We see thatµB/A(eε) = ε · 1. To show thate2ε = ε · eε we use theB ⊗A B-linearity ofuε to compute

e2ε = eε · uε(1) = uε(µB/A(eε) · 1) = uε(µB/A(eε)) = ε · eε.Next take any almost elementx of IB/A and compute

x · eε = x · uε(1) = uε(µB/A(x) · 1) = 0.

This establishes (3.1.5). Next let us take any otherδ ∈ m and a corresponding almost elementeδ.Bothε·1−eε andδ·1−eδ are elements ofIB/A∗, hence we have(δ·1−eδ)·eε = 0 = (ε·1−eε)·eδwhich implies

δ · eε = ε · eδ for all ε, δ ∈ m.(3.1.6)

Let us define a mapeB/A : m⊗V m→ B ⊗A B∗ by the rule

ε⊗ δ 7→ δ · eε for all ε, δ ∈ m.(3.1.7)

To show that (3.1.7) does indeed determine a well defined morphism, we need to check thatδ · v · eε = δ · ev·ε andδ · eε+ε′ = δ · (eε + eε′) for all ε, ε′, δ ∈ m and allv ∈ V . However, bothidentities follow easily by a repeated application of (3.1.6). It is easy to see thateB/A defines analmost element with the required properties.

Conversely, suppose an almost elementeB/A of B ⊗A B is given with the stated properties.We defineu : B → B ⊗A B by b 7→ eB/A · (1⊗ b) (b ∈ B∗) andv := µB/A. Then (iii) says thatu is aB ⊗A B-linear morphism and (ii) shows thatv u = 1B. Hence, by lemma 2.4.15,φ isunramified.

Remark 3.1.8. The proof of proposition 3.1.4 shows that ifI is an ideal in an almostV -algebraA, thenA/I is almost projective overA if and only if I is generated by an idempotent ofA∗.This idempotent is uniquely determined.

Corollary 3.1.9. Under the hypotheses and notation of the proposition, the ideal IB/A has anatural structure ofB ⊗A B-algebra, with unit morphism given by1 := 1B⊗AB/A

− eB/A andwhose multiplication is the restriction ofµB⊗AB/A to IB/A. Moreover the natural morphism

B ⊗A B → IB/A ⊕B x 7→ (x · 1⊕ µB/A(x))

is an isomorphism ofB ⊗A B-algebras.


Proof. Left to the reader as an exercise.

3.2. Nilpotent deformations of almost algebras and modules.Throughout the following,the terminology “epimorphism ofV a-algebras” will refer to a morphism ofV a-algebras thatinduces an epimorphism on the underlyingV a-modules.

Lemma 3.2.1. LetA → B be an epimorphism of almostV -algebras with kernelI. LetU betheA-extension0 → I/I2 → A/I2 → B → 0. Then the assignmentf 7→ f ∗ U defines anatural isomorphism

HomB(I/I2,M)∼→ ExalA(B,M).(3.2.2)

Proof. LetX := (0 → M → Ep→ B → 0) be anyA-extension ofB byM . The composition

g : A→ Ep→ B of the structural morphism forE followed byp coincides with the projection

A→ B. Thereforeg(I) ⊂M andg(I2) = 0. Henceg factors throughA/I2; the restriction ofgto I/I2 defines a morphismf ∈ HomB(I/I2,M) and a morphism ofA-extensionsf ∗U → X.In this way we obtain an inverse for (3.2.2).

3.2.3. Now consider any morphism ofA-extensions

B :

f

0 // I //

u

B //

f

B0//

f0

0

C : 0 // J // C // C0// 0.

(3.2.4)

The morphismu induces by adjunction a morphism ofC0-modules

C0 ⊗B0 I → J(3.2.5)

whose image is the idealI · C, so that the square diagram of almost algebras defined byf iscofibred (i.e.C0 ≃ C ⊗B B0) if and only if (3.2.5) is an epimorphism.

Lemma 3.2.6. Let f : B → C be a morphism ofA-extensions as above, such that the corre-sponding square diagram of almost algebras is cofibred. Thenthe morphismf : B → C is flatif and only iff0 : B0 → C0 is flat and(3.2.5)is an isomorphism.

Proof. It follows directly from the (almost version of the) local flatness criterion (see [54, Th.22.3]).

We are now ready to put together all the work done so far and begin the study of deformationsof almost algebras.

3.2.7. The morphismu : I → J is an element inHomB0(I, J); by lemma 3.2.1 the latter groupis naturally isomorphic toExalB(B0, J). On the other hand, in view of proposition 2.5.42 andlemma 2.4.14(i) we have natural isomorphisms:

ExtiC0!!(LC0/B0

,M!) ≃ ExtiC0(La

C0/B0,M)(3.2.8)

for every i ∈ N and everyC0-moduleM . By applying transitivity (theorem 2.5.32) to the

sequence of morphismsB → B0f0→ C0 we deduce an exact sequence of abelian groups

ExalB0(C0, J)→ ExalB(C0, J)→ HomB0(I, J)∂→ Ext2C0

(LaC0/B0

, J).

Hence we can form the elementω(B, f0, u) := ∂(u) ∈ Ext2C0(La

C0/B0, J). The proof of the next

result goes exactly as in [46, III.2.1.2.3].

Proposition 3.2.9. Let theA-extensionB, theB0-linear morphismu : I → J and the mor-phism ofA-algebrasf0 : B0 → C0 be given as above.


(i) There exists anA-extensionC and a morphismf : B → C completing diagram(3.2.4)ifand only ifω(B, f0, u) = 0. (i.e. ω(B, f0, u) is the obstruction to the lifting ofB overf0.)

(ii) Assume that the obstructionω(B, f0, u) vanishes. Then the set of isomorphism classes ofA-extensionsC as in(i) forms a torsor under the group:

ExalB0(C0, J) ≃ Ext1C0(La

C0/B0, J).

(iii) The group of automorphisms of anA-extensionC as in (i) is naturally isomorphic toDerB0(C0, J) (≃ Ext0C0

(LaC0/B0

, J)).

3.2.10. The obstructionω(B, f0, u) depends functorially onu. More exactly, if we denote by

ω(B, f0) ∈ Ext2C0(La

C0/B0, C0 ⊗B0 I)

the obstruction corresponding to the natural morphismI → C0 ⊗B0 I, then for any othermorphismu : I → J we have

ω(B, f0, u) = v! ω(B, f0)

wherev is the morphism (3.2.5). Taking lemma 3.2.6 into account we deduce

Corollary 3.2.11. Suppose thatB0 → C0 is flat. Then

(i) The classω(B, f0) is the obstruction to the existence of a flat deformation ofC0 overB,i.e. of aB-extensionC as in (3.2.4)such thatC is flat overB andC ⊗B B0 → C0 is anisomorphism.

(ii) If the obstructionω(B, f0) vanishes, then the set of isomorphism classes of flat deforma-tions ofC0 overB forms a torsor under the groupExalB0(C0, C0 ⊗B0 I).

(iii) The group of automorphisms of a given flat deformation ofC0 overB is naturally isomor-phic toDerB0(C0, C0 ⊗B0 I).

3.2.12. Now, suppose we are given twoA-extensionsC1, C2 with morphisms ofA-extensions

B :

f i

0 // I //

ui

B //

f i

B0//

f i0

0

Ci : 0 // J i // Ci // Ci0

// 0

and morphismsv : J1 → J2, g0 : C10 → C2

0 such that

u2 = v u1 and f 20 = g0 f 1

0 .(3.2.13)

We consider the problem of finding a morphism ofA-extensions

C1 :

g

0 // J1 //

v

C1 //

g

C10

//

g0

0

C2 : 0 // J2 // C2 // C20

// 0

(3.2.14)

such thatf 2 = g f 1. Let us denote bye(Ci) ∈ Ext1Ci0(La

Ci0/B

, J i) the classes defined by the

B-extensionsC1, C2 via the isomorphisms (2.5.31) and (3.2.8), and by

v∗ : Ext1C10(La

C10/B

, J1)→ Ext1C10(La

C10/B

, J2)

∗g0 : Ext1C20(La

C20/B

, J2)→ Ext1C20(C2

0 ⊗C10

LaC1

0/B, J2)

the canonical morphisms defined byv andg0. Using the natural isomorphism

Ext1C10(La

C10/B

, J2) ≃ Ext1C20(C2

0 ⊗C10

LC10/B

, J2)


we can identify the target of bothv∗ and∗g with Ext1C10(La

C10/B

, J2). It is clear that the problem

admits a solution if and only if theA-extensionsv ∗ C1 and C2 ∗ g0 coincide, i.e. if andonly if v ∗ e(C1) − e(C2) ∗ g0 = 0. By applying transitivity to the sequence of morphismsB → B0 → C1

0 we obtain an exact sequence

Ext1C10(La

C10/B0

, J2) → Ext1C10(La

C10/B

, J2)→ HomC10(C1

0 ⊗B0 I, J2).

It follows from (3.2.13) that the image ofv∗e(C1)−e(C2)∗g0 in the groupHomC10(C1

0⊗B0I, J2)

vanishes, therefore

v ∗ e(C1)− e(C2) ∗ g0 ∈ Ext1C10(La

C10/B0

, J2).(3.2.15)

In conclusion, we derive the following result as in [46, III.2.2.2].

Proposition 3.2.16.With the above notations, the class(3.2.15)is the obstruction to the exis-tence of a morphism ofA-extensionsg : C1 → C2 as in(3.2.14)such thatf 2 = gf 1. When theobstruction vanishes, the set of such morphisms forms a torsor under the groupDerB0(C

10 , J

2)(the latter being identified withExt0C2

0(C2

0 ⊗C10

LaC1

0/B0, J2)).

3.2.17. For a given almostV -algebraA, we define the categoryA-w.Et (resp.A-Et) as thefull subcategory ofA-Alg consisting of all weakly etale (resp. etale)A-algebras. Notice that,by lemma 3.1.2(iv) all morphisms inA-w.Et are weakly etale.

Theorem 3.2.18.LetA be aV a-algebra.

(i) LetB be a weaklyetaleA-algebra,C anyA-algebra andI ⊂ C a nilpotent ideal. Thenthe natural morphism

HomA-Alg(B,C)→ HomA-Alg(B,C/I)

is bijective.(ii) Let I ⊂ A a nilpotent ideal andA′ := A/I. Then the natural functor

A-w.Et→ A′-w.Et (φ : A→ B) 7→ (1A′ ⊗A φ : A′ → A′ ⊗A B)

is an equivalence of categories.(iii) The equivalence of(ii) restricts to an equivalenceA-Et→ A′-Et.

Proof. By induction we can assumeI2 = 0. Then (i) follows directly from proposition 3.2.16and theorem 2.5.36. We show (ii) : by corollary 3.2.11 (and again theorem 2.5.36) a givenweakly etale morphismφ′ : A′ → B′ can be lifted to auniqueflat morphismφ : A → B.We need to prove thatφ is weakly etale,i.e. thatB is B ⊗A B-flat. However, it is clear thatµB′/A′ : B′ ⊗A′ B′ → B′ is weakly etale, hence it has a flat liftingµ : B ⊗A B → C. Thenthe compositionA → B ⊗A B → C is flat and it is a lifting ofφ′. We deduce that there is anisomorphism ofA-algebrasα : B → C lifting 1B′ and moreover the morphismsb 7→ µ(b⊗ 1)andb 7→ µ(1 ⊗ b) coincide withα. Claim (ii) follows. To show (iii), suppose thatA′ → B′

is etale and letIB′/A′ denote as usual the kernel ofµB′/A′ . By corollary 3.1.9 there is a naturalmorphism of almost algebrasB′ ⊗A′ B′ → IB′/A′ which is clearly etale. HenceIB′/A′ lifts toa weakly etaleB ⊗A B-algebraC, and the isomorphismB′ ⊗A′ B′ ≃ IB′/A′ ⊕ B′ lifts to anisomorphismB ⊗A B ≃ C ⊕ B of B ⊗A B-algebras. It follows thatB is an almost projectiveB ⊗A B-module,i.e.A→ B is etale, as claimed.

We conclude with some results on deformations of almost modules. These can be establishedindependently of the theory of the cotangent complex, alongthe lines of [46, IV.3.1.12].


3.2.19. We begin by recalling some notation fromloc. cit. LetR be a ring andJ ⊂ R an idealwith J2 = 0. SetR′ := R/J ; an extension ofR-modulesM := (0 → K → M

p→ M ′ → 0)whereK andM ′ are killed byJ , defines a natural morphism ofR′-modulesu(M) : J⊗R′M ′ →K such thatu(M)(x ⊗m′) = xm for x ∈ J , m ∈ M andp(m) = m′. By the local flatnesscriterion ([54, Th. 22.3])M is flat overR if and only if M ′ is flat overR′ andu(M) is anisomorphism. One can then show the following.

Proposition 3.2.20. (cp. [46, IV.3.1.5]) With the notation of(3.2.19)we have:(i) GivenR′-modulesM ′ andK and a morphismu′ : J ⊗R′ M ′ → K there exists an ob-

structionω(R, u′) ∈ Ext2R′(M ′, K) whose vanishing is necessary and sufficient for the

existence of an extension ofR-modulesM ofM ′ byK such thatu(M) = u′.(ii) Whenω(R, u′) = 0, the set of isomorphism classes of such extensionsM forms a torsor

underExt1R′(M ′, K); the group of automorphisms of such an extension is isomorphic to

HomR′(M ′, K).

Lemma 3.2.21.Let R be a ring,M a finitely generatedR-module such thatAnnRM is anilpotent ideal. ThenR admits a filtration0 = Jm ⊂ ... ⊂ J1 ⊂ J0 = R such that eachJi/Ji+1

is a quotient of a direct sum of copies ofM .

Proof. This is [41, 1.1.5]; for the convenience of the reader we reproduce the proof. LetI :=F0(M) ⊂ R; if M is generated byk elements, we have(AnnRM)k ⊂ I ⊂ AnnRM , especiallyI is nilpotent.

Claim 3.2.22. It suffices to show thatR := R/I admits a filtration as above.

Proof of the claim:Indeed, if0 = J ′0 ⊂ J ′1 ⊂ ... ⊂ J ′n−1 ⊂ J ′n = R/I is such a filtration, wededuce filtrations0 ⊂ J ′1(I

t/I t+1) ⊂ ... ⊂ J ′n−1(It/I t+1) ⊂ (I t/I t+1) for everyt ∈ N; the

graded module associated to this filtration is a direct sum ofquotients of modules of the form(J ′k/J

′k+1)⊗R (I t/I t+1), so the claim follows easily.

Let F0(M/IM) be the0-th Fitting ideal of theR-moduleM/IM ; we haveF0(M/IM) =F0(M) · R = 0, and(AnnRM/IM)k ⊂ F0(M/IM), i.e. AnnRM/IM is a nilpotent ideal.Thus, thanks to claim 3.2.22 we can replaceR andM by R andM/IM , and thereby reduceto the case whereF0(M) = 0. We claim that the filtration0 = F0(M) ⊂ F1(M) ⊂ ... ⊂Fk+1(M) = R will do in this case. Indeed, letL1

φ→ L0 → M → 0 be a presentation ofM ,whereL0 andL1 are freeR-modules and the rank ofL0 equalsk; by definitionFj(M) is theimage of the mapΛk−j

R L1 ⊗R ΛjRL0 → Λk

RL0 ≃ R defined by the rulex⊗ y 7→ Λk−jR φ(x) ∧ y.

We deduce easily that the induced surjectionΛk−jR L1 ⊗R Λj

RL0 → Fj(M)/Fj−1(M) factorsthrough the moduleΛk−j

R L1 ⊗R ΛjRM ; however the latter is a quotient of sums ofM , at least

whenj ≥ 1, so the claim follows.

Lemma 3.2.23.Let A → B be a finite morphism of almost algebras with nilpotent kernel.There existsm ≥ 0 such that the following holds. For everyA-linear morphismφ : M → N ,setφB := φ⊗A 1B : M ⊗A B → N ⊗A B; then :

(i) AnnA(Coker φB)m ⊂ AnnA(Cokerφ).(ii) (AnnV (KerφB) · AnnV (TorA1 (B,N)) · AnnV (Coker φ))m ⊂ AnnA(Kerφ).

If B = A/I for some nilpotent idealI, andIn = 0, then we can takem = n in (i) and(ii) .

Proof. Under the assumptions, we can find a finitely generatedA∗-moduleQ such thatm ·B∗ ⊂Q ⊂ B∗. By lemma 3.2.21 there exists a finite filtration0 = Jm ⊂ ... ⊂ J1 ⊂ J0 = A∗ suchthat eachJi/Ji+1 is a quotient of a direct sum of copies ofQ. This implies that

AnnA(M ⊗A B)m ⊂ AnnA(M)(3.2.24)


for everyA-moduleM ; (i) follows easily. Notice that ifB = A/I and In = 0, then we

can takem = n in (3.2.24). For (ii) letC• := Coneφ. We estimateH := H−1(C•L

⊗A B)in two ways. By the first spectral sequence of hyperhomology we have an exact sequenceTorA1 (N,B)→ H → KerφB. By the second spectral sequence for hyperhomology we have anexact sequenceTorA2 (Coker φ,B) → Ker(φ) ⊗A B → H. HenceKer(φ)⊗A B is annihilatedby the product of the three annihilators in (ii) and the result follows by applying (3.2.24) withM := Kerφ.

Lemma 3.2.25.Keep the assumptions of lemma3.2.23and letM be anA-module.

(i) If A→ B is an epimorphism,M is flat andMB := B ⊗AM is almost projective overB,thenM is almost projective overA.

(ii) If MB is an almost finitely generatedB-module thenM is an almost finitely generatedA-module.

(iii) If TorA1 (B,M) = 0 andMB is almost finitely presented overB, thenM is almost finitelypresented overA.

Proof. (i) : we have to show thatExt1A(M,N) is almost zero for everyA-moduleN . Let

I := Ker(A → B); by assumptionI is nilpotent, so by the usual devissage we may assumethat IN = 0. If χ ∈ Ext1

A(M,N) is represented by an extension0 → N → Q → M → 0then after tensoring byB and using the flatness ofM we get an exact sequence ofB-modules0 → N → B ⊗A Q → MB → 0. Thusχ comes from an element ofExt1

B(MB, N) which isalmost zero by assumption.

(ii) : for a given finitely generated subidealm0 ⊂ m, let N ⊂ MB be a finitely generatedB-submodule such thatm0MB ⊂ N . Since the induced mapM∗⊗A∗B∗ → (MB)∗ is almost sur-jective, we can find a finitely generatedA-submoduleN0 ⊂M such thatm0N ⊂ Im((N0)B →MB); by lemma 3.2.23(i) it follows thatm2n

0 (M/N0) = 0 for somen ≥ 0 depending only onB, whence the claim.

(iii) : Let m0 be as above. By (ii),M is almost finitely generated overA, so we can choosea morphismφ : Ar → M such thatm0 · Coker φ = 0. ConsiderφB := φ ⊗A 1B : Br →MB. By claim 2.3.12, there is a finitely generated submoduleN of KerφB containingm2

0 ·KerφB. Notice thatKer(φ)⊗AB maps ontoKer(Br → Im(φ)⊗AB) andKer(Im(φ)⊗AB →MB) ≃ TorA1 (B,Cokerφ) is annihilated bym0. Hencem0 ·KerφB is contained in the image ofKerφ and therefore we can lift a finite generating setx′1, ..., x′n for m2

0N to almost elementsx1, ..., xn of Kerφ. If we quotientAr by the span of thesexi, we get a finitely presentedA-moduleF with a morphismφ : F → M such thatKer(φ ⊗A B) is annihilated bym4

0 andCoker φ is annihilated bym0. By lemma 3.2.23(ii) we derivem5m

0 ·Ker φ = 0 for somem ≥ 0.Sincem0 is arbitrary, this proves the result.

Remark 3.2.26. (i) Inspecting the proof, one sees that parts (ii) and (iii) of lemma 3.2.25 holdwhenever (3.2.24) holds. For instance, ifA → B is any faithfully flat morphism, then (3.2.24)holds withm := 1.

(ii) Consequently, ifA → B is faithfully flat andM is anA-module such thatMB is flat(resp. almost finitely generated, resp. almost finitely presented) overB, thenM is flat (resp.almost finitely generated, resp. almost finitely presented)overA.

(iii) On the other hand, we do not know whether a general faithfully flat morphismA → Bdescends almost projectivity. However, using (ii) and proposition 2.4.18 we see that if theB-moduleMB is almost finitely generated projective, thenM has the same property.

(iv) Furthermore, ifB is faithfully flat and almost finitely presented as anA-module, thenA → B does descend almost projectivity, as can be easily deduced from lemma 2.4.31(i) andproposition 2.4.18(ii).


3.2.27. We denote byA-Etafp the full subcategory ofA-Et consisting of all etaleA-algebrasB such thatB is almost finitely presented as anA-module.

Theorem 3.2.28.Let I ⊂ A be a nilpotent ideal, and setA′ := A/I.(i) Suppose thatm is a (flat)V -module of homological dimension≤ 1. LetP ′ be an almost

projectiveA′-module.(a) There is an almost projectiveA-moduleP withA′ ⊗A P ≃ P ′.(b) If P ′ is almost finitely presented, thenP is almost finitely presented.

(ii) The equivalence of theorem3.2.18(ii)restricts to an equivalenceA-Etafp → A′-Etafp.

Proof. (i.a): as usual we reduce toI2 = 0. Then proposition 3.2.20(i) applies withR := A∗,J := I∗, R′ := A∗/I∗,M ′ := P ′! , K := I∗ ⊗R′ P ′! andu′ := 1K . We obtain a classω(A∗, u

′) ∈Ext2

R′(P ′! , I∗ ⊗R′ P ′! ) which gives the obstruction to the existence of a flatA∗-moduleF liftingP ′! . SinceP ′! is almost projective, lemma 2.4.14(ii) says thatExt2

R′(P ′! , I∗ ⊗R′ P ′! ) = 0, sosuchF can always be found, and then theA-moduleP = F a is a flat lifting ofP ′; by lemma3.2.25(i) we see thatP is almost projective. Now (i.b) follows from (i.a), lemma 3.2.25(ii) andproposition 2.4.18(i).

(ii): in view of theorem 3.2.18(iii), we only have to show that an etaleA-algebraB is almostfinitely presented as anA-module wheneverB ⊗A A′ is almost finitely presented as anA′-module. However, the assertion is a direct consequence of lemma 3.2.25(iii).

Remark 3.2.29. (i) According to proposition 2.1.12(ii.b), theorem 3.2.28(i) applies especiallywhenm is countably generated as aV -module.

(ii) For P andP ′ as in theorem 3.2.28(i.b) letσP : P → P ′ be the projection. It is naturalto ask whether the pair(P, σP ) is uniquely determined up to isomorphism,i.e. whether, forany other pair(Q, σQ : Q → P ′) for which theorem 3.2.28(i.b) holds, there exists anA-linear isomorphismφ : P → Q such thatσQ φ = σP . The answer is negative in general.Consider the caseP ′ := A′. TakeP := Q := A and letσP be the natural projection, whileσQ := (u′ · 1A′) σP , whereu′ is a unit inA′∗. Then the uniqueness question amounts towhether every unit inA′∗ lifts to a unit ofA∗. The following counterexample is related to thefact that the completion of the algebraic closureQp of Qp is not maximally complete. LetV := Zp, the integral closure ofZp in Qp. ThenV is a non-discrete valuation ring of rankone, and we take form the maximal ideal ofV , A := (V/p2V )a andA′ := A/pA. Choosea compatible system of roots ofp. An almost element ofA′ is just aV -linear morphismφ :colimn>0

p1/n!V → V/pV . Such aφ can be represented (in a non-unique way) by an infinite series

of the form∑∞

n=1 anp1−1/n! (an ∈ V ). The meaning of this expression is as follows. For

everym > 0, scalar multiplication by the element∑m

n=1 anp1−1/n! ∈ V defines a morphism

φm : p1/m!V → V/pV . Form′ > m, let jm,m′ : p1/m!V → p1/m′!V be the imbedding. Then wehaveφm′ jm,m′ = φm, so that we can defineφ := colim

m>0φm. Similarly, every almost element of

A can be represented by an expression of the forma0 +∑∞

n=1 anp2−1/n!. Now, if σ : A→ A′ is

the natural projection, the induced mapσ∗ : A∗ → A′∗ is given by:a0 +∑∞

n=1 anp2−1/n! 7→ a0.

In particular, its image is the subringV/p ⊂ (V/p)∗ = A′∗. For instance, the unit∑∞

n=1 p1−1/n!

of A′∗ does not lie in the image of this map.

In the light of remark 3.2.29, the best one can achieve in general is the following result.

Proposition 3.2.30.Assume (A) (see(2.1.6)) and keep the notation of theorem3.2.28(i). Sup-pose that(Q, σQ : Q→ P ′) and(P, σP : P → P ′) are two pairs as in remark3.2.29(ii). Thenfor everyε ∈ m there existA-linear morphismstε : P → Q andsε : Q→ P such that

PQ(ε)σQ tε = ε · σP σP sε = ε · σQsε tε = ε2 · 1P tε sε = ε2 · 1Q.


Proof. Since bothQ andP are almost projective andσP , σQ are epimorphisms, there existmorphismstε : P → Q andsε : Q → P such thatσQ tε = ε · σP andσP sε = ε · σQ.Then we haveσP (sε tε − ε2 · 1P ) = 0 andσQ (tε sε − ε2 1Q) = 0, i.e. the morphismuε := ε2 ·1P −sε tε (resp.vε := ε2 ·1Q−tε sε) has image contained in the almost submoduleIP (resp.IQ). SinceIm = 0 this impliesumε = 0 andvmε = 0. Hence

ε2m · 1P = (ε21P )m − umε = (

m−1∑

a=0

ε2aum−1−aε ) sε tε.

Defines(2m−1)ε := (∑m−1

a=0 ε2aum−1−a

ε ) sε. Notice thats(2m−1)ε = sε (∑m−1

a=0 ε2avm−1−a

ε ).This implies the equalitiess(2m−1)ε tε = ε2m · 1P andtε s(2m−1)ε = ε2m · 1Q. Then the pair(s(2m−1)ε, ε

2(m−1) · tε) satisfiesPQ(ε2m−1). Under (A), every element ofm is a multiple of anelement of the formε2m−1, therefore the claim follows for arbitraryε ∈ m.

3.3. Nilpotent deformations of torsors. For the considerations that follow it will be conve-nient to extend yet further our basic setup. Namely, supposethatT is any topos; we can defineabasic setup relative toT as a pair(V,m) consisting of aT -ring V and an idealm ⊂ V satisfy-ing the usual assumptions (2.1.1). Then most of the discussion of chapter 2 extendsverbatimtothis relative setting. Accordingly, we generally continueto use the same notation as inloc.cit.;however, if it is desirable for clarity’s sake, we may sometimes stress the dependance onT bymentioning it explicitly. For instance, an almostT -ring is an object of the categoryV a-Alg ofassociative, commutative and unitary monoids of the abelian tensor category ofV a-modules,and sometimes the same category is denoted(T, V,m)a-Alg.

3.3.1. LetT be a topos and(V,m) a basic setup relative toT . For every objectU of T , we canconsider the restriction(V/U ,m/U ) of (V,m) to T/U , which is a basic setup relative to the lattertopos. As usual, for every objectX → U of T/U , one definesV/U(X) := HomT/U

(X, V ) andsimilarly for m/U (X). The restriction functorV -Mod → V/U -Mod clearly preserves almostisomorphisms, whence a restriction functor

(T, V,m)a-Mod→ (T/U , V/U ,m/U)a-Mod M 7→M/U .

Similar functors exist for the categories ofV a-algebras, and more generally, forA-algebras,whereA is any(T, V,m)a-algebra.

3.3.2. For every almostT -moduleM , we define a functorM∗ : T o → Z-Mod by the rule:U 7→ HomV a

/U(V a

/U ,M/U). LetN be aV -module; using the natural isomorphism

HomV/U(m/U , N/U)

∼→ Na∗ (U)

(cp. (2.2.4)) one sees thatM∗ is a sheaf for the canonical topology ofT , hence it representableby an abelian group object ofT , which we denote by the same name. It is then easy to checkthatM∗ is aV -module, and that the functorM 7→ M∗ is right adjoint to the localization func-tor V -Mod → V a-Mod. We can then generalize to the case of the localization functorsA-Mod → Aa-Mod andA-Alg → Aa-Alg, for an arbitraryV -algebraA. Likewise, the leftadjoint functors to localizationM 7→ M! andB 7→ B!! are obtained in the same way as in theearlier treatment of the one-point topos.

3.3.3. LetT and(V,m) be as in (3.3.1) and letR be anyV -algebra. AnaffineR-schemeisan object of the categoryR-Algo. An affine almostR-scheme(or anaffineRa-scheme) is anobject of the categoryRa-Algo. If X is an affineRa-scheme, then we may writeOX in placeof theRa-algebraXo, and anXo-module will also be called aquasi-coherentOX-module. Amorphismφ : X → Y of affine almost schemes is the same as the morphism of almost algebras


φ♯ := φo : OY → OX ; moreover,φ induces pullback (i.e. tensor product−⊗OYOX) and direct

image functors (i.e. restriction of scalars), which we will denote

φ∗ : OY -Mod→ OX-Mod and φ∗ : OX-Mod→ OY -Mod(3.3.4)

respectively. In the same vein, ifA is anyRa-algebra, we may writeSpecA to denote the objectAo represented byA in the opposite categoryRa-Algo. We say thatX is flat overRa if OX isa flatRa-algebra. Clearly the category of affineRa-schemes admits arbitrary products. Hence,we can define anaffineRa-group schemeas a group object in the category of affineRa-schemes(and likewise for the notion of affineR-group scheme).

3.3.5. The functorsB 7→ B∗ andB 7→ B!! induce functors on almost schemes, which wedenote in the same way. Notice that the functorX 7→ X∗ (resp.X 7→ X!!) from affineRa-schemes toR-schemes (resp. toRa

!!-schemes) commutes with all colimits (resp. with all limits).Especially, ifG is an affineRa-group scheme, thenG!! is an affineRa

!!-group scheme.

3.3.6. Throughout the rest of this section we fix aV a-algebraA and letS := SpecA.Let X an affineA-scheme,G an affineA-group scheme. Aright action of G on X is a

morphism ofS-schemesρ : G ×S X → X fulfilling the usual conditions. To the datum(X,G, ρ) one assigns itsnerveG•X which is a simplicial affineS-scheme whose component indegreen isGn

X := Gn ×S X, and whose face and degeneracy morphisms

∂i : Gn+1X → Gn

X σj : GnX → Gn+1

X i = 0, ..., n+ 1; j = 0, ..., n

are defined for everyn ∈ N as in [47, Ch.VI,§2.5].

3.3.7. Let(X,G, ρ) be as in (3.3.6), and letM be a quasi-coherentOX-module. AG-actiononM is a morphism of quasi-coherentOG1

X-modules

β : ∂∗0M → ∂∗1M(3.3.8)

such that the following diagram commutes:

∂∗0∂∗0M

∂∗0β // ∂∗0∂∗1M ∂∗2∂

∗0M

∂∗2β

∂∗1∂

∗0M

∂∗1β // ∂∗1∂∗1M ∂∗2∂

∗1M

(3.3.9)

and such that

σ∗0β = 1M .(3.3.10)

One also says that(M,β) is aG-equivariantOX-module. One defines in the obvious waythe morphisms ofG-equivariantOX-modules, and we denote byOX-ModG the category ofG-equivariantOX-modules.

Lemma 3.3.11.For everyG-equivariantOX-moduleM , the morphism(3.3.8) is an isomor-phism.

Proof. We let τ : G1X → G2

X be the morphism given onT -points by the rule:(g, x) 7→(g, g−1, x). Working out the identifications, one checks easily thatτ ∗(3.3.9) boils down tothe diagram

∂∗0Mβ // ∂∗1M

∂∗1M.τ∗∂∗0β

ccGGGGGGGGG ∂∗1σ∗0β

;;wwwwwwwww


However,∂∗1σ∗0β = 1∂∗1M , in view of (3.3.10), henceβ is an epimorphism andτ ∗∂∗0β is an

monomorphism. Since∂0 τ is an isomorphism, we deduce thatβ is already a monomorphism,whence the claim.

Lemma 3.3.12. If G is a flat affineS-group scheme acting on an affineS-schemeX, then thecategory ofG-equivariant quasi-coherentOX-modules is abelian.

Proof. We need to verify that every (G-equivariant) morphismφ : M → N of G-equivariantOX-modules admits a kernel and a cokernel (and then it will be clear that the kernel ofN →Coker φ equals the cokernel ofKerφ → M , since the same holds forOX-modules). Theobvious candidates are the kernelK and cokernelC taken in the category ofOX-modules, andone has only to show that theG-actions ofM andN induceG-actions onK andC. Thisis always the case forC (even whenG is not flat). To deal withK, one remarks that bothmorphisms∂0, ∂1 : G1

X → X are flat; indeed, this is clear for∂1. Then the same holds for∂0 := ∂1 ω, whereω := (ρ, 1X) : G1

X → G1X is the isomorphism deduced from the action

ρ : G1X → X. It follows thatKer (∂∗i φ) ≃ ∂∗i (Kerφ) for i = 0, 1, whence the claim.

3.3.13. Letφ : X → Y be a morphism of affineS-schemes. Asquare zero deformationof XoverY is a datum of the form(j : X → X ′,I , β), consisting of:

(a) a morphism ofY -schemesj : X → X ′ such that the induced morphism ofOY -algebrasj♯ : OX′ → OX is an epimorphism andJ := Ker j♯ is a square zero ideal, and

(b) a quasi-coherentOX-moduleI with an isomorphism ofOX-modulesβ : j∗J∼→ I .

The square zero deformations form a categoryExalY (X,I ), with morphisms defined in theobvious way. As in the case of the one-point topos, we can compute the isomorphism classesof square zero deformations ofX in terms of an appropriate cotangent complex. And, just as inthe earlier treatment, we have to make sure that we are dealing with exact algebras, hence theright definition of the cotangent complex ofφ is:

LX/Y := ι∗L(X∐SpecV a)!!/(Y ∐SpecV a)!!

whereι : X!! → (X ∐ SpecV a)!! is the natural morphism of schemes. This is an object ofD(OX!!

-Mod).

3.3.14. Next, letG be a flat affineS-group scheme acting onX andY , in such a way thatφ isG-equivariant; it follows from lemma 3.3.12 thatJ has a naturalG-action. AG-equivariantsquare zero deformationsof X overY , is a datum(j : X → X ′,I , β) as above, such that,additionally,X ′ andI are endowed with aG-action and bothj andβ areG-equivariant. Letus denote byExalY (X/G,I ) the category of suchG-equivariant deformations. We aim toclassify the isomorphism classes of objects ofExalY (X/G,I ), and more generally, studytheG-equivariant deformation theory ofX by means of an appropriate cotangent complexcohomology. This is achieved by the following device.

3.3.15. LetI be a (small) category; recall ([47, Ch.VI,§5.1]) that afibred topos overI is apseudo-functorX of I in the2-category of of topoi,i.e. the datum consisting of:

(a) for every objecti of I, a toposXi

(b) for every arrowf : i → j in I, a morphism of topoiXf : Xi → Xj (sometimes denotedf )

(c) for every compositionif→ j

g→ k, a transitivity isomorphismXf,g : Xg Xf∼→ Xgf ,

submitted to certain compatibility conditions (in practice, we will omit from the notationthe transitivity isomorphisms).

Given a fibred toposX overI, one denotes byTop(X) the following category, which is easilyseen to be a topos ([47, Ch.VI,§5.2]). An object ofE of Top(X) is the datum of


(a) for everyi ∈ I, an objectEi of Xi

(b) for every arrowf : i→ j, a morphismEf : f ∗Ej → Ei such that, for every composition

if→ j

g→ k, one hasEgf = Ef f ∗Ej , provided one identifies(gf)∗Ek with f ∗g∗Ek viaXf,g.

As an example we have the toposs.T whose objects are the cosimplicial objects ofT ; indeedthis is the toposTop(F ) associated to a fibred toposF , whose indexing category is the category∆o defined in (2.2) : to every object[n] of ∆o one assignsF[n] := T and for every morphismf : [n]→ [m] of ∆o, one takesFf := 1T , the identity functor of the toposT .

There is an obvious functorT → s.T that assigns to any objectZ of T the constant cosim-plicial T -object s.Z associated toZ. Especially, if (T, V,m) is a basic setup forT , then(s.T, s.V, s.m) is a basic setup relative tos.T .

3.3.16. Suppose now thatM• is a cosimplicialV a-module. By applying termwise the functorN 7→ N∗ we deduce a cosimplicialV -module(M•)∗, whence an object ofs.T which we denoteby the same name. Clearly(M•)∗ is as.V -module and we can therefore take its image in thelocalized category(s.V )a-Mod. This defines a functor

s.(V a-Mod)→ (s.V )a-Mod(3.3.17)

and it is not difficult to see that (3.3.17) is an equivalence of categories. Similar equivalencesthen follow for categories of cosimplicialV a-algebras (a.k.a. simplicialV a-schemes) and thelike. For instance, for(X,G, ρ) as in (3.3.6) we can regard the nerveG•X as an affines.S-scheme.

3.3.18. More generally, letX• be any simplicialS-scheme. A quasi-coherentOX•-module isthe same as a cosimplicialOS-moduleM•, such thatMn is anOXn-module for everyn ∈ N,and the coface (resp. codegeneracy) morphisms∂i : Mn → Mn+1 (resp.σj : Mn → Mn−1)are∂i-linear (resp.σj-linear), i.e. they induceMn-linear morphismsMn → ∂i∗Mn+1 (resp.Mn → σj∗Mn−1) (notation of (3.3.4)). It is convenient to introduce the following notation: foreveryi, n ∈ N we let

∂i : ∂∗iMn → Mn+1

theMn+1-linear morphism deduced from∂i by extension of scalars (and likewise forσj).

3.3.19. For everyS-schemeX on whichG acts, and for everyn ∈ N, setπX,n := ∂1 ∂2 ... ∂n : Gn

X → X. For anyG-equivariant quasi-coherentOX-moduleM , we define a quasi-coherentOG•

X-moduleπ∗XM as follows. According to (3.3.16), this is the same as defining a

moduleπ∗XM over the cosimplicial almost algebraOG•X

; then we setπ∗XMn := π∗X,nM for everyobject[n] of ∆o. Next, we remark thatπX,n−1 ∂i = πX,n for everyi, n > 0, hence we havenatural isomorphismsπ∗XMn

∼→ ∂∗i π∗XMn−1, from which we deduce the coface morphisms

∂i : π∗XMn−1 → π∗XMn for every i, n > 0. Finally we use the morphism (3.3.8) and thecartesian diagram:

GnX

∂0 //

τ :=∂2...∂n

Gn−1X

πX,n−1

G1X

∂0 // X

to define∂0 as the composition:

Mn−1// ∂∗0Mn−1 = ∂∗0π

∗X,n−1M = τ ∗∂∗0M

τ∗β // π∗X,nM = Mn.


We leave to the reader the task of defining the codegeneracy morphisms; using the cocyclerelation encoded in (3.3.9) one can then verify the requiredcosimplicial identities. In this waywe obtain a functor

OX-ModG → OG•X

-Mod M 7→ π∗XM.(3.3.20)

Proposition 3.3.21.The functor(3.3.20) is fully faithful, and its essential image is the fullsubcategory of all quasi-coherentOG•

X-modules(Mn | n ∈ N) such that∂n : ∂∗nMn−1 → Mn

is an isomorphism for everyn ∈ N (notation of (3.3.18)).

Proof. Let φ• : π∗XM → φ∗XN be a morphism ofOG•X

-modules. SinceMn = π∗X,nM for everyn ∈ N, we see thatφ• is already determined by its componentφ0 : M → N , whence the fullfaithfulness of (3.3.20). On the other hand, letM• be a quasi-coherentOG•

X-module satisfying

the condition of the lemma, and setM := M0. We define a morphismβ : ∂∗0M → ∂∗1M as the

composition∂∗0M∂0

→M1(∂1)−1

→ ∂∗1M .

Claim3.3.22. β defines an action ofG onM .

Proof of the claim:The identityσ∗0β = 1M is immediate, hence it suffices to show that (3.3.9)commutes. This is the same as showing the commutativity of the following diagram:

∂∗0∂∗1M = ∂∗2∂

∗0M

∂∗0 (∂1)

vvnnnnnnnnnnnnn∂∗2 (∂0)

((PPPPPPPPPPPPP

∂∗0M1

∂0

((PPPPPPPPPPPPPPP∂∗2M1

∂2

vvnnnnnnnnnnnnnnn

∂∗0∂∗0M

∂∗0 (∂0)

OO

M2 ∂∗2∂∗1M

∂∗2 (∂1)

OO

∂∗1∂∗0M

∂∗1 (∂0)// ∂∗1M1

∂1

OO

∂∗1∂∗1M

∂∗1 (∂1)oo

which can be checked separately on each of its three quadrangular subdiagrams. Let us verifyfor instance the commutativity of the bottom left diagram. By linearity, we can simplify downto the diagram:

M1∂0

// M2

M

∂0

OO

∂0 // M1

∂1

OO

whose commutativity expresses one of the identities defining the cosimplicial moduleM•.

To conclude the proof, it suffices to exhibit an isomorphismγ• : π∗XM∼→ M•. For every

n ∈ N, let γn : π∗X,nM → Mn be the morphism induced by the composition∂n ∂n−1 ... ∂1;under our assumption,γn is an isomorphism, and we leave to the reader the verificationthat(γn | n ∈ N) defines a morphism of cosimplicial modules.

3.3.23. In caseG is flat overS, I is G-equivariant, andφ : X → Y is an equivariantmorphism of affineS-schemes on whichG acts, we deduce a natural functor:

ExalY (X/G,I )→ ExalG•Y(G•X , π

∗XI ).(3.3.24)

Namely, to anyG-equivariant square zero deformationD := (j : X → X ′,I , β) one assignsthe datumGD := (Gj : G•X → G•X′ , π∗XI , π∗Xβ). The flatness ofG ensures thatKer(OGn

X′→

OGnX) ≃ π∗X,nI for everyn ∈ N, which means thatGD is indeed a deformation ofG•X over

G•Y .


The basic observation is contained in the following:

Lemma 3.3.25.Under the assumptions of(3.3.23), the functor(3.3.24)is an equivalence ofcategories.

Proof. Let (j : G•X → X ′•, π∗XI , β) be a deformation ofG•X . For everyn ∈ N we have a

commutative diagram of affineS-schemes:

X ′n∂ //

X ′0

GnY

πY // Y

where∂ := ∂1 ∂2 ...∂n andπY,n is the natural projection as in (3.3.19). We deduce a uniquemorphismαn : X ′n → Gn

Y ×Y X ′0 ≃ GnX′

0. By construction,α♯n fits into a commutative diagram

0 // OGn ⊗OSI // OGn ⊗OS

OX′0

α♯n

// OGn ⊗OSOX

// 0

0 // OGn ⊗OSI // OX′

n// OGn ⊗OS

OX// 0

henceαn is an isomorphism for everyn ∈ N. To conclude, it suffices to verify that the systemof morphisms(αn | n ∈ N) defines a morphism of simplicialS-schemes:α• : X ′• → G•X′

0.

This amounts to showing that theαn commute with the face and degeneracy morphisms, whichhowever is easily checked from the definition.

3.3.26. Combining lemmata 3.3.25 and 2.5.17 (which holdsverbatimin the present context)we derive a natural equivalence of categories:

ExalY (X/G,I )→ ExalG•Y !!

(G•X!!, (π∗XI )∗).

This enables us to use the usual theory of the cotangent complex to classify theG-equivariantdeformations ofX.

3.3.27. According to [47, Ch.VI,§5.3], for everyn ∈ N we have a morphism of topoi

[n]T : T → s.T

calledrestriction to then-th level. It is given by a pair of adjoint functors([n]∗T , [n]T∗) such that[n]∗T : s.T → T is the functor that assigns to any cosimplicial object(X[k] | k ∈ N) the objectX[n] of T . For everyk ∈ N setNX,k := Gk

X ∐ Spec V a; the system(NX,k | k ∈ N) defines asimplicial SpecV a-schemeNX (and likewise we defineNY ). In view of [46, Ch.II, (1.2.3.5)],we deduce natural isomorphisms of simplicial complexes of flatONX,n!!

-modules:

[n]∗TLNX!!/NY !!

∼→ LNX,n!!/NY,n!!for everyn ∈ N

whence natural isomorphisms of simplicial complexes of flatOGnX!!

-modules:

[n]∗TLG•X/G

•Y

:= [n]∗T ι∗•LNX!!/NY !!

∼→ ι∗[n][n]∗TLNX!!/NY !!

∼→ ι∗[n]LNX,n!!/NY,n!!=: LGn

X/GnY

whereι• : G•X!! → NX!! is the morphism of simplicial schemes defined as in (3.3.13).Inother words,LG•

X/G•Y

is a mixed simplicial-cosimplicial moduleL•• whose rowsL•n are the


cotangent complexes of the morphismsGφ : GnX → Gn

Y . Furthermore, for everyn ∈ N andeveryi ≤ n+ 1 we have a cartesian diagram

Gn+1X

∂i //

Gn+1φ

GnX

Gnφ

Gn+1Y

∂i // GnY

whose horizontal arrows are flat morphisms (sinceG is S-flat by assumption). Whence, takinginto account theorem 2.5.35, a natural isomorphism inD(ONX,n+1!!

-Mod) :

∂iL : ∂∗i LGnX/G

nY

∼→ LGn+1X /Gn+1

Y

and, by unwinding the definitions, one sees that∂iL is induced by thei-th coface morphismL•n → L•n+1 of the double complexL••.

Definition 3.3.28. LetG be an affineS-group scheme,φ : X → Y aG-equivariant morphismof affineS-schemes on whichG acts. We say thatφ is aG-torsor overY if the action ofG onY is trivial (i.e. ρ : G×S Y → Y is the natural projection) and there exists a cartesian diagramof affineS-schemes

G×S Zg //

pZ

X

φ

Z

f // Y

(3.3.29)

such thatf is faithfully flat, g isG-equivariant for the action onG ×S Z deduced fromG, andpZ is the natural projection.

3.3.30. Suppose thatG is a flat affineS-group scheme andX → Y is aG-torsor overY .Then the groupoidG×S X //

// X defined by theG-action onX is an effective equivalencerelation (cp. (4.5.1)) andY ≃ X/G. Furthermore, letX×Y n := X×Y X×Y ....×Y X (then-thfold cartesian power ofX overY ); there are natural identificationsGn

X ≃ X×Y n+1, amountingto an isomorphism of (semi-)simplicialS-schemes augmented overY :

G•X∼→ [Y |X].

(cp. the notation of [5, Exp.Vbis, (1.2.7)]). We can regard the cosimplicialT -ring OG•X

as aring of the toposΓ(∆× T ) deduced from the simplicial topos∆× T (notation of [5, Exp.Vbis,(1.2.8)]), whence an augmentation of fibred topoi

θ : (∆× T,OG•X)→ (∆× T,OY )(3.3.31)

and it follows from the foregoing and from remark 3.1.3 that (3.3.31) is anaugmentation of2-cohomological descent(see [5, Exp.Vbis, Def.2.2.6]). Denote by

θ∗ : (T,OY )-Mod→ (Γ(∆× T ),OG•X)-Mod

the morphism obtained as the composition of the constant functor ε∗ : (T,OY )-Mod →(Γ(∆ × T ),OY )-Mod and the functorΓ(θ∗) : (Γ(∆ × T ),OY )-Mod → OG•

X-Mod. Since

the morphisms∂iL are isomorphisms, it then follows from general cohomological descent ([5,Exp.Vbis, Prop.2.2.7]) that, for everyk ∈ N, the truncated system(τ[−kLGn

X/GnY| n ∈ N) is

in the essential image of the functorL+θ∗. In the following we will be only interested in thecase where the cotangent complex is concentrated in degree zero, in which case one can avoidthe recourse to cohomological descent, and rather appeal tomore down-to-earth faithfully flat


descent. In any case, the foregoing shows that there exists auniquely determined pro-object(LG

X/Y,k | k ∈ N) of D(OY -Mod) such that

L+θ∗(LGX/Y,k) ≃ τ[−kL

aG•

X/G•Y

for everyk ∈ N.

By trivial duality there follow natural isomorphisms for everyk ∈ N and every complexK• ∈D+(OG•

X-Mod) such thatK = τ[−kK :

ExtkOG•X

(LaG•

X/G•Y, K•) ≃ ExtkOY

(LGX/Y,k, R

+θ∗K•).(3.3.32)

Definition 3.3.33. Let G be an affineS-scheme ande : S → G its unit section. Thecoliecomplexof G is the complex ofOS-modulesℓG/S := e∗La

G/S.

Proposition 3.3.34.LetG be a flatS-group scheme,φ : X → Y be aG-torsor overY andπY : Y → S the structure morphism. Then, for everyk ∈ N, the complexLG

X/Y,k is locallyisomorphic toπ∗Y τ[−kℓG/S in the fpqc topology ofY .

Proof. We will exhibit more precisely a faithfully flat morphismf : Z → Y such that, foreveryk ∈ N there exist isomorphismsR+f ∗LG

X/Y,k ≃ R+f ∗π∗Y τ[−kℓG/S in D(OZ-Mod). LetπG : G→ S be the structure morphism; first of all we remark:

Claim 3.3.35. LaG/S ≃ π∗GℓG/S.

Proof of the claim: Indeed,πG is trivially a G-torsor overS, hence we have a compatiblesystem of isomorphismsτ[−kLa

G/S ≃ π∗Y LGG/S,k. If e : S → G is the unit section, we deduce:

τ[−kℓG/S ≃ e∗τ[−kLaG/S ≃ LG

G/S,k for everyk ∈ N. After takingπ∗G of the two sides, the claimfollows.

Let now f : Z → Y be a faithfully flat morphism that trivializes the givenG-torsorφ :X → Y , so we have a cartesian diagram (3.3.29). Denoting bypG : G×S Z → G the naturalprojection, we deduce (sinceG isS-flat) an isomorphismg∗LX/Y ≃ p∗GLG/S. Whence, in viewof claim 3.3.35 :

p∗Zf∗LG

X/Y,k ≃ p∗Gπ∗Gτ[−kℓG/S ≃ p∗Zπ

∗Zτ[−kℓG/S ≃ p∗Zf

∗π∗Y τ[−kℓG/S.(3.3.36)

Let eZ := e×S 1Z : Z → G×S Z; the claim follows after applyinge∗Z to (3.3.36).

3.3.37. Finally we can wrap up this section with a discussionof equivariant deformations oftorsors. Hence, letφ : X → Y be aG-torsor overY , andjY : Y → Y ′ a morphism of affineS-schemes such thatI := Ker(j♯Y : OY ′ → OY ) is a square zero ideal ofOY ′ . We wish toclassify thesquare zero deformations of the torsorφ overY ′, that is, the isomorphism classesof cartesian diagrams

X

φ

jX // X ′

φ′

Y

jY // Y ′

such thatφ′ is aG-torsor overY ′ andjX isG-equivariant.

Theorem 3.3.38.Suppose thatG is flat overS, thatHi(ℓG/S) = 0 for i > 0 and thatH0(ℓG/S)is an almost finitely generated projectiveOS-module. Furthermore, suppose that the homologi-cal dimension ofm is≤ 1. Then, in the situation of(3.3.37), we have:

(i) The pro-object(LGX/Y,k | k ∈ N) is constant, isomorphic to a complex ofD(OY -Mod)

concentrated in degree zero, that we shall denote byLGX/Y .

(ii) The set of isomorphism classes of square zero deformations of the torsorφ : X → Y overY ′ is a torsor under the groupExt1

OY(LG

X/Y ,I ).


(iii) The group of automorphisms of a square zero deformationφ′ : X ′ → Y ′ as in(3.3.37)isnaturally isomorphic toExt0

OY(LG

X/Y ,I ).

Proof. (i) follows easily from proposition 3.3.34; moreover it follows from remark 3.2.26(iii)thatH0(LG

X/Y )a is an almost finitely generated projectiveOY -module. LetJ := Ker(j♯ :

OX′ → OX); by flatness, the natural morphismφ∗I → J is aG-equivariant isomorphism.We notice that, for every quasi-coherentOY -moduleM , there is a natural isomorphism

θ∗M!∼→ (π∗Xφ

∗M)!.

By cohomological descent (or else, by plain old-fashioned faithfully flat descent) it follows thatthe counit of the adjunction:

I! → R+θ∗π∗XJ!

is an isomorphism, whence, in light of (3.3.32), natural isomorphisms:

ExtkOG•X

(LaG•

X/G•Y, π∗XJ ) ≃ ExtkOY

(H0(LGX/Y ),I ) for everyk ∈ N.(3.3.39)

Claim3.3.40. The diagram of functors:

OY -Mod(−)! //

φ∗

OY !!-Mod

φ∗!!

OX-Mod(−)! // OX!!-Mod

is essentially commutative.

Proof of the claim:Indeed, both of the two possible composition of arrows represent a functorOY -Mod→ OX!!-Mod which is left adjoint to the functorM 7→ φ∗M

a.

Claim3.3.41. The counit of the adjunction:ε : H0(LGX/Y )→ H0(LG

X/Y )a! is an isomorphism.

Proof of the claim: It follows easily from claim 3.3.40 and lemma 2.5.28 thatφ∗(ε) is anisomorphism. Sinceφ is faithfully flat, the assertion follows.

Combining (3.3.32), claim 3.3.41 and lemma 2.4.14(i),(ii), we deduce that

Ext2OG•X

(LG•X/G

•Y, π∗XJ ) = 0.

However, by (3.3.26) and the usual arguments (cp. section 3.2) one knows that the obstruction todeforming the torsorφ is a class in the latter group; since the obstruction vanishes, one deduces(ii). (iii) follows in the same way.

3.4. Descent. Faithfully flat descent in the almost setting presents no particular surprises: sincethe functorA 7→ A!! preserves faithful flatness of morphisms (see remark 3.1.3)many well-known results for usual rings and modules extendverbatimto almost algebras.

3.4.1. So for instance, faithfully flat morphisms are of universal effective descent for the fibredcategoriesF : V a-Alg.Modo → V a-Algo andG : V a-Alg.Morpho → V a-Algo (see defi-nition 2.5.22: for an almostV -algebraB, the fibreFB (resp.GB) is the opposite of the categoryof B-modules (resp.B-algebras)). Then, using remark 3.2.26, we deduce also universal effec-tive descent for the fibred subcategories of flat (resp. almost finitely generated, resp. almostfinitely presented, resp. almost finitely generated projective) modules. Likewise, a faithfullyflat morphism is of universal effective descent for the fibredsubcategoriesEto → V a-Algo ofetale (resp.w.Eto → V a-Algo of weakly etale) algebras.


3.4.2. More generally, since the functorA 7→ A!! preserves pure morphisms in the sense of[58], and since, by a theorem of Olivier (loc. cit.), pure morphisms are of universal effectivedescent for modules, the same holds for pure morphisms of almost algebras.

3.4.3. Non-flat descent is more delicate. Our results are notas complete here as it could bewished, but nevertheless, they suffice for the further studyof etale and unramified morphismsthat shall be taken up in chapter 5. (and they also cover the cases needed in [34]). Our firststatement is the almost version of a theorem of Gruson and Raynaud ([42, Part II, Th. 1.2.4]).

Proposition 3.4.4. A finite monomorphism of almost algebras descends flatness.

Proof. Let φ : A → B be such a morphism. Under the assumption, we can find a finiteA∗-moduleQ such thatmB∗ ⊂ Q ⊂ B∗. One sees easily thatQ is a faithfulA∗-module, so by [42,Part II, Th. 1.2.4 and lemma 1.2.2],Q satisfies the following condition :

If (0 → N → L → P → 0) is an exact sequence ofA∗-modules withL flat, suchthatIm(N ⊗A∗ Q) is a pure submodule ofL⊗A∗ Q, thenP is flat.(3.4.5)

Now letM be anA-module such thatM ⊗AB is flat. Pick an epimorphismp : F → M with Ffree overA. ThenY := (0→ Ker(p⊗A 1B)→ F ⊗AB → M ⊗AB → 0) is universally exactoverB, hence overA. Consider the sequenceX := (0→ Im(Ker(p)! ⊗A∗ Q)→ F! ⊗A∗ Q→M!⊗A∗Q→ 0). ClearlyXa ≃ Y . However, it is easy to check that a sequenceE ofA-modulesis universally exact if and only if the sequenceE! is universally exact overA∗. We concludethatX = (Xa)! is a universally exact sequence ofA∗-modules, hence, by condition (3.4.5),M!

is flat overA∗, i.e.M is flat overA as required.

Corollary 3.4.6. LetA→ B be a finite morphism of almost algebras, with nilpotent kernel. IfC is a flatA-algebras such thatC ⊗A B is weaklyetale (resp.etale) overB, thenC is weaklyetale (resp.etale) overA.

Proof. In the weakly etale case, we have to show that the multiplication morphismµ : C ⊗AC → C is flat. AsN := Ker(A → B) is nilpotent, the local flatness criterion reduces thequestion to the situation overA/N . So we may assume thatA→ B is a monomorphism. ThenC⊗AC → (C⊗AC)⊗AB is a monomorphism, butµ⊗C⊗AC 1(C⊗AC)⊗AB is the multiplicationmorphism ofC ⊗A B, which is flat by assumption. Therefore, by proposition 3.4.4,µ is flat.

For the etale case, we have to show thatC is almost finitely presented as aC⊗AC-module. ByhypothesisC⊗AB is almost finitely presented as aC⊗AC⊗AB-module and we know alreadythatC is flat as aC ⊗A C-module, so by lemma 3.2.25(iii) (applied to the finite morphismC ⊗A C → C ⊗A C ⊗A B) the claim follows.

3.4.7. Suppose now that we are given a cartesian diagramD of almost algebras

A0f2 //

f1

A2

g2

A1g1 // A3

(3.4.8)

such that one of the morphismsAi → A3 (i = 1, 2) is an epimorphism. DiagramD induces anessentially commutative diagram for the corresponding categoriesAi-Mod, where the arrowsare given by the “extension of scalars” functors. We define the category ofD-modulesas the2-fibre productD-Mod := A1-Mod ×A3-Mod A2-Mod. Recall (see [8, Ch.VII§3] or [43,Ch.I] for generalities on2-categories and2-fibre products) thatD-Mod is the category whoseobjects are the triples(M1,M2, ξ), whereMi is anAi-module (i = 1, 2) andξ : A3 ⊗A1 M1

∼→A3 ⊗A2 M2 is anA3-linear isomorphism. There follows a natural functor

π : A0-Mod→ D-Mod.


Given an object(M1,M2, ξ) of D-Mod, let us denoteM3 := A3 ⊗A2 M2; we have a naturalmorphismM2 → M3, andξ gives a morphismM1 → M3, so we can form the fibre productT (M1,M2, ξ) := M1 ×M3 M2. In this way we obtain a functorT : D-Mod → A0-Mod, andwe leave to the reader the verification thatT is right adjoint toπ. Let us denote byε : 1M0 →T π andη : π T → 1D-Mod the unit and counit of the adjunction.

Lemma 3.4.9. The functorπ induces an equivalence of full subcategories :

X ∈ Ob(A0-Mod) | εX is an isomorphism π→ Y ∈ Ob(D-Mod) | ηY is an isomorphismhavingT as quasi-inverse.

Proof. General nonsense.

Lemma 3.4.10.LetM be anyA0-module. ThenεM is an epimorphism. IfTorA01 (M,A3) = 0,

thenεM is an isomorphism.

Proof. Indeed,εM : M → (A1⊗A0 M)×A3⊗A0M (A2⊗A0 M) is the natural morphism. So, the

assertions follow by applying−⊗A0 M to the short exact sequence ofA0-modules

0→ A0f→ A1 ⊕A2

g→ A3 → 0(3.4.11)

wheref(a) := (f1(a), f2(a)) andg(a, b) := g1(a)− g2(b).

3.4.12. There is another case of interest, in whichεM is an isomorphism. Namely, supposethat one of the morphismsAi → A3 (i = 1, 2), sayA1 → A3, has a section. Then also themorphismA0 → A2 gains a sections : A2 → A0 and we have the following :

Lemma 3.4.13. In the situation of(3.4.12), suppose that theA0-moduleM arises by extensionof scalars from anA2-moduleM ′, via the sections : A2 → A0. ThenεM is an isomorphism.

Proof. Indeed, in this case (3.4.11) is split exact as a sequence ofA2-modules, and it remainssuch after tensoring byM ′.

Lemma 3.4.14.η(M1,M2,ξ) is an isomorphism for all objects(M1,M2, ξ).

Proof. To fix ideas, suppose thatA1 → A3 is an epimorphism. Consider anyD-module(M1,M2, ξ). LetM := T (M1,M2, ξ); we deduce a natural morphism

φ : (M ⊗A0 A1)×M⊗A0A3 (M ⊗A0 A2)→ M1 ×M3 M2

such thatφ εM = 1M . It follows thatεM is injective, hence it is an isomorphism, by lemma3.4.10. We derive a commutative diagram with exact rows :

0 // M // (M ⊗A0 A1)⊕ (M ⊗A0 A2)

φ1⊕φ2

// M ⊗A0 A3//

φ3

0

0 // M // M1 ⊕M2// M3

// 0.

From the snake lemma we deduce(∗) Ker(φ1)⊕Ker(φ2) ≃ Ker(φ3)(∗∗) Coker(φ1)⊕ Coker(φ2) ≃ Coker(φ3).

SinceM3 ≃ M1 ⊗A1 A3 we haveA3 ⊗A1 Coker φ1 ≃ Cokerφ3. But by assumptionA1 → A3

is an epimorphism, so alsoCoker φ1 → Coker φ3 is an epimorphism. Then(∗∗) implies thatCoker φ2 = 0. Butφ3 = 1A3 ⊗A2 φ2, thusCoker φ3 = 0 as well. We look at the exact sequence

0 → Kerφ1 → M ⊗A0 A1φ1→ M1 → 0 : applyingA3 ⊗A1 − we obtain an epimorphism

A3 ⊗A1 Kerφ1 → Kerφ3. From (∗) it follows that Kerφ2 = 0. In conclusion,φ2 is anisomorphism. Hence the same is true forφ3 = 1A3 ⊗A2 φ2, and again(∗), (∗∗) show thatφ1 isan isomorphism as well, which implies the claim.


Lemma 3.4.15. In the situation of(3.4.7), letM be anyA0-module andn ≥ 1 an integer. Thefollowing conditions are equivalent:

(a) TorA0j (M,Ai) = 0 for everyj ≤ n andi = 1, 2, 3.

(b) TorAij (Ai ⊗A0 M,A3) = 0 for everyj ≤ n andi = 1, 2.

Proof. There is a base change spectral sequence

E2pq := TorAi

p (TorA0q (M,Ai), A3)⇒ TorA0

p+q(M,A3).(3.4.16)

If now (a) holds, we deduce thatTorp(Ai ⊗A0 M,A3) ≃ Torp(M,A3) wheneverp ≤ n, andthen (b) follows. Conversely, suppose that (b) holds; we show (a) by induction onn. Say thatthe morphismA1 → A3 is an epimorphism, so that the same holds for the morphismA0 → A2,and denote byI the common kernel of these morphisms. Forn = 1 andi = 1, the assumption isequivalent to saying that the natural morphismI ⊗A1 (A1 ⊗A0 M)→ (A1 ⊗A0 M) is injective.It follows that the same holds for the morphismI ⊗A0 M → M , which already shows thatTorA0

1 (M,A2) = 0. Next, the assumption fori = 2 means that the termE210 of the spectral

sequence (3.4.16) vanishes, whence an isomorphismTorA01 (M,A3) ≃ TorA0

1 (M,A2)⊗A2A3 ≃0. Finally, we use the long exact Tor sequence arising from theshort exact sequence (3.4.11)to deduce that alsoTorA0

1 (M,A1) vanishes. Let nown > 1 and suppose that assertion (a) isalready known for allj < n. We choose a presentation

0→ R→ F → M → 0(3.4.17)

with F flat overA0; using the long exact Tor sequence we deduce thatTorA0j (M,Ai) ≃

TorA0j−1(R,Ai) for everyj > 1 and everyi ≤ 3. Moreover, assertion (a) taken withj = 1

shows that the sequenceAi⊗A0(3.4.17) is again exact fori ≤ 3, thereforeTorA01 (R,Ai) van-

ishes as well fori ≤ 3. In other words, theA0-moduleR fulfills condition (b) for everyj < n,hence the inductive assumption shows thatTorA0

j (R,Ai) = 0 for everyj < n andi = 1, 2, 3;in turns this implies the sought vanishing forM .

The following lemma 3.4.18(iii) generalizes a result of D.Ferrand ([35, lemma]).

Lemma 3.4.18.LetM be anyA0-module. We have:

(i) AnnA0(M ⊗A0 A1) · AnnA0(M ⊗A0 A2) ⊂ AnnA0(M).(ii) M admits a three-step filtration0 ⊂ Fil0M ⊂ Fil1M ⊂ Fil2M = M such thatFil0M

andgr2M areA2-modules andgr1M is anA1-module.(iii) If (A1 × A2)⊗A0 M is flat overA1 ×A2, thenM is flat overA0.

Proof. To fix ideas, suppose thatA1 → A3 is an epimorphism, and letI be its kernel; let alsoMi := Ai ⊗A0 M for i = 1, 2, 3.

(i): clearly I ≃ Ker(A0 → A2), thereforeaM ⊂ IM for everya ∈ AnnA0(M2). On theother hand, the natural morphismI ⊗A1 M1

∼→ I ⊗A0 M → IM is obviously an epimorphism,whence the assertion.

(ii): we setFil0M := Ker(εM : M → M1 ×M3 M2); using the short exact sequence (3.4.16)we see thatFil0M ≃ TorA0

1 (M,A3)/TorA01 (A1⊕A2,M). ObviouslyI annihilatesFil0M (since

it annihilates alreadyTorA01 (M,A3)), hence the latter is anA2-module. By lemma 3.4.10, we

haveM ′ := M/Fil0M ≃ M1 ×M3 M2. We can then filter the latter module by definingFil0M

′ := 0, Fil1M′ := Ker(M ′ → M2) ≃ Ker(M1 → M3), which is aA1-module, and

Fil2M′ := M ′. Sincegr2M

′ ≃M2, the assertion follows.(iii): in view of lemma 3.4.15, it suffices to show the following:

Claim 3.4.19. M is flat overA0 if and only ifMi isAi-flat andTorA01 (M,Ai) = 0 for i ≤ 2.


Proof of the claim:It suffices to prove the non-obvious implication, and in viewof (ii) we are re-duced to showing thatTorA0

1 (M,L) = 0 wheneverL is anAi-module, fori = 1, 2. However, foranyAi-moduleL we have a base change spectral sequenceE2

pq := TorAip (TorA0

q (M,Ai), L)⇒TorA0

p+q(M,L). If TorA01 (M,Ai) = 0, this yieldsTorA0

1 (M,L) ≃ TorAi1 (Mi, L), which vanishes

whenMi isAi-flat.

3.4.20. For anyV a-algebraA, letA-Modfl (resp.A-Modproj, resp.A-Modafpr) denote thefull subcategory ofA-Mod consisting of all flat (resp. almost projective, resp. almost finitelygenerated projective)A-modules. For any integern ≥ 1, letA0-Modn be the full subcategoryof all A0-modules satisfying condition (a) of lemma 3.4.15; let alsoAi-Modn (for i = 1, 2) bethe full subcategory ofAi-Mod consisting of allAi-modulesM such thatTorAi

j (M,A3) = 0for everyj ≤ n. Finally, LetA-Algfl be the full subcategory ofA-Alg consisting of all flatA-algebras.

Proposition 3.4.21. In the situation of(3.4.7), the natural essentially commutative diagram:

A0-Mod?//

A2-Mod?

A1-Mod?

// A3-Mod?

is 2-cartesian (i.e. cartesian in the2-category of categories, cp.[43, Ch.I]) whenever? = “fl”or ? = “proj” or ? = “afpr”, or ? = n, for any integern ≥ 1.

Proof. The assertion for flat almost modules follows directly from lemmata 3.4.9, 3.4.10, 3.4.14and 3.4.18(iii). Similarly the assertion for the categories Ai-Modn follows from the samelemmata and from lemma 3.4.15. SetB := A1 × A2. To establish the assertion for projectivemodules, it suffices to show that, ifP is anA0-module such thatB ⊗A0 P is almost projectiveoverB, thenP is almost projective overA0, or which is the same, thatalExtiA0

(P,N) ≃ 0 forall i > 0 and anyA0-moduleN . We know already thatP is flat. LetM be anyA0-module and

N anyB-module. The standard isomorphismRHomB(BL

⊗A0M,N) ≃ RHomA0(M,N) yieldsa natural isomorphismalExtiB(B ⊗A0 M,N) ≃ alExtiA0

(M,N), wheneverTorA0j (B,M) = 0

for everyj > 0. In particular, we havealExtiA0(P,N) ≃ 0 wheneverN comes from either an

A1-module, or anA2-module. In view of lemma 3.4.18(ii), we deduce that the sought vanishingholds in fact for everyA0-moduleN . Finally, suppose thatP ⊗A B is almost finitely generatedprojective overB; we have to show thatP is almost finitely generated. To this aim, noticethatAnnA0(P ⊗A B)2 ⊂ AnnA0(P ), in view of lemma 3.4.18(i); then the claim follows fromremark 3.2.26(i).

Corollary 3.4.22. In the situation of(3.4.7), the natural essentially commutative diagram:

A0-C //

A2-C

A1-C // A3-C

is2-cartesian wheneverC is one of the categoriesAlgfl, Et, w.Et, Etafp (notation of(3.2.27)).

3.4.23. Next we want to reinterpret the equivalences of proposition 3.4.21 in terms of descentdata. IfF : C → V a-Algo is a fibred category over the opposite of the category of almost alge-bras, and ifX → Y is a given morphism of almost algebras, we shall denote byDesc(C , Y/X)the category of objects of the fibre categoryFY , endowed with a descent datum relative to the


morphismX → Y (cp. [39, Ch.II§1]). In the arguments hereafter, we consider morphismsof almost algebras and modules, and one has to reverse the direction of the arrows to pass tomorphisms in the relevant fibred category. Denote bypi : Y → Y ⊗X Y (i = 1, 2), resp.pij : Y ⊗X Y → Y ⊗X Y ⊗X Y (1 ≤ i < j ≤ 3) the usual morphisms.

3.4.24. As an example,Desc(V a-Alg.Modo, Y/X) consists of the pairs(M,β) whereM isaY -module andβ is aY ⊗X Y -linear isomorphismβ : p∗2(M)

∼→ p∗1(M) such that

p∗12(β) p∗23(β) = p∗13(β).(3.4.25)

3.4.26. Let nowI ⊂ X be an ideal, and setX := X/I, Y := Y/IY . For anyF : C →V a-Algo as in (3.4.23), one has an essentially commutative diagram:

Desc(C , Y/X) //

Desc(C , Y /X)

FY // FY .

(3.4.27)

This induces a functor :

Desc(C , Y/X)→ Desc(C , Y /X)×FYFY .(3.4.28)

Lemma 3.4.29. In the situation of(3.4.26), suppose moreover that the natural morphismI →IY is an isomorphism. Then diagram(3.4.27)is 2-cartesian wheneverC is one of the fibredcategoriesV a-Alg.Modo, V a-Alg.Morpho, Eto, w.Eto.

Proof. For anyn > 0, denote byY ⊗n (resp.Y ⊗n) then-fold tensor product ofY (resp.Y ) withitself overX (resp.X), and byρn : Y ⊗n → Y ⊗n the natural morphism. First of all we claimthat, for everyn > 0, the natural diagram of almost algebras

Y ⊗nρn //

µn

Y ⊗n

µn

Y

ρ1 // Y

(3.4.30)

is cartesian (whereµn andµn aren-fold multiplication morphisms). For this, we need to verifythat, for everyn > 0, the induced morphismKer ρn → Ker ρ1 (defined by multiplicationof the first two factors) is an isomorphism. It then suffices tocheck that the natural morphismKer ρn → Ker ρn−1 is an isomorphism for alln > 1. Indeed, consider the commutative diagram

I ⊗X Y ⊗n−1p // IY ⊗n−1 i //

ψ

Y ⊗n−1

φ⊗1Y ⊗n−1

1Y ⊗n−1

((QQQQQQQQQQQQQ

I ⊗X Y ⊗n−1

p′// Ker ρn

i′// Y ⊗n

µY/X⊗1Y ⊗n−2

// Y ⊗n−1.

FromIY = φ(Y ), it follows thatp′ is an epimorphism. Hence alsoψ is an epimorphism. Sincei is a monomorphism, it follows thatψ is also a monomorphism, henceψ is an isomorphismand the claim follows easily.

We consider first the caseC := V a-Alg.Modo; we see that (3.4.30) is a diagram of the kindconsidered in (3.4.8), hence, for everyn > 0, we have the associated functorπn : Y ⊗n-Mod→Y ⊗n-Mod ×Y -Mod Y -Mod and also its right adjointTn. Denote bypi : Y → Y ⊗2 (i =1, 2) the usual morphisms, and similarly definepij : Y ⊗2 → Y ⊗3. Suppose there is givena descent datum(M,β) for M , relative toX → Y . The cocycle condition (3.4.25) implieseasily thatµ∗2(β) is the identity onµ∗2(p

∗iM) = M . It follows that the pair(β, 1M) defines


an isomorphismπ2(p∗1M)

∼→ π2(p∗2M) in the categoryY ⊗2-Mod ×Y -Mod Y -Mod. Hence

T2(β, 1M) : T2 π2(p∗1M) → T2 π2(p

∗2M) is an isomorphism. However, we remark that

either morphismpi yields a section forµ2, hence we are in the situation contemplated in lemma3.4.13, and we derive an isomorphismβ : p∗2(M)

∼→ p∗1(M). We claim that(M,β) is an objectof Desc(C , Y/X), i.e. thatβ verifies the cocycle condition (3.4.25). Indeed, we can compute:π3(p

∗ijβ) = (ρ∗3(p

∗ijβ), µ∗3(p

∗ijβ)) and by construction we haveρ∗3(p

∗ijβ) = p∗ij(β) andµ∗3(p

∗ijβ) =

µ∗2(β) = 1M . Therefore, the cocycle identity forβ implies the equalityπ3(p∗12β) π3(p

∗23β) =

π3(p∗13β). If we now apply the functorT3 to this equality, and then invoke again lemma 3.4.13,

the required cocycle identity forβ will ensue. Clearlyβ is the only descent datum onM liftingβ. This proves that (3.4.28) is essentially surjective. The same sort of argument also shows thatthe functor (3.4.28) induces bijections on morphisms, so the lemma follows in this case. Next,the caseC := V a-Alg.Morpho can be deduced formally from the previous case, by applyingrepeatedly natural isomorphisms of the kindp∗i (M ⊗Y N) ≃ p∗i (M) ⊗Y⊗XY p

∗i (N) (i = 1, 2).

Finally, the “etaleness” of an object ofDesc(V a-Alg.Morpho, Y/X) can be checked on itsprojection ontoY -Algo, hence also the casesC := w.Eto andC := Eto follow directly.

3.4.31. Now, letB := A1×A2; to an object(M,β) in Desc(V a-Alg.Modo, B/A) we assigna D-module(M1,M2, ξ) (notation of (3.4.7)) as follows. SetMi := Ai ⊗B M (i = 1, 2) andAij := Ai ⊗A0 Aj . We can writeB ⊗A0 B =

∏2i,j=1Aij andβ gives rise to theAij-linear

isomorphismsβij : Aij ⊗B⊗A0B p∗2(M)

∼→ Aij ⊗B⊗A0B p∗1(M). In other words, we obtain

isomorphismsβij : Ai ⊗A0 Mj → Mi ⊗A0 Aj. However, we have a natural isomorphismA12 ≃ A3 (indeed, suppose thatA1 → A3 is an epimorphism with kernelI; thenI is also anideal ofA0 andA0/I ≃ A2; now the claim follows by remarking thatIA1 = I). Hence we canchooseξ = β12. In this way we obtain a functor :

Desc(V a-Alg.Modo, B/A0)→ D-Modo.(3.4.32)

Proposition 3.4.33.The functor(3.4.32)is an equivalence of categories.

Proof. Let us say thatA1 → A3 is an epimorphism with kernelI. ThenI is also an idealof B and we haveB/I ≃ A3 × A2 andA0/I ≃ A2. We intend to apply lemma 3.4.29 tothe morphismA0 → B. However, the induced morphismB := B/I → A0 := A0/I inV a-Algo has a section, and hence it is of universal effective descentfor every fibred category.Thus, we can replace in (3.4.28) the categoryDesc(V a-Alg.Modo, B/A0) byA0-Modo, andthereby, identify (up to equivalence) the target of (3.4.28) with the 2-fibred product(A1-Mod×A2-Mod)o×(A3-Mod×A2-Mod)oA2-Modo. The latter is equivalent to the categoryD-Modo andthe resulting functorDesc(V a-Alg.Modo, B/A0) → D-Modo is canonically isomorphic to(3.4.32), whence the claim.

Putting together propositions 3.4.21 and 3.4.33 we obtain the following :

Corollary 3.4.34. In the situation of(3.4.8), the morphismA0 → A1×A2 is of effective descentfor the fibred categories of flat modules and of almost projective modules.

3.4.35. Next we would like to give sufficient conditions to ensure that a morphism of almostalgebras is of effective descent for the fibred categoryw.Eto → V a-Algo of weakly etalealgebras (resp. for etale algebras). To this aim we are led to the following :

Definition 3.4.36. A morphismφ : A → B of almost algebras is said to bestrictly finite ifKerφ is nilpotent andB ≃ Ra, whereR is a finiteA∗-algebra.

Theorem 3.4.37.Letφ : A→ B be a strictly finite morphism of almost algebras. Then :

(i) For everyA-algebraC, the induced morphismC → C ⊗A B is again strictly finite.


(ii) If M is a flatA-module andB ⊗A M is almost projective overB, thenM is almostprojective overA.

(iii) A → B is of universal effective descent for the fibred categories of weaklyetale (resp.etale) almost algebras.

Proof. (i): suppose thatB = Ra for a finiteA∗-algebraR; thenS := C∗ ⊗A∗ R is a finiteC∗-algebra andSa ≃ C. It remains to show thatKer(C → C ⊗A B) is nilpotent. Suppose thatRis generated byn elements as anA∗-module and letFA∗(R) (resp.FC∗(S)) be the Fitting idealof R (resp.ofS); we haveAnnC∗(S)n ⊂ FC∗(S) ⊂ AnnC∗(S) (see [50, Ch.XIX Prop.2.5]); onthe other handFC∗(S) = FA∗(R) · C∗, so the claim is clear.

(iii): we shall consider the fibred categoryF : w.Eto → V a-Algo; the same argumentapplies also to etale almost algebras. We begin by establishing a very special case :

Claim 3.4.38. Assertion (iii) holds whenB = (A/I1) × (A/I2), whereI1 andI2 are ideals inA such thatI1 ∩ I2 is nilpotent.

Proof of the claim:First of all we remark that the situation considered in the claim is stableunder arbitrary base change, therefore it suffices to show thatφ is of F -2-descent in this case.Then we factorφ as a compositionA→ A/Kerφ→ B and we remark thatA→ A/Kerφ is ofF -2-descent by theorem 3.2.18; since a composition of morphisms ofF -2-descent is again ofF -2-descent, we are reduced to show thatA/Kerφ→ B is of F -2-descent,i.e. we can assumethatKerφ ≃ 0. However, in this case the claim follows easily from corollary 3.4.34.

Claim 3.4.39. More generally, assertion (iii) holds whenB =∏n

i=1A/Ii, whereI1, ..., In areideals ofA, such that

⋂ni=1 Ii is nilpotent.

Proof of the claim:We prove this by induction onn, the casen = 2 being covered by claim3.4.38. Therefore, suppose thatn > 2, and setB′ := A/(

⋂n−1i=1 Ij). By induction, the morphism

B′ → ∏n−1i=1 A/Ii is of universalF -2-descent. However, according to [39, Ch.II Prop.1.1.3],

the sieves of universalF -2-descent form a topology onV a-Algo; for this topology,A,B is acovering family ofA×B and(A→ B′× (A/In))

o is a covering morphism, henceB′, A/Inis a covering family ofA, and then, by composition of covering families,∏n−1

i=1 A/Ii, A/Inis a covering family ofA, which is equivalent to the claim.

Now, let A → B be a general strictly finite morphism, so thatB = Ra for some fi-nite A∗-algebraR. Pick generatorsf1, ..., fm of theA∗-moduleR, and monic polynomialsp1(X), ..., pm(X) such thatpi(fi) = 0 for i = 1, ..., m.

Claim 3.4.40. There exists a finite and faithfully flat extensionC of A∗ such that the images inC[X] of p1(X),...,pm(X) split as products of monic linear factors.

Proof of the claim:This extensionC can be obtained as follows. It suffices to find, for eachi = 1, ..., m, an extensionCi that splitspi(X), because thenC := C1 ⊗A∗ ...⊗A∗ Cm will splitthem all, so we can assume thatm = 1 andp1(X) = p(X); moreover, by induction on thedegree ofp(X), it suffices to find an extensionC ′ such thatp(X) factors inC ′[X] as a productof the formp(X) = (X − α) · q(X), whereq(X) is a monic polynomial of degreedeg(p)− 1.Clearly we can takeC ′ := A∗[T ]/(p(T )).

Given aC as in claim 3.4.40, we remark that the morphismA → Ca is of universalF -2-descent. Considering again the topology of universalF -2-descent, it follows thatA → Bis of universalF -2-descent if and only if the same holds for the induced morphism Ca →Ca ⊗A B. Therefore, in proving assertion (iii) we can replaceφ by 1C ⊗A φ and assumefrom start that the polynomialspi(X) factor inA∗[X] as product of linear factors. Now, letdi := deg(pi) andpi(X) :=

∏di

j (X−αij) (for i = 1, ..., m). We get a surjective homomorphismof A∗-algebrasD := A∗[X1, ..., Xm]/(p1(X1), ..., pm(Xm)) → R by the ruleXi 7→ fi (i =


1, ..., m). Moreover, any sequenceα := (α1,j1, α2,j2, ..., αm,jm) yields a homomorphismψα :D → A∗, determined by the assignmentXi 7→ αi,ji. A simple combinatorial argument showsthat

∏α Kerψα = 0, whereα runs over all the sequences as above. Hence the product map∏

α ψα : D → ∏αA∗ has nilpotent kernel. We notice that theA∗-algebra(

∏αA∗) ⊗D R is a

quotient of∏

αA∗, hence it can be written as a product of rings of the formA∗/Iα, for variousidealsIα. By (i), the kernel of the induced homomorphismR→ ∏

αA∗/Iα is nilpotent, hencethe same holds for the kernel of the compositionA→∏

αA/Iaα, which is therefore of the kind

considered in claim 3.4.39. HenceA → ∏αA/I

aα is of universalF -2-descent. Since such

morphisms form a topology, it follows that alsoA → B is of universalF -2-descent, whichconcludes the proof of (iii).

Finally, letM be as in (ii) and pick againC as in the proof of claim 3.4.40. By remark3.2.26(iv),M is almost projective overA if and only ifCa ⊗AM is almost projective overCa;hence we can replaceφ by 1Ca ⊗A φ, and by arguing as in the proof of (iii), we can assumefrom start thatB =

∏nj=1A/Ij for idealsIj ⊂ A, j = 1, ..., n such thatI :=

⋂nj=1 Ij is

nilpotent. By an easy induction, we can furthermore reduce to the casen = 2. We factorφ asA → A/I → B; by proposition 3.4.21 it follows that(A/I) ⊗A M is almost projective overA/I, and then lemma 3.2.25(i) says thatM itself is almost projective.

Remark 3.4.41. It is natural to ask whether theorem 3.4.37 holds if we replace everywhere“strictly finite” by “finite with nilpotent kernel” (or even by “almost finite with nilpotent ker-nel”). We do not know the answer to this question.

3.4.42. On the categoryV a-Alg (taken in some universe) consider the topologiesτe (resp.τw)of universal effective descent for the fibred categoryEto (resp.w.Eto). For a ringR denote byIdemp(R) the set of idempotents ofR.

Proposition 3.4.43.With the notation of(3.4.42)we have:

(i) The presheafA 7→ Idemp(A∗) is a sheaf for bothτe andτw.(ii) If f : A→ B is anetale (resp. weaklyetale) morphism of almostV -algebras and there is

a covering family(A → Aα)o for τe (resp.τw) such thatAα → Aα ⊗A B is an almost

projective epimorphism for allα, thenf is an almost projective epimorphism.(iii) τe is finer thanτw.

Proof. (i): use descent of morphisms and the bijection

HomA-Alg(A×A,A)∼→ Idemp(A∗) φ 7→ φ∗(1, 0).

(ii): by remark 3.1.8,Ker(Aα → Aα⊗AB) is generated byeα ∈ Idemp(Aα∗). eα andeβ agreein (Aα ⊗A Aβ)∗, so by (i) there is an idempotente ∈ A∗ that restricts toeα in Idemp(Aα∗),for eachα. TheA-algebrasB andA/eA become isomorphic after applying− ⊗A Aα; theseisomorphisms are unique and are compatible onAα⊗AAβ, hence they patch to an isomorphismB ≃ A/eA.

(iii): we have to show that ifA is an almostV -algebra,R a sieve of(V a-Alg)o/A andRis of universalw.Eto-2-descent, thenR is of universalEto-2-descent. Since the assumption isstable under base change, it suffices to show thatR is of Eto-2-descent. Descent of morphismsis clear. LetR be the sieve generated by a family of morphisms(A → Aα)

o. Any descentdatum consisting of etaleAα-algebrasBα and isomorphismsAα⊗ABβ ≃ Bα⊗AAβ satisfyingthe cocycle condition, becomes effective when we pass tow.Eto. So one has to verify that ifBis a weakly etaleA-algebra such thatB ⊗A Aα is etale overAα for all α, thenB is etale overA. Indeed, an application of (ii) gives thatB ⊗A B → B is almost projective.


3.4.44. We conclude with a digression to explain the relationship between our results andknown facts that can be extracted from the literature. So, wenow place ourselves in the “clas-sical limit” m := V (cp. example 2.1.2(ii)). In this case, weakly etale morphisms had alreadybeen considered in some earlier work, and they were called “absolutely flat” morphisms. A ringhomomorphismA→ B is etale in the usual sense of [40] if and only if it is absolutely flat andof finite presentation. Let us denote byu.Eto the fibred category overV -Algo, whose fibreover aV -algebraA is the opposite of the category of etaleA-algebras in the usual sense. Weclaim that, if a morphismA → B of V -algebras is of universal effective descent for the fibredcategoryw.Eto (resp. Eto), then it is a morphism of universal effective descent foru.Eto.Indeed, letC be an etaleA-algebra (in the sense of definition 3.1.1) and such thatC ⊗A B isetale overB in the usual sense. We have to show thatC is etale in the usual sense,i.e. that it isof finite presentation overA. This amounts to showing that, for every filtered inductive system(Aλ)λ∈Λ of A-algebras, we havecolim

λ∈ΛHomA-Alg(C,Aλ) ≃ HomA-Alg(C, colim

λ∈ΛAλ). Since, by

assumption, this is known after extending scalars toB and toB ⊗A B, it suffices to show that,for anyA-algebraD, the natural sequence

HomA-Alg(C,D) // HomB-Alg(CB, DB) //// HomB⊗AB-Alg(CB⊗AB, DB⊗AB)

is exact. For this, note thatHomA-Alg(C,D) = HomD-Alg(CD, D) (and similarly for the otherterms) and by hypothesis(D → D ⊗A B)o is a morphism of 1-descent for the fibred categoryw.Eto (resp.Eto).

As a consequence of these observations and of theorem 3.4.37, we see that any finite ringhomomorphismφ : A→ B with nilpotent kernel is of universal effective descent forthe fibredcategory of etale algebras. This fact was known as follows.By [40, Exp.IX, 4.7],Spec(φ) is ofuniversal effective descent for the fibred category of separated etale morphisms of finite type.One has to show that ifX is such a scheme overA, such thatX⊗AB is affine, thenX is affine.This follows by reduction to the noetherian case and [27, Ch.II, 6.7.1].

3.5. Behaviour of etale morphisms under Frobenius.We consider the following categoryBof basic setups. The objects ofB are the pairs(V,m), whereV is a ring andm is an ideal ofVwith m = m2 andm is flat. The morphisms(V,mV )→ (W,mW ) between two objects ofB arethe ring homomorphismsf : V →W such thatmW = f(mV ) ·W .

3.5.1. We have a fibred and cofibred categoryB-Mod → B (see [40, Exp.VI§5,6,10] forgeneralities on fibred categories). An object ofB-Mod (which we may call a “B-module”)consists of a pair((V,m),M), where(V,m) is an object ofB andM is aV -module. Giventwo objectsX := ((V,mV ),M) andY := ((W,mW ), N), the morphismsX → Y are the pairs(f, g), wheref : (V,mV )→ (W,mW ) is a morphism inB andg : M → N is anf -linear map.

3.5.2. Similarly one has a fibred and cofibred categoryB-Alg → B of B-algebras. We willalso need to consider the fibred and cofibred categoryB-Mon→ B of non-unitary commuta-tive B-monoids: an object ofB-Mon is a pair((V,m), A) whereA is aV -module endowedwith a morphismA ⊗V A → A subject to associativity and commutativity conditions, asdis-cussed in section 2.2. The fibre over an object(V,m) of B, is the category ofV -monoidsdenoted(V,m)-Mon or simplyV -Mon.

3.5.3. The almost isomorphisms in the fibres ofB-Mod → B give a multiplicative systemΣ in B-Mod, admitting a calculus of both left and right fractions. The “locally small” condi-tions are satisfied (see [67, p.381]), so that one can form thelocalised categoryBa-Mod :=Σ−1(B-Mod). The fibres of the localised category over the objects ofB are the previ-ously considered categories of almost modules. Similar considerations hold forB-Alg andB-Mon, and we get the fibred and cofibred categoriesBa-Mod → B, Ba-Alg → B and


Ba-Mon → B. In particular, for every object(V,m) of B, we have an obvious notion ofalmostV -monoid and the category consisting of these is denotedV a-Mon.

3.5.4. The localisation functors

B-Mod→ Ba-Mod : M 7→Ma B-Alg→ Ba-Alg : B 7→ Ba

have left and right adjoints. These adjoints can be chosen asfunctors of categories overB suchthat the adjunction units and counits are morphisms over identity arrows inB. On the fibresthese induce the previously considered left and right adjointsM 7→ M!, M 7→ M∗, B 7→ B!!,B 7→ B∗. We will use the same notation for the corresponding functors on the larger categories.Then it is easy to check that the functorM 7→ M! is cartesian and cocartesian (i.e. it sendscartesian arrows to cartesian arrows and cocartesian arrows to cocartesian arrows),M 7→ M∗andB 7→ B∗ are cartesian, andB 7→ B!! is cocartesian.

3.5.5. LetB/Fp be the full subcategory ofB consisting of all objects(V,m) whereV is anFp-algebra. Define similarlyB-Alg/Fp, B-Mon/Fp andBa-Alg/Fp, Ba-Mon/Fp, so thatwe have again fibred and cofibred categoriesBa-Alg/Fp → B/Fp andBa-Alg/Fp → B/Fp(resp. the same for non-unitary monoids). We remark that thecategoriesBa-Alg/Fp andBa-Mon/Fp have small limits and colimits, and these are preserved by the projection toB/Fp.Especially, ifA→ B andA→ C are two morphisms inBa-Alg/Fp or Ba-Mon/Fp, we candefineB ⊗A C as such a colimit.

3.5.6. IfA is a (unitary or non-unitary)B-monoid overFp, we denote byΦA : A → A theFrobenius endomorphism: x 7→ xp. If (V,m) is an object ofB/Fp, it follows from proposition2.1.7(ii) thatΦV : (V,m) → (V,m) is a morphism inB. If B is an object ofB-Alg/Fp (resp.B-Mon/Fp) overV , then the Frobenius map induces a morphismΦB : B → B in B-Alg/Fp(resp. B-Mon/Fp) over ΦV . In this way we get a natural transformation from the identityfunctor ofB-Alg/Fp (resp.B-Mon/Fp) to itself that induces a natural transformation on theidentity functor ofBa-Alg/Fp (resp.Ba-Mon/Fp).

3.5.7. Using the pull-back functors, any objectB of B-Alg overV defines new objectsB(m)

of B-Alg (m ∈ N) overV , whereB(m) := (ΦmV )∗(B), which is justB considered as aV -

algebra via the homomorphismVΦm

−→ V → B. These operations also induce functorsB 7→B(m) on almostB-algebras.

Definition 3.5.8. Let (V,m) be an object ofB/Fp.(i) We say that a morphismf : A → B of almostV -algebras (resp. almostV -monoids) is

invertible up toΦm if there exists a morphismf ′ : B → A in Ba-Alg (resp.Ba-Mon)overΦm

V , such thatf ′ f = ΦmA andf f ′ = Φm

B .(ii) We say that an almostV -monoidI (e.g.an ideal in aV a-algebra) isFrobenius nilpotentif

ΦI is nilpotent.

3.5.9. Notice that a morphismf of V a-Alg (or V a-Mon) is invertible up toΦm if and onlyif f∗ : A∗ → B∗ is so as a morphism ofFp-algebras.

Lemma 3.5.10.Let(V,m) be an object ofB/Fp and letf : A→ B, g : B → C be morphismsof almostV -algebras or almostV -monoids.

(i) If f (resp.g) is invertible up toΦn (resp.Φm), theng f is invertible up toΦm+n.(ii) If f (resp.g f ) is invertible up toΦn (resp.Φm), theng is invertible up toΦm+n.(iii) If g (resp.g f ) is invertible up toΦn (resp.Φm), thenf is invertible up toΦm+n.(iv) The Frobenius morphisms induceΦV -linear morphisms (i.e. morphisms inBa-Mod over

ΦV ) Φ′ : Ker f → Ker f andΦ′′ : Coker f → Coker f , andf is invertible up to somepower ofΦ if and only if bothΦ′ andΦ′′ are nilpotent.


(v) Consider a map of short exact sequences of almostV -monoids :

0 // A′ //

f ′

A //

f

A′′ //

f ′′

0

0 // B′ // B // B′′ // 0

and suppose that two of the morphismsf ′, f, f ′′ are invertible up to a power ofΦ. Thenalso the third morphism has this property.

Proof. (i): if f ′ is an inverse off up toΦn andg′ is an inverse ofg up toΦm, thenf ′ g′ isan inverse ofg f up to Φm+n. (ii): given an inversef ′ of f up to Φn and an inverseh′ ofh := g f up toΦm, let g′ := Φn

B f h′. We compute :

g g′ = g ΦnB f h′ = Φn

C g f h = ΦnC Φm

C

g′ g = ΦnB f h′ g = f h′ g Φn

B = f h′ g f f ′= f Φm

A f ′ = ΦmB f f ′ = Φm

B ΦnB.

(iii) is similar and (iv) is an easy diagram chasing left to the reader. (v) follows from (iv) andthe snake lemma.

Lemma 3.5.11.Let (V,m) be an object ofB/Fp.(i) If f : A → B is a morphism of almostV -algebras, invertible up toΦn, then so isA′ →

A′ ⊗A B for every morphismA→ A′ of almost algebras.(ii) If f : (V,mV )→ (W,mW ) is a morphism inB/Fp, the functors

f∗ : (V,mV )a-Alg→ (W,mW )a-Alg and f ∗ : (W,mW )a-Alg→ (V,mV )a-Alg

preserve the class of morphisms invertible up toΦn.

Proof. (i): given f ′ : B → A(m), construct a morphismA′ ⊗A B → A′(m) using the morphismA′ → A′(m) coming fromΦm

A′ andf ′. (ii): the assertion forf ∗ is clear, and the assertion forf∗follows from (i).

Remark 3.5.12. Statements like those of lemma 3.5.11 hold for the classes offlat, (weakly)unramified, (weakly) etale morphisms.

Theorem 3.5.13.Let (V,m) be an object ofB/Fp andf : A→ B a weaklyetale morphism ofalmostV -algebras.

(i) If f is invertible up toΦn (n ≥ 0), then it is an isomorphism.(ii) For every integerm ≥ 0 the natural square diagram

Af //

ΦmA

B

ΦmB

A(m)

f(m) // B(m)

(3.5.14)

is cocartesian.

Proof. (i): we first show thatf is faithfully flat. Sincef is flat, it remains to show that ifMis anA-module such thatM ⊗A B = 0, thenM = 0. It suffice to do this forM := A/I,for an arbitrary idealI of A. After base change byA → A/I, we reduce to show thatB = 0impliesA = 0. However,A∗ → B∗ is invertible up toΦn, soΦn

A∗= 0 which meansA∗ = 0. In

particular,f is a monomorphism, hence the proof is complete in case thatf is an epimorphism.

In general, consider the compositionB1B⊗f−→ B ⊗A B

µB/A−→ B. From lemma 3.5.11(i) it followsthat1B⊗f is invertible up toΦn; then lemma 3.5.10(ii) says thatµB/A is invertible up toΦn. The


latter is also weakly etale; by the foregoing we derive thatit is an isomorphism. Consequently1B ⊗ f is an isomorphism, and finally, by faithful flatness,f itself is an isomorphism.

(ii): the morphismsΦmA andΦm

B are invertible up toΦm. By lemma 3.5.11(i) it follows that1B ⊗ Φm

A : B → B ⊗A A(m) is invertible up toΦm; hence, by lemma 3.5.10(ii), the morphismh : B⊗AA(m) → B(m) induced byΦm

B andf(m) is invertible up toΦ2m (in fact one verifies thatit is invertible up toΦm). But h is a morphism of weakly etaleA(m)-algebras, so it is weaklyetale, so it is an isomorphism by (i).

Remark 3.5.15. Theorem 3.5.13(ii) extends a statement of Faltings ([34, p.10]) for his notionof almost etale extensions.

3.5.16. We recall (cp. [39, Ch.0, 3.5]) that a morphismf : X → Y of objects in a site iscalledbicoveringif the induced map of associated sheaves of sets is an isomorphism; if f issquarable (“quarrable” in French), this is equivalent to the condition that bothf and the diagonalmorphismX → X ×Y X are covering morphisms.

3.5.17. LetF → E be a fibered category andf : P → Q a squarable morphism ofE.Consider the following condition:

for every base changeP×QQ′ → Q′ of f , the inverse image functorFQ′ → FP×QQ′

is an equivalence of categories.(3.5.18)

Inspecting the arguments in [39, Ch.II,§1.1] one can show:

Lemma 3.5.19.With the above notation, letτ be the topology of universal effective descentrelative toF → E. Then we have :

(i) if (3.5.18)holds, thenf is a covering morphism for the topologyτ .(ii) f is bicovering forτ if and only if (3.5.18)holds both forf and for the diagonal morphism

P → P ×Q P .

Remark 3.5.20. In [39, Ch.II, 1.1.3(iv)] it is stated that “la reciproque est vraie sii = 2”,meaning that (3.5.18) is equivalent to the condition thatf is bicovering forτ . (Actually thecited statement is given in terms of presheaves, but one can show that (3.5.18) is equivalent tothe corresponding condition for the fibered categoryF+ → EU considered inloc.cit.) However,this fails in general : as a counterexample we can give the following. LetE be the categoryof schemes of finite type over a fieldk; setP = A1

k, Q = Spec k. Finally letF → E be thediscretely fibered category defined by the presheafX 7→ H0(X,Z). Then it is easy to showthatf satisfies (3.5.18) but the diagonal map does not, sof is not bicovering. The mistake inthe proof is in [39, Ch.II, 1.1.3.5], where one knows thatF+(d) is an equivalence of categories(notation ofloc.cit.) but one needs it also after base changes ofd.

Lemma 3.5.21. (i) Let f : A→ B be a morphism ofV a-algebras.

(i) If f is invertible up toΦm, then the induced functorsA-Et → B-Et andA-w.Et →B-w.Et are equivalences of categories.

(ii) If f is weaklyetale andC → D is a morphism ofA-algebras invertible up toΦm, then theinduced map:HomA-Alg(B,C)→ HomA-Alg(B,D) is bijective.

Proof. We first consider (i) for the special case wheref := ΦmA : A → A(m). The functor

(ΦmV )∗ : V a-Alg→ V a-Alg induces a functor(−)(m) : A-Alg→ A(m)-Alg, and by restriction

(see remark 3.5.11) we obtain a functor(−)(m) : A-Et → A(m)-Et; by theorem 3.5.13(ii), thelatter is isomorphic to the functor(Φm)∗ : A-Et→ A(m)-Et of the lemma. Furthermore, from


remark 2.1.4(ii) and (2.2.4) we derive a natural ring isomorphismω : A(m)∗ ≃ A∗, hence anessentially commutative diagram

A-Et//

(Φm)∗

A-Algα //

(−)(m)

(A∗,m · A∗)a-Alg

ω∗

A(m)-Et // A(m)-Algβ // (A(m)∗,m · A(m)∗)

a-Alg

whereα andβ are the equivalences of remark 2.2.12. Clearlyα andβ restrict to equivalenceson the corresponding categories of etale algebras, hence the lemma follows in this case.

For the general case of (i), letf ′ : B → A(m) be a morphism as in definition 3.5.8. Di-agram (3.5.14) induces an essentially commutative diagramof the corresponding categoriesof algebras, so by the previous case, the functor(f ′)∗ : B-Et → A(m)-Et has both a leftquasi-inverse and a right quasi-inverse; these quasi-inverses must be isomorphic, sof∗ hasa quasi-inverse as desired. Finally, we remark that the map in (ii) is the same as the mapHomC-Alg(B ⊗A C,C) → HomD-Alg(B ⊗A D,D), and the latter is a bijection in view of(i).

Remark 3.5.22. Notice that lemma 3.5.21(ii) generalises the lifting theorem 3.2.18(i) (in caseV is anFp-algebra). Similarly, it follows from lemmata 3.5.21(i) and 3.5.10(iv) that, in caseV isanFp-algebra, one can replace “nilpotent” in theorem 3.2.18(ii),(iii) by “Frobenius nilpotent”.

3.5.23. In the following,τ will denote indifferently the topology of universal effective descentdefined by either of the fibered categoriesw.Eto → V a-Algo or Eto → V a-Algo.

Proposition 3.5.24. If f : A → B is a morphism of almostV -algebras which is invertible upto Φm, thenf o is bicovering for the topologyτ .

Proof. In light of lemmata 3.5.19(ii) and 3.5.21(i), it suffices to show thatµB/A is invertible up

to a power ofΦ. For this, factor the identity morphism ofB asB1B⊗f−→ B ⊗A B

µB/A−→ B andargue as in the proof of theorem 3.5.13.

Proposition 3.5.25.LetA→ B be a morphism of almostV -algebras andI ⊂ A an ideal. SetA := A/I andB := B/IB. Suppose that either

(a) I → IB is an epimorphism with nilpotent kernel, or(b) V is anFp-algebra andI → IB is invertible up to a power ofΦ.

Then we have :

(i) conditions(a)and(b) are stable under any base changeA→ C.(ii) (A→ B)o is covering (resp. bicovering) forτ if and only if(A→ B)o is.

Proof. Suppose first thatI → IB is an isomorphism; in this case we claim thatIC → I(C ⊗AB) is an epimorphism andKer(IC → I(C⊗AB))2 = 0 for anyA-algebraC. Indeed, since byassumptionI ≃ IB, C ⊗A B acts onC ⊗A I, henceKer(C → C ⊗A B) annihilatesC ⊗A I,hence annihilates its imageIC, whence the claim. If, moreover,V is anFp-algebra, lemma3.5.10(iv) implies thatIC → I(C ⊗A B) is invertible up to a power ofΦ.

In the general case, consider the intermediate almostV -algebraA1 := A ×B B equippedwith the idealI1 := 0 ×B (IB). In case (a),I1 = IA1 andA → A1 is an epimorphism withnilpotent kernel, hence it remains such after any base changeA→ C. To prove (i) in case (a), itsuffices then to consider the morphismA1 → B, hence we can assume from start thatI → IBis an isomorphism, which is the case already dealt with. To prove (i) in case (b), it suffices toconsider the cases of(A, I) → (A1, I1) and(A1, I1) → (B, IB). The second case is treatedabove. In the first case, we do not necessarily haveI1 = IA1 and the assertion to be checked is


that, for everyA-algebraC, the morphismIC → I1(A1⊗A C) is invertible up to a power ofΦ.We apply lemma 3.5.10(v) to the commutative diagram with exact rows:

0 // I //

A //

A/I // 0

0 // IB // A1// A/I // 0

to deduce thatA→ A1 is invertible up to some power ofΦ, hence so isC → A1 ⊗A C, whichimplies the assertion.

As for (ii), we remark that the “only if” part is trivial; and we assume therefore that(A→ B)o

is τ -covering (resp.τ -bicovering). Consider first the assertion for “covering”.We need to showthat(A→ B)o is of universal effective descent forF , whereF is either one of our two fiberedcategories. In light of (i), this is reduced to the assertionthat(A → B)o is of effective descentfor F . We notice that(A → A1)

o is bicovering forτ (in case (a) by theorem 3.2.18 andlemma 3.5.19(ii), in case (b) by proposition 3.5.24). As(A → A1/I1)

o is an isomorphism,the assertion is reduced to the case whereI → IB is an isomorphism. In this case, by lemma3.4.29, there is a natural equivalence:Desc(F,B/A)

∼→ Desc(F,B/A) ×FBFB. Then the

assertion follows easily from corollary 3.4.22. Finally suppose that(A → B)o is bicovering.The foregoing already says that(A→ B)o is covering, so it remains to show that(B ⊗A B →B)o is also covering. The above argument again reduces to the case whereI → IB is anisomorphism. Then, as in the proof of lemma 3.4.29, the induced morphismI(B ⊗A B)→ IBis an isomorphism as well. Thus the assertion for “bicovering” is reduced to the assertion for“covering”.

We conclude this section with a result of a more special nature, which can be interpreted asan easy case of almost purity in positive characteristic.

3.5.26. We suppose now that the basic setup(V,m) consists of aperfectFp-algebraV , i.e.such that the Frobenius endomorphismΦV : V → V is bijective; moreover we assume thatthere exists a non-zero-divisorε ∈ m such thatm =

⋃n>0 ε

1/pmV . Let us denote byV a-Etuafp

(resp.V [ε−1]-u.Etfp) the category of uniformly almost finite projective etaleV a-algebras (resp.of finite etaleV [ε−1]-algebra in the usual sense of [40]). We will be concerned with the naturalfunctor:

V a-Etuafp → V [ε−1]-u.Etfp : A 7→ A∗[ε−1].(3.5.27)

Theorem 3.5.28.Under the assumptions of(3.5.26), the functor(3.5.27)is an equivalence ofcategories.

Proof. Let R be a finite etaleV [ε−1]-algebra. SinceV [ε−1] is perfect, the same holds forR,in view of theorem 3.5.13(ii) (applied in the classical limit case of example 2.1.2(ii)). Let uschoose a finiteV -algebraR0 ⊂ R such thatR0[ε

−1] = R and defineR1 :=⋃n∈N Φ−nR (R0).

Claim3.5.29. TheV a-algebraRa1 does not depend on the choice ofR0.

Proof of the claim:Let R′0 ⊂ R be another finiteV -algebra such thatR′0[ε−1] = R; clearly

we haveεmR0 ⊂ R′0 ⊂ ε−mR0 for m ∈ N sufficiently large. It follows thatεm/pnΦ−nR (R0) ⊂

Φ−nR (R′0) ⊂ ε−m/pnΦ−nR (R0) for everyn ∈ N. The claim readily follows.

Claim3.5.30. Ra1 is an unramifiedV a-algebra.

Proof of the claim:Let e ∈ R ⊗V [ε−1] R be the idempotent provided by proposition 3.1.4; form ∈ N large enough,εm · e is contained in the subringR0 ⊗V R0. Hence, for everyn ∈ N,


εm/pn · e ∈ Φ−nR (R0)⊗V Φ−nR (R0), soe defines an almost element in(R1⊗V R1)

a which fulfillsthe conditions (i)-(iii) of proposition 3.1.4, and the claim follows.

Claim 3.5.31. Ra1 is a uniformly almost finiteV a-algebra.

Proof of the claim:For large enoughm ∈ N we have:R0 ⊂ Φ−1R (R0) ⊂ ε−mR0, therefore

Φ−nR (R0) ⊂ Φ−(n+1)R (R0) ⊂ ε−m/p

nΦ−nR (R0) for everyn ∈ N. By an easy induction we

deduce:Φ−(n+k)R (R0) ⊂

∏kj=0 ε

−m/pn+j · Φ−nR (R0) ⊂ ε−m/pn−1

Φ−nR (R0) for everyn, k ∈ N.

Finally, this implies thatR1 ⊂ ε−m/pn−1

Φ−nR (R0) for everyn ∈ N and the claim follows.

Claim 3.5.32. Let S be the integral closure ofV in R; thenRa1 = Sa.

Proof of the claim: Let us endowR with the unique ring topologyτ such that the inducedsubspace topology onR0 is theε-adic topology andR0 is open inR. It is easy to check thatSconsists of power-bounded elements ofR relative to the topologyτ . Since clearlyR1 ⊂ S, itsuffices therefore to show that(Ra

1)∗ ⊂ R is the subring of all power-bounded elements ofR.However,(Ra

1)∗ can be characterized as the subring of allx ∈ R such thatm · x ⊂ R1; thisalready implies that(Ra

1)∗ consists of power-bounded elements. On the other hand, ifx ∈ Ris power-bounded, it follows thatδ · x is topologically nilpotent for everyδ ∈ m; sinceR0 isopen inR, it follows that, for everyδ ∈ m there existsn0 ∈ N such that(δ · x)n ∈ R0 for everyn > n0. By takingn := pk for sufficiently largek ∈ N, we deduce thatΦk

R(δ · x) ∈ R0, that isδ · x ∈ R1, and the claim follows.

Claim 3.5.33. Ra1 is an almost projectiveV a-algebra.

Proof of the claim:As a special case of claim 3.5.32, letW be the integral closure ofV inV [ε−1]; then:

W a = V a.(3.5.34)

Next, letTrR/V [ε−1] : R → V [ε−1] be the trace map of the finite etale extensionV [ε−1] → R;recall thatTrR/V [ε−1] sends elements integral overV to elements integral overV (to see this, wecan assume thatR has constant rankn overV [ε−1]; then the assertion can be checked after afaithfully flat base changeV [ε−1] → S, so we can further suppose thatR ≃ V [ε−1]n, in whichcase everything is clear); it then follows from claim 3.5.32and (3.5.34) thatTraR/V [ε−1] restrictsto a morphismT : Ra

1 → V a. Furthermore, lete ∈ R ⊗V [ε−1] R be the idempotent definingthe diagonal imbedding; by claim 3.5.30, for everyδ ∈ m we can writeδ · e =

∑ni xi ⊗ yi for

certainxi, yi ∈ R1. By remark 4.1.17 (whose proof does not use theorem 3.5.28) we deducethe identity:δ · b =

∑ni xi · T (b · yi) for everyb ∈ (Ra

1)∗. This allows us to define morphismsα : Ra

1 → (V a)n, β : (V a)n → Ra1 with β α = δ ·1Ra

1, namelyα(b) = (T (b · y1), ..., T (b · yn))

andβ(v1, ..., vn) =∑n

i xi · vi for everyb ∈ Ra1 andv1, ..., vn ∈ V a

∗ . By lemma 2.4.15, the claimfollows.

Claim 3.5.35. The functor (3.5.27) is fully faithful.

Proof of the claim:First of all, it is clear that, for every flatV a-algebrasA, B, the natural map

HomV a-Alg(A,B)→ HomV [ε−1]-Alg(A∗[ε−1], B∗[ε

−1])(3.5.36)

is injective, sinceA∗ ⊂ A∗[ε−1] and similarly forB. Suppose now thatA andB are etale and

almost finite overV a; thenΦA andΦB are automorphisms, due to theorem 3.5.13(ii) and theassumption thatV is perfect. Letψ : A∗[ε

−1]→ B∗[ε−1] be any map ofV [ε−1]-algebras; since

A is almost finite, we haveψ(A∗) ⊂ ε−mB∗ for m ∈ N large enough. Since Frobenius com-mutes with every ring homomorphism, we deduceψ(A∗) = ψ(Φ−nA∗

(A∗)) ⊂ ε−m/pnΦ−nB∗

(B∗) =


ε−m/pnB∗ for everyn ∈ N, soψ induces a morphismψa : A → B, which shows that (3.5.36)

is surjective.It now follows from claims 3.5.29, 3.5.30, 3.5.31, 3.5.33 that the assignmentR→ Ra

1 definesa quasi-inverse to 3.5.27; together with claim 3.5.35, thisconcludes the proof of the theorem.


4. FINE STUDY OF ALMOST PROJECTIVE MODULES

An alternative title for this chapter could have been “Everything you can do with traces”.Right at the outset we find the definition of the trace map of an almost projective almost finitelygeneratedA-module. The whole purpose of the chapter is to showcase the versatility of thisconstruction, a real swiss-knife of almost linear algebra.For instance, we apply it to characterizeetale morphisms (theorem 4.1.14); more generally, it is used to define thedifferent idealof analmost finiteA-algebra. In section 4.3 it is employed in an essential way tostudy the importantclass ofA-modules offinite rank, i.e. those almost projectiveA-modulesP such thatΛi

AP = 0for sufficiently largei ∈ N. A rather complete and satisfactory description is achieved forsuchA-modules (proposition 4.3.27). This is further generalized in theorem 4.3.28, to arbitraryA-modules so called ofalmost finite rank(see definition 4.3.9(ii)). The interest of the latterclass is that it contains basically all the almost projective modules found in nature; indeed, wecannot produce a single example of an almost projective module that is almost finitely generatedbut has not almost finite rank (but we suspect that they do exist). In any case, almost finitelygenerated modules whose rank is not almost finite are certainly rather weird beasts : some clueabout their looks can be gained by analyzing the structure ofinvertiblemodules : we do this atthe end of section 4.4.

The other main construction of chapter 4 is thesplitting algebraof an almost projectivemodule, introduced in section 4.4 : with its aid we show thatA-modules of finite rank arelocally free in the flat topology ofA. It should be clear that this is a very pleasant and usefulculmination for our study of almost projective modules; we put it to use right away in thefollowing section 4.5, where we show that an etale groupoidof finite rank over the categoryof affine almost schemes (more prosaically, the opposite of the category of almost algebras)is universally effective, that is, it admits a good quotient, as in the classical algebro-geometricsetting.

4.1. Almost traces. LetA be aV a-algebra.

Definition 4.1.1. Let P be an almost finitely generated projectiveA-module. ThenωP/A is anisomorphism by lemma 2.4.29(b). Thetrace morphismof P is theA-linear morphism

trP/A := evP/A ω−1P/A : EndA(P )a → A.

We letζP be the unique almost element ofP ⊗A P ∗ such thatωP/A(ζP ) = 1P .

Lemma 4.1.2. LetM , N be almost finitely generated projectiveA-modules, andφ : M → N ,ψ : N →M twoA-linear morphisms. Then :

(i) trM/A(ψ φ) = trN/A(φ ψ).(ii) If ψ φ = a · 1M andφ ψ = a · 1N for somea ∈ A∗, and if, furthermore, there exist

u ∈ EndA(M), v ∈ EndA(N) such thatv φ = φu, thena · (trM/A(u)− trN/A(v)) = 0.

Proof. (i) : by lemma 2.4.29(i), the natural morphismN⊗AalHomA(M,A)→ alHomA(M,N)is an isomorphism (and similarly when we exchange the roles of M andN). ByA-linearity, wecan therefore assume thatφ (resp.ψ) is of the formx 7→ n ·α(x) for somen ∈ N∗, α : M → A(resp. of the formx 7→ m · β(x) for somem ∈ M∗, β : N → A). Then a simple computationyields:

φ ψ = ωN/A(n · α(m)⊗ β) ψ ψ = ωM/A(m · β(n)⊗ α)

and the claim follows directly from the definition of the trace morphism. For (ii) we computeusing (i) :a·trM/A(u) = trM/A(ψφu) = trM/A(ψvφ) = trN/A(vφψ) = a·trN/A(v).

Lemma 4.1.3. LetM ,N be two almost finitely generated projectiveA-modules,φ ∈ EndA(M)andψ ∈ EndA(N). ThentrM⊗AN/A(φ⊗ ψ) = trM/A(φ) · trN/A(ψ).


Proof. As usual we can suppose thatφ = ωM/A(m⊗ α), ψ = ωN/A(n⊗ β) for someα ∈ M∗,β ∈ N∗. Thenφ ⊗ ψ = ωM⊗AN/A((m ⊗ n) ⊗ (α ⊗ β)) and the sought identity follows byexplicit calculation.

Proposition 4.1.4. Let M = (0 → M1i→ M2

p→ M3 → 0) be an exact sequence of al-most finitely generated projectiveA-modules, and letu = (u1, u2, u3) : M → M be anendomorphism ofM , given by endomorphismsui : Mi → Mi (i = 1, 2, 3). Then we havetrM2/A(u2) = trM1/A(u1) + trM3/A(u3).

Proof. Suppose first that there exists a splittings : M3 → M2 for p, so that we can viewu2 as a

matrix

(u1 v0 u3

), wherev ∈ HomA(M3,M1). By additivity of the trace, we are then reduced

to show thattrM2/A(i v p) = 0. By lemma 4.1.2(i), this is the same astrM3/A(p i v),which obviously vanishes. In general, for anya ∈ m we consider the morphismµa = a · 1M3

and the pull back morphismM ∗ µa →M :

0 // M1i // M2

p // M3// 0

0 // M1// P

p′ //

φ

OO

M3//

µa

OO

0.

Pick a morphismj : M3 → M2 such thatp j = a · 1M3; the pair(j, 1M3) determines amorphismσ : M3 → P such thatσ p′ = 1M3, i.e. the sequenceM ∗ µa is split exact; thissequence also inherits the endomorphismu ∗ µa = (u1, v, u3), for a certainv ∈ EndA(P ).The pair of morphisms(a · 1M2, p) determines a morphismψ : M2 → P , and it is easy tocheck thatφ ψ = a · 1M2 andψ φ = a · 1P . We can therefore apply lemma 4.1.2 todeduce thata · (trP/A(v) − trM/A(u2)) = 0. By the foregoing we know thattrP/A(v) =trM1/A(u1) + trM3/A(u3), so the claim follows.

Lemma 4.1.5. LetA be aV a-algebra.

(i) If P := M ⊗A N is an almost projective and faithful (resp. and almost finitely generated)A-module, then so areM andN .

(ii) If M ⊗A N ≃ A, then the evaluation mapevM : M ⊗AM∗ → A is an isomorphism.(iii) An invertibleA-module is faithful and almost finitely generated projective.(iv) An epimorphismφ : M → N of invertibleA-modules is an isomorphism.

Proof. Clearly (iii) is just a special case of (i). We show (i): by proposition 2.4.28(iv) we knowthatEP/A = A; however, one checks easily thatEP/A ⊂ EN/A, whence

EN/A = A.(4.1.6)

ThereforeN will be faithful, as soon as it is shown to be almost projective, again by virtueof proposition 2.4.28(iv). In any case, (4.1.6) means that,for everyε ∈ m, we can find analmost element of the form

∑ni=1 xi ⊗ φi ∈ N ⊗A N∗, such that

∑ni=1 φi(xi) = ε. We use

such an element to define morphismsA → Nn → A whose composition equalsε · 1N . Aftertensoring byM , we obtain morphismsM → P →M whose composition isε ·1M . Then, sinceP is almost projective, it follows easily that so must beM ; similarly, if P is almost finitelygenerated, the same follows forM . By symmetry, the same holds forN .

(ii): notice that, by (i), we know already thatM andN are almost finitely generated pro-jective. By lemma 4.1.3 we deduce thattrM/A(1M) · trN/A(1N) = 1, so both factors are in-vertible inA∗. It follows that the morphismA → EndA(M) given bya 7→ a · 1M providesa splitting for the trace morphism (and similarly forN in place ofM). Thus we can writeEndA(M) ≃ A ⊕X, EndA(N) ≃ A ⊕ Y for someA-modulesX, Y . However, on one hand


we have a natural isomorphismEndA(M) ⊗A EndA(N) ≃ A; on the other hand, we have adecompositionEndA(M)⊗A EndA(N) ≃ A⊕X ⊕ Y ⊕ (X ⊗A Y ); working out the identifi-cations, one sees that the induced isomorphismA ⊕X ⊕ Y ⊕ (X ⊗A Y ) ≃ A restricts to theidentity morphism on the direct summandA; it follows thatX = Y = 0, which readily impliesthe claim.

(iv): in view of (ii) we can replaceφ by φ ⊗A 1M∗, and thereby assume thatM = A. ThenN ≃ A/Ker(φ); it is clear that such a module is faithful if and only ifKer(φ) = 0. By (iii), theclaim follows.

Lemma 4.1.5 explains why we do not insist, in the definition ofan invertibleA-module, thatit should be almost projective or almost finitely generated:both conditions can be deduced.

4.1.7. Suppose now thatB is an almost finite projectiveA-algebra. For anyb ∈ B∗, denoteby µb : B → B theB-linear morphismb′ 7→ b · b′. The mapb 7→ µb defines aB-linearmonomorphismµ : B → EndA(B)a. The composition

TrB/A := trB/A µ : B → A

will also be called the almost trace morphism of theA-algebraB.

Proposition 4.1.8. LetA andB be as in(4.1.7).

(i) If φ : A→ B is an isomorphism, thenTrB/A = φ−1.(ii) If C any otherA-algebra, thenTrC⊗AB/C = 1C ⊗A TrB/A.

(iii) If C is an almost finite projectiveB-algebra, thenTrC/A = TrB/A TrC/B.

Proof. (i) and (ii) are left as exercises for the reader. We verify (iii). It comes down to checkingthat the following diagram commutes:

C ⊗B alHomB(C,B) //

evC/B

C ⊗B alHomA(C,B)∼ // C ⊗A alHomA(C,A)

evC/A

B

TrB/A // A.

Therefore, pickc ∈ C∗ andφ ∈ HomB(C,B). For everyε ∈ m we can find elementsb1, ..., bk ∈B∗ andφ1, ..., φk ∈ HomA(C,A) such thatε · φ(x) =

∑i bi · φi(x) for everyx ∈ C∗. TheB-

linearity ofφ translates into the identity:∑

i

bi · φi(b · x) =∑

i

b · bi · φi(x) for all b ∈ B∗, x ∈ C∗.(4.1.9)

Thenε · evC/B(c⊗ φ) =∑

i bi · φi(c) and we need to show that

TrB/A(∑

i

bi · φi(c)) =∑

i

φi(c · bi).(4.1.10)

For everyi ≤ k, let µi : A → B be the morphisma 7→ bi · a (for all a ∈ A∗); furthermore, letµc : B → C be the morphismb 7→ c · b (for all b ∈ B∗). In view of (4.1.9), the left-hand side of(4.1.10) is equal totrB/A(

∑i µi φi µc). By lemma 4.1.2(i), we havetrB/A(µi φi µc) =

trA/A(φi µc µi) = φi(c · bi) for everyi ≤ k. The claim follows.

Corollary 4.1.11. LetA→ B be a faithfully flat almost finitely presented andetale morphismof almostV -algebras. ThenTrB/A : B → A is an epimorphism.

Proof. Under the stated hypotheses,B is an almost projectiveA-module (by proposition 2.4.18).Let C = Coker(TrB/A) andTrB/B⊗AB the trace morphism for the morphism of almostV -algebrasµB/A. By faithful flatness, the natural morphismC → C ⊗A B = Coker(TrB⊗AB/B)is a monomorphism, hence it suffices to show thatTrB⊗AB/B is an epimorphism (hereB ⊗A B


is considered as aB-algebra via the second factor). However, from proposition4.1.8(i) and (iii)we see thatTrB/B⊗AB is a right inverse forTrB⊗AB/B. The claim follows.

4.1.12. It is useful to introduce theA-linear morphism

tB/A := TrB/A µB/A : B ⊗A B → A.

We can viewtB/A as a bilinear form; it induces anA-linear morphism

τB/A : B → B∗ = alHomA(B,A)

characterized by the equalitytB/A(b1⊗ b2) = τB/A(b1)(b2) for all b1, b2 ∈ B∗. We say thattB/Ais a perfect pairingif τB/A is an isomorphism.

Lemma 4.1.13.Let A → B be an almost finite projective morphism ofV a-algebras andCanyA-algebra. Denote byηB,C : C ⊗A alHomA(B,A) → alHomC(C ⊗A B,C) the naturalisomorphism provided by lemma2.4.31(i). Then :

(i) τB/A isB-linear (for the naturalB-module structure ofB∗ defined in remark2.4.25);(ii) ηB,C isC ⊗A B-linear;(iii) ηB,C (1C ⊗ τB/A) = τC⊗AB/C .

Proof. For anyb ∈ B∗, let ξb : B → A theA-linear morphism defined by the ruleb′ 7→TrB/A(b′ · b) for all b′ ∈ B∗. Then, directly from the definition we can compute:(ηB,C (1C ⊗τB/A))(c⊗ b)(c′⊗ b′) = ηB,C(c⊗ ξb)(c′⊗ b′) = c · c′ ·TrB/A(b′ · b) for all b, b′ ∈ B∗, c, c′ ∈ C∗.But by proposition 4.1.8(ii), the latter expression can be rewritten asτC⊗AB/C(c ⊗ b)(c′ ⊗ b′),which shows (iii). The proofs of (i) and (ii) are similar direct verifications: we show (i) andleave (ii) to the reader. Let us pick anyb, b′, b′′ ∈ B∗; then(b · τB/A(b′))(b′′) = τB/A(b′)(bb′′) =TrB/A(bb′b′′) = (τB/A(bb′))(b′′).

Theorem 4.1.14.An almost finite projective morphismφ : A → B of almostV -algebras isetale if and only if the trace formtB/A is a perfect pairing.

Proof. By lemma 4.1.13, we have a commutative diagram:

(B ⊗A B)⊗B B ∼ //

1B⊗AB⊗BτB

B ⊗A B1B⊗AτB

τB⊗AB/B

))RRRRRRRRRRRRRR

(B ⊗A B)⊗B B∗ ∼ // B ⊗A B∗ηB,B // alHomB(B ⊗A B,B)

(4.1.15)

in which all the morphisms areB⊗AB-linear (here we take theB-module structure onB⊗ABgiven by multiplication on the right factor). Suppose now that φ is etale; then, by corollary3.1.9, there is an isomorphism ofB-algebras:B ⊗A B ≃ IB/A ⊕B. It follows thatτB⊗AB/B =τB/B ⊕ τIB/A/B. Especially,1B ⊗B⊗AB τB⊗AB/B is the identity morphism ofB (by proposition4.1.8(i)). This means that in the diagramB⊗B⊗AB (4.1.15) all the arrows are isomorphisms. Inparticular,τB/A is an isomorphism, as claimed.

To prove the converse, we consider the almost elementζB of theB ⊗A B-moduleB ⊗AB∗. Viewing B∗ as aB-module in the natural way (cp. remark 2.4.25), we also get a scalarmultiplication morphismσB∗/B : B ⊗A B∗ → B∗ (see (2.2.5)).

Claim4.1.16. With the above notation we have:IB/A · ζB = 0 andσB∗/B(ζB) = TrB/A.

Proof of the claim:Notice thatωB/A is alsoB ⊗A B-linear for theB ⊗A B-module structureon EndA(B) such that((b ⊗ b′) · φ)(b′′) = b′ · φ(b · b”) for every b, b′, b” ∈ B∗ and everyφ ∈ EndA(B). We computeωB/A((b ⊗ b′) · ζB)(b”) = ((b ⊗ b′) · ωB/A(ζB))(b”) = b · b′ · b”.Whencex · ζB = µB/A(x) · ζB for everyx ∈ B ⊗A B∗ which implies the first claimed identity.Next we compute:σB∗/B(ζB)(b) = evB((1 ⊗ b) · ζB) = (trB/A ωB/A)((1 ⊗ b) · ζB) =


trB/A((1 ⊗ b) · ωB/A(ζB)) = trB/A((1 ⊗ b) · 1B) = TrB/A(b) for everyb ∈ B∗. The claimfollows.

Suppose now thatτB/A is an isomorphism. Then we can definee := (1B ⊗ τ−1B/A)(ζB). From

claim 4.1.16 and lemma 4.1.13(i) we derive thatIB/A · e = 0 andτB/A(σB/B(e)) = TrB/A.The latter equality implies thatσB/B(e) = 1, in other wordsµB/A(e) = 1. We see thereforethate satisfies conditions (ii) and (iii) of proposition 3.1.4 andtherefore also condition (i), sincethe latter is an easy consequence of the other two. ThusA → B is an etale morphism, asclaimed.

Remark 4.1.17. By inspection of the proof of theorem 4.1.14, we see that the following hasbeen shown. LetA→ B be an etale morphism ofV a-algebras. Then(1B ⊗ τB/A)(eB/A) = ζB.

Definition 4.1.18. Thenilradical of an almost algebraA is the idealnil(A) = nil(A∗)a (where,

for a ringR, we denote bynil(R) the ideal of nilpotent elements inR). We say thatA is reducedif nil(A) ≃ 0.

4.1.19. Notice that, ifR is a V -algebra, then every nilpotent ideal inRa is of the formIa,whereI is a nilpotent ideal inR (indeed, it is of the formIa whereI is an ideal, andm · I isseen to be nilpotent). It follows easily thatnil(A) is the colimit of the nilpotent ideals inA;moreovernil(R)a = nil(Ra). Using this one sees thatA/nil(A) is reduced.

Proposition 4.1.20.LetA → B be anetale almost finitely presented morphism of almost al-gebras. IfA is reduced thenB is reduced as well.

Proof. Under the stated hypothesis,B is an almost projectiveA-module (by virtue of proposi-tion 2.4.18(ii)). Hence, for givenε ∈ m, pick a sequence of morphismsB

uε→ Anvε→ B such

thatvε uε = ε · 1B; let µb : B → B be multiplication byb ∈ B∗ and defineνb : An → An

by νb = vε µb uε. One verifies easily thatνmb = εm−1 · νbm for all integersm > 0. Now,suppose thatb ∈ nil(B∗). It follows thatbm = 0 for m sufficiently large, henceνmb = 0 for msufficiently large. Letp be any prime ideal ofA∗; let π : A∗ → A∗/p be the natural projectionandF the fraction field ofA∗/p. TheF -linear morphismνb∗ ⊗A∗ 1F is nilpotent on the vectorspaceF n, henceπ trAn

∗ /A∗(νb∗) = trFn/F (νb∗ ⊗A∗ 1F ) = 0. This shows thattrAn∗/A∗(νb∗)

lies in the intersection of all prime ideals ofA∗, hence it is nilpotent. Since by hypothesisA isreduced, we gettrAn

∗ /A∗(νb∗) = 0, whencetrAn/A(νb) = 0. Using lemma 4.1.2(i) we deduceε · trB/A(µb) = 0, and finally,trB/A(b) = 0. Now, for anyb′ ∈ B∗, the almost elementbb′ willbe nilpotent as well, so the same conclusion applies to it. This shows thatτB/A(b) = 0. But byhypothesisB is etale overA, hence theorem 4.1.14 yieldsb = 0, as required.

Remark 4.1.21. LetM be anA-module. We say that an almost elementa of A isM-regularif the multiplication morphismm 7→ am : M → M is a monomorphism. Assume (A) (see(2.1.6)) and suppose furthermore thatm is generated by a multiplicative systemS which is acofiltered semigroup under the preorder structure(S ,≻) induced by the divisibility relation inV . We say thatS is archimedeanif, for all s, t ∈ S there existsn > 0 such thatsn ≻ t.Suppose thatS is archimedean and thatA is a reduced almost algebra. ThenS consists ofA-regular elements. Indeed, by hypothesisnil(A∗)

a = 0; since the annihilator ofS in A∗ is 0we getnil(A∗) = 0. Suppose thats ·a = 0 for somes ∈ S anda ∈ A∗. Let t ∈ S be arbitraryand pickn > 0 such thattn ≻ s. Then(ta)n = 0 henceta = 0 for all t ∈ S , hencea = 0.

Definition 4.1.22. Let φ : A→ B be an almost finite projective morphism ofV a-algebras. By(4.1.12), we can assign toφ aB-linear trace morphismτB/A : B → B∗. Thedifferent idealofthe morphismφ is the idealDB/A := AnnB(Coker τB/A) ⊂ B.


Lemma 4.1.23.LetM1φ→M2

ψ→ M3 be twoA-linear morphisms of invertibleA-modulesMi

(i ≤ 3) andC anA-algebra. Then:

(i) AnnA(Coker(ψ φ)) = AnnA(Coker φ) · AnnA(Cokerψ).(ii) AnnC(Coker(1C ⊗A ψ)) = C · AnnA(Cokerψ).

Proof. (i): Since, by lemma 4.1.5(iii),M3 is faithfully flat, lemma 2.4.6 yields

AnnA(Cokerψ) = AnnA(Coker(ψ ⊗A 1M∗3))(4.1.24)

and likewise forφ; hence we can replaceMi byMi⊗AM∗3 and suppose thatM3 = A. Moreover,sinceM2 is invertible,evM2/A is an isomorphism, by lemma 4.1.5(ii). LetevM2/A : M∗2 ⊗AM2 → A be the map given by the rule:φ ⊗ x 7→ evM2/A(x ⊗ φ), for everyφ ∈ (M∗2 )∗ andx ∈M2. Setλ := evM2/A(φ⊗A1M∗

2) : M1⊗AM∗2 → A; thenφ(1M1⊗A evM2/A) = λ⊗A1M2,

so thatAnnA(φ) = AnnA(λ ⊗A 1M2) = AnnA(λ). Thus, we can replaceφ by λ ⊗A 1M2 andthen we have to show that

AnnA(Cokerψ (λ⊗A 1M2)) = AnnA(Cokerψ) · AnnA(Coker λ).

However, quite generally we have:

AnnA(Coker(M → A)) = Im(M → A)(4.1.25)

for anyA-linear morphismM → A. Hence we compute:AnnA(Cokerψ (λ ⊗A 1M2)) =Im(ψ(λ⊗A1M2)) = ψ(Im(λ⊗A1M2)) = ψ(Im(λ)·M2) = Im(λ)·Im(ψ) = AnnA(Coker λ)·AnnA(Cokerψ).

(ii): again, using (4.1.24) we reduce to the case whereM3 = A; then the claim follows easilyfrom (4.1.25).

Lemma 4.1.26.Letφ : A → B be a morphism ofV a-algebras as in definition4.1.22. LetCbe anA-algebra. Suppose that eitherC is flat overA, or B∗ is an invertibleB-module for itsnaturalB-module structure. ThenDC⊗AB/C = DB/A · (C ⊗A B).

Proof. Under the stated assumptions,alHomA(B,A) is an almost finitely generated projectiveA-module. In particular,Coker τB/A is almost finitely generated; IfC is flat overA, it followsby lemma 2.4.6 thatAnnC⊗AB(C⊗ACoker τB/A) = DB/A · (C⊗AB); if B∗ is an invertibleB-module, the same holds by virtue of lemma 4.1.23(ii). However, by lemma 4.1.13(iii), the tracepairing is preserved under arbitrary base changes, so:C⊗ACoker τB/A ≃ Coker(1C⊗AτB/A) ≃Coker τC⊗A/B, which shows the claim.

Proposition 4.1.27.Let B → C be a morphism ofA-algebras, and suppose thatB (resp.C) is an almost finite projectiveA-algebra (resp.B-algebra). Suppose moreover thatB∗ :=alHomA(B,A) (resp.C∗ := alHomB(C,B)) is an invertibleB-module (resp.C-module) forits naturalB-module (resp.C-module) structure. Then

DC/A = DC/B ·DB/A.

Proof. LetC∗/A := alHomA(C,A) and define aC-linear morphismξ : alHomB(C,B∗)→ C∗/Aby the rule:φ 7→ (c 7→ φ(c)(1)) for everyφ ∈ HomB(C,B∗) andc ∈ C∗.Claim4.1.28. C∗/A is an invertibleC-module andξ is an isomorphism.

Proof of the claim:By lemma 2.4.29(i), the natural morphismλ : C∗⊗BB∗ → alHomB(C,B∗)is aC-linear isomorphism. It suffices therefore to show thatξ λ−1 : C∗ ⊗B B∗ → C∗/A is anisomorphism. One verifies easily thatξ λ−1 is defined by the rule:φ ⊗ ψ 7→ ψ φ, and thenthe claim follows from lemma 2.4.31(iii).


Unwinding the definitions, one verifies that the following diagram commutes:

CτC/A //

τC/B

C∗/A

C∗alHomB(C,τB/A)

// alHomB(C,B∗).

ξ

OO

Thus, taking into account claim 4.1.28, and lemma 4.1.23(i), we haveAnnC(Coker τC/A) =AnnC(Coker τC/B)·AnnC(Coker(alHomB(C, τB/A))). However,Coker(alHomB(C, τB/A)) ≃C∗⊗B Coker τB/A by lemma 2.4.29(b). By lemma 4.1.5(iii),C∗ is faithfully flat; consequently:

AnnC(Coker(alHomB(C, τB/A))) = AnnC(Coker τB/A)

which implies the assertion.

Lemma 4.1.29.Letφ : A→ B be a morphism ofV a-algebras as in definition4.1.22. Supposemoreover thatB∗ is an invertibleB-module for its naturalB-module structure. Thenφ is etaleif and only ifDB/A = B.

Proof. By theorem 4.1.14 it follows easily thatDB/A = B wheneverφ is etale. Conversely,suppose thatDB/A = B; it then follows thatτB/A is an epimorphism. Again by theorem 4.1.14,we need only show thatτB/A is an isomorphism. This follows from lemma 4.1.5(iv).

The following lemma will be useful when we will compute the different ideal in situationssuch as those contemplated in proposition 6.3.11.

Lemma 4.1.30.LetA be aV a-algebra,B an almost finite almost projectiveA-algebra, andlet Bα | α ∈ J be a net ofA-subalgebras ofB, withBα almost finite projective overA foreveryα ∈ J , such thatlim

α∈JBα = B in IA(B). Thenlim

α∈JDBα/A = DB/A.

Proof. For givenα ∈ J , let ε ∈ V such thatεB ⊂ Bα; lemma 4.1.2(ii) implies thatε ·TrBα/A(b) = ε · TrB/A(b) for everyb ∈ Bα∗. Hence the diagrams:

Bµε //

ε·τB/A

Bα

τBα/A

Bα

τBα/A

// B

ε·τB/A

B∗ // B∗α B∗α

µ∗ε // B∗

commute. The rightmost diagram implies thatDBα/A · Im µ∗ε ⊂ Im(ε · τB/A) ⊂ Im τB/A. Henceε · DBα/A ⊂ DB/A, so finally AnnV (B/Bα) · DBα/A ⊂ DB/A. From the leftmost diagramwe deduce thatε · DB/A (which is an ideal inBα) annihilatesCoker(ε · τB/A : B → B∗)and on the other handAnnV (B/Bα) obviously annihilatesCoker(B∗ → B∗α); we deduce thatAnnV (B/Bα)

2 ·DB/A ⊂ DBα/A, whence the claim.

4.2. Endomorphisms of Gm. This section is dedicated to a discussion of the universal ringthat classifies endomorphisms of the formal groupGm. The results of this section will be usedin sections 4.3 and 4.4.

4.2.1. For every ringR and every integern ≥ 0 we introduce the ”n-truncated” version ofGm,R. This is the schemeGm,R(n) := SpecR[T ]/(T n+1), endowed with the multiplicationmorphism which is associated to the co-multiplication map

R[T ]/(T n+1)→ R[T, S]/(T, S)n+1 T 7→ T + S + T · S.Then in the category of formal schemes we have a natural identificationGm,R ≃ colim

n∈NGm,R(n).


4.2.2. In the terminology of [52,§II.4], Gm(n) is then-bud of Gm. We will be mainly in-terested in the endomorphisms ofGm(n), but before we can get to that, we will need somecomplements on buds over artinian ring. Therefore, supposewe have a cartesian diagram ofartinian rings

R3//

R1

R2

// R0

(4.2.3)

such that one of the two mapsRi → R0 (i = 1, 2) is surjective. For any ringR, we definethe categoryBud(n, d, R) of n-buds overR whose underlyingR-algebra is isomorphic toR[T1, ..., Td]/(T1, ..., Td)

n+1.

Lemma 4.2.4. LetS be a finite flat augmented algebra over a local noetherian ringR; let I bethe augmentation ideal, and suppose thatIn+1 = 0. Letκ be the residue field ofR, and supposethatS ⊗R κ ≃ κ[t1, ..., td]/(t1, ..., td)

n+1. ThenS ≃ R[T1, ..., Td]/(T1, ..., Td)n+1.

Proof. Let ε : S → R be the augmentation map. For everyi = 1, ..., d, pick a liftingT ′i ∈ S ofti; setTi = T ′i − ε(T ′i ). By Nakayama’s lemma, the monomialsT a11 · ... · T ad

d with∑d

i=1 ai ≤ ngenerate theR-moduleS. Furthermore, under the stated hypothesis,S is a freeR-module, andits rank is equal todimκ S⊗R κ; hence the above monomials form anR-basis ofS. Clearly theelementsTi lie in the augmentation ideal ofS, therefore every product ofn + 1 of them equalszero; in other words, the natural morphismR[X1, ..., Xd] → S given byXi 7→ Ti is surjective,with kernel containingJ := (X1, ..., Xd)

n+1; but by comparing the ranks overR we see thatthis kernel cannot be larger thanJ . The assertion follows.

Proposition 4.2.5. In the situation of(4.2.3), the natural functor

Bud(n, d, R3)→ Bud(n, d, R1)×Bud(n,d,R0) Bud(n, d, R2)

is an equivalence of categories.

Proof. Let Mi,proj (i = 0, ..., 3) be the category of projectiveRi-modules. By our previousdiscussion on descent, we already know that (4.2.3) inducesa natural equivalence betweenM3,proj and the2-fibered productM1,proj×M0,proj

M2,proj. It is easy to see that this equivalencerespects the rank ofRi-modules, hence induces a similar equivalence for the categoriesMi,f.f. offreeRi-modules of finite rank. Given two objectsM := (M1,M2, α : M1⊗R1R0

∼→ M2⊗R2R0)andN := (N1, N2, β : N1 ⊗R1 R0

∼→ N2 ⊗R2 R0), define the tensor productM ⊗ N :=(M1⊗R1 N1,M2⊗R2 N2, α⊗R0 β). Then one checks easily that the above equivalences respecttensor products. It follows formally that one has analogousequivalences for the categories offinite flatRi-algebras. From there, one further obtains equivalences onthe categories of suchRi-algebras that are augmented overRi, and even on the subcategoriesRi-Alg

(n)aug.fl. of those

augmentedRi-algebras such that the(n+1)-th power of the augmentation ideal vanishes. Thesecategories admit finite coproducts, that are constructed asfollows. For augmentedRi-algebrasεA : A → Ri andεB : B → Ri, set(A → Ri) ⊗ (B → Ri) := (A ⊗Ri

B/Ker(εA ⊗Ri

εB)n+1 → Ri); this is a coproduct ofA andB. By formal reasons, the foregoing equivalencesof categories respect these coproducts. Finally, an objectof Bud(n, d, Ri) can be defined as acommutative group object in(Ri-Alg

(n)aug.fl.)

o, such that its underlyingRi-algebra is isomorphictoRi[T1, ..., Td]/(T1, ..., Td)

n+1. By formal categorical considerations we see that the foregoingequivalence induces equivalences on the commutative groupobjects in the respective categories.It remains to check that anR3-algebraS such thatS ⊗R3 Ri ≃ Ri[T1, ..., Td]/(T1, ..., Td)

n+1,(for i = 1, 2) is itself of the formRi[T1, ..., Td]/(T1, ..., Td)

n+1. However, this follows readilyfrom lemma 4.2.4 and the fact that one of the mapsR3 → Ri (i = 1, 2) is surjective.


4.2.6. For a given ringR, the endomorphisms ofGm,R(n) are all the polynomialsf(T ) :=a0+a1 ·T+...+an ·T n such thatf(T )+f(S)+f(T )·f(S)≡ f(T+S+T ·S) (mod(T, S)n+1).This relationship translates into a finite set of polynomialidentities for the coefficientsa0, ..., an,and using these identities we can therefore define a quotientGn of the ring inn indeterminatesZ[X1, ..., Xn] which will be the “universal ring of endomorphisms” ofGm(R), i.e., such thatX1 ·T +X2 ·T 2 + ...+Xn ·T n is an endomorphism ofGm,Gn(n) and such that, for every ringR,and everyf(T ) as above, the mapZ[X1, ..., Xn] → R given byXi 7→ ai (i = 1, ..., n) factorsthrough a (necessarily unique) mapGn → R. One of the main results of this section will be asimple and explicit description of the ringGn.

Proposition 4.2.7. Gn is a smoothZ-algebra.

Proof. We know already thatGn is of finite type overZ, therefore it suffices to show that, forevery prime idealp of Gn, the local ringGn,p is formally smooth for thep-adic topology (see[30, Ch.IV, Prop.17.5.3]). Therefore, letR1 → R0 be a surjective homomorphism of localartinian rings; we need to show that the natural mapEnd(Gm,R1(n)) → End(Gm,R0(n)) issurjective. Letf ∈ End(Gm,R0(n)); we define an automorphismχ of Gm,R0 ×R0 Gm,R0 :=

R0[T, S]/(T, S)n+1, then-bud of Gm × Gm, by setting(T, S) 7→ (T, f(T ) + S + f(T ) · S).Then, thanks to proposition 4.2.5, we obtain ann-budXn overR2 := R1 ×R0 R1, by gluingtwo copies ofGm,R1 ×Gm,R1 along the automorphismχ.

Claim 4.2.8. Then-budXn is isomorphic toGm,R2 × Gm,R2 if and only if χ lifts to an auto-morphism ofGm,R1 ×Gm,R1 .

Proof of the claim:Taking into account the decription ofB(n, d, R2) as2-fibered product ofcategories, the proof amounts to a simple formal verification, which is best left to the reader.

Claim 4.2.9. There exists a compatible system ofk-budsXk overR2 for everyk > n, such thatXk reduces toXk−1 overR2, and specializes toG2

m,R1(k) over the quotientR1 of R2.

Proof of the claim:In caseR2 is a torsion-freeZ-algebra, this follows from [52, Ch.II,§4.10]and an easy induction. IfR2 is a general artinian ring, choose a torsion-freeZ-algebraR3 with asurjective homomorphismR3 → R2. By loc. cit. (and an easy induction) we can find ann-budYn overR3 such thatYn specialises toXn on the quotientR2, andYn reduces toG2

m,R3(1) over

R3. Then, again byloc. cit, we can find a compatible system ofk-budsYk on R3 for everyk > n, such thatYk reduces toYk−1 overR3 and specializes toG2

m,R1(k) over the quotientR1

of R3. The claim holds if we takeXk equal to the specialization ofYk overR2.

The direct limit (in the category of formal schemes) of the system(Xk)k≥n is a formal groupX overR2, such thatX⊗R2 R1 ≃ Gm,R1 × Gm,R1 . This formal group gives rise to ap-divisiblegroup(X(n))n≥0, whereX(n) is the kernel of multiplication bypn in X. For everym ∈ N,X(m) is a finite flat group scheme overR2, such thatX(m)×R2 R1 ≃ µpm,R1×µpm,R1. Denoteby X(m)∗ the Cartier dual ofX(m) (cp. [56,§III.14]). ThenX(m)∗×R2 R1 ≃ (Z/pmZ)2

R1, in

particular it hasp2m connected components. Since the pair(R2, R1) is henselian, it follows thatX(m)∗ must havep2m connected components as well, and consequentlyX(m)∗ ≃ (Z/pmZ)2

R2.

Finally, this shows thatX(n) ≃ µpm,R2 × µpm,R2 , whenceX ≃ Gm,R2 × Gm,R2 . Fromclaim 4.2.8, we deduce thatχ lifts to an automorphismχ′ of Gm,R1(n) × Gm,R1(n). Leti : Gm,R1(n)→ Gm,R1(n)×Gm,R1(n), π : Gm,R1(n)×Gm,R1(n)→ Gm,R1(n) be respectivelythe imbedding of the first factor, and the projection onto thesecond factor; clearlyπ χ′ iyields a lifting off(T ), as required.

4.2.10. Next, let us remark that, for everyn ∈ N, the polynomial(1 + T )X − 1 := X · T +(X2

)·T 2 + ...+

(Xn

)·T n ∈ Q[X, T ] is an endomorphism ofGm,Q[X](n). As a consequence, there


is a unique ring homomorphismGn → Z[X,(X2

), ...,

(Xn

)] representing this endomorphism. The

following theorem will show that this homomorphism is an isomorphism.

Theorem 4.2.11.The functor

Z-Alg→ Set R 7→ EndR(Gm,R(n))

is represented by the ringZ[X,(X2

), ...,

(Xn

)].

Proof. The above discussion has already furnished us with a naturalsurjective mapρ : Gn →Z[X,

(X2

), ...,

(Xn

)]. Therefore, it suffices to show that this map is injective.

Claim4.2.12. ρ⊗Z 1Q is an isomorphism.

Proof of the claim:First of all, the mapρ can be characterized in the following way. The identitymapGn → Gn determines an endomorphismf(T ) := a0 + a1 · T + ... + an · T n of Gm,Gn(n);thenρ is the unique ring homomorphism such thatρ(f) := f(a0)+f(a1) ·T + ...+f(an) ·T n =(1 + T )X − 1. On the other hand, the ringGn ⊗Z Q represents endomorphisms ofGm(n) inthe category ofQ-algebras. However, for everyn ∈ N and for everyQ-algebraR, there is anisomorphism

log : Gm,R(n)∼−→ Ga,R(n)

to then-bud of the additive formal groupGa,R. The endomorphism group ofGa,R(n) is easilycomputed, and found to be isomorphic toR. In other words, the universal ring representing en-domorphisms ofGa(n) overQ-algebras is justQ[X], and the bijectionHomQ-Alg(Q[X], R) ≃End(Ga,R(n)) assigns to a homomorphismφ : Q[X] → R, the endomorphismgφ(T ) :=φ(X) ·T . It follows that, for anyQ-algebraR there is a natural bijectionHomQ-Alg(Q[X], R) ≃End(Gm,R(n)) given by:(φ : Q[X]→ R) 7→ exp(φ(X) · log(1 + T ))− 1 = (1 + T )φ(X) − 1.Especially,f(T ) can be written in the form(1 + T )ψ(X) − 1 for a unique ring homomorphismψ : Q[X]→ Gn ⊗Z Q. Clearlyψ is inverse toρ⊗Z 1Q.

In view of claim 4.2.12, we are thus reduced to show thatGn is a flatZ-algebra, which followsreadily from proposition 4.2.7.

4.2.13. Furthermore,Gn is endowed with a co-addition,i.e. a ring homomomorphismGn →Gn⊗ZGn satisfying the usual co-associativity and co-commutativity conditions. The co-additionis given by the rule:

coadd : Gn → Gn ⊗Z Gn

(X

k

)7→

∑

i+j=k

(X

i

)⊗(X

j

).

Moreover, for everyk ∈ Z, we have a ring homomorphismπk : Gn → Z, which corresponds tothe endomorphism ofGm,Z(n) given by the rule:T 7→ (1 + T )k − 1 (raising to thek-th powerin Gm,Z(n)). Hence we derive, for everyk ∈ Z, a ring homomorphism

Gncoadd // Gn ⊗Z Gn

1Gn⊗πk // Gn.(4.2.14)

Remark 4.2.15. (i) On Gn ⊗Z Q = Q[X], (4.2.14) is the unique map such that(Xi

)7→(X+ki

)

for all i ≤ n, therefore we see that(X+ki

)∈ Gn for all k ∈ Z, n ≥ 0 and0 ≤ i ≤ n. Moreover,

(4.2.14) is clearly an automorphism for everyk ∈ Z.(ii) It is also interesting (though it will not be needed in this work) to remark thatGn is

endowed additionally with a co-composition structure, so thatGn is actually a co-ring, and it


represents the functorR 7→ End(Gm,R(n)) from Z-algebras to unitary rings. One can checkthat the co-composition map is given by the rule:

(X

k

)7→∑

φ

(X

φ

)⊗∏

j∈N∗

(Y

j

)φ(j)

whereφ ranges over all the functionsφ : N∗ := N \ 0 → N subject to the condition that∑j∈N∗ j · φ(j) = k, and

(Xφ

):=

X(X−1)·...·(X−∑

j∈N∗ φ(j)+1)∏j∈N∗ φ(j)!

. To show that(Xφ

)∈ Gn, one notices

that(Xφ

)=∏

j∈N∗

(X−

∑j−1i=1 φ(i)

φ(j)

)and then uses (i).

4.2.16. For the rest of this section we fix a prime numberp and we letvp : Q→ Z ∪ ∞ bethep-adic valuation.

Lemma 4.2.17.The ringGn,(p) := Gn⊗Z Z(p) is theZ(p)-algebra generated by the polynomialsX,(Xp

),(Xp2

),...,(Xpk

), wherek is the unique integer such thatpk ≤ n < pk+1.

Proof. We proceed by induction onn. It suffices to prove that(Xn

)is contained in theZ(p)-

algebraR := Z(p)[(Xp

),(Xp2

), ...,

(Xpk

)]. We will use the following (easily verified) identity which

holds inQ[X] for everyi, j ∈ N :(

X

i+ j

)=

(X

i

)·(X − ij

)·(i+ j

j

)−1

.(4.2.18)

Suppose first thatn is a multiple ofpk, and writen = (b + 1)pk for someb < p− 1. If b = 0,there is nothing to prove, so we can even assume thatb > 0. We apply (4.2.18) withi = b · pkandj = pk. By remark 4.2.15(i),

(X−b·pk

pk

)is in R, and so is

(Xb·pk

), by induction. The claim

will therefore follow in this case, if we show that((b+1)pk

pk

)is invertible inZ(p). However, this is

clear, sincevp(i) = vp(i + b · pk) for everyi = 1, ..., pk. Finally, it remains consider the casewheren = b · pk + a for someb > 0 and0 < a < pk. This is dealt with in the same way: apply(4.2.18) withi = b · pk andj = a and use the previous case.

Lemma 4.2.19.Let k ∈ N. If R is a flat Z(p)-algebra andf ∈ R, then the following twoconditions are equivalent:

(i)(fpi

)∈ R for everyi = 1, ..., k;

(ii) locally onSpecR there existsj ∈ Z such thatf ≡ j (mod pk).

Proof. We may assume thatR is local. Fork = 0 there is nothing to prove. Fork = 1 we have(fp

)= u ·p−1 ·∏p−1

i=0 (f − i) for a unitu of R. Then the assertion holds since all but possibly oneof thef − i are invertible. Fork > 1, by induction we can writef = i + p · g for someg ∈ Rand0 ≤ i < p. Sincevp(pk!) = 1 + p+ p2 + ...+ pk−1, we have

(f

pk

)= u · p−1−p−p2−...−pk−1 ·

∏

j≡i (mod p)

0≤j<pk

(f − j) = u′ ·(

g

pk−1

)

for some unitsu, u′ ∈ R. The claim follows.

4.2.20. For every integerk ≥ 0, we construct a schemeXk by gluing the affine schemesSpec Z(p)[

X−ipk ] (0 ≤ i < pk) along their general fibres. For everyk ∈ N and everyi ∈ N with

0 ≤ i < pk+1 there is an obvious imbeddingZ(p)[X−ipk ] ⊂ Z(p)[

X−ipk+1 ]. By gluing the duals of

these imbeddings, we obtain, for everyk ∈ N, a morphism of schemesρk : Xk+1 → Xk. Letalsoξk : Spec Gpk+1 → Spec Gpk be the morphism which is dual to the imbeddingGpk ⊂ Gpk+1.


Proposition 4.2.21.With the notation of(4.2.20)we have:

(i) For givenn > 0, let k be the unique integer such thatpk ≤ n < pk+1. Then there is anatural isomorphism of schemes:πk : Xk

∼−→ Spec Gn ⊗Z Z(p).(ii) For everyk ∈ N the diagram of schemes:

Xk+1

πk+1 //

ρk

Spec Gpk+1 ⊗Z Z(p)

ξk⊗ZZ(p)

Xk

πk // Spec Gpk ⊗Z Z(p)

commutes.

Proof. By lemma 4.2.17 we may assume thatn = pk. By lemma 4.2.19, we see that bothXk

andSpec Gpk ⊗Z Z(p) represent the same functor from the category of flatZ(p)-schemes to thecategory of sets. Since both schemes are flat overSpec Z(p), (i) follows. It is similarly clear thatξk ⊗Z Z(p) andρk represent the same natural transformation of functors, so (ii) follows.

Corollary 4.2.22. (i) For givenn ∈ N, let k be the unique integer such thatpk−1 ≤ n < pk.Then there is a natural ring isomorphism

Z/pkZ∼−→ End (Gm,Fp(n)) i 7→ (1 + T )i − 1

(ii) LetR be a ring such thatFp ⊂ R. Then there is a natural ring isomorphism

C 0(SpecR,Zp)∼−→ EndR (Gm,R) β 7→ (1 + T )β − 1.

Proof. (i): by lemma 4.2.17 we can assumen = pk − 1. In this case, it is clear that thepolynomials(1 + T )i − 1 are all distinct fori = 0, ..., pk − 1 and they form a subring ofEnd(Gm,Fp(n)). However, an endomorphism ofGm,Fp(n) corresponds to a unique point inSpec Gn(Fp). From proposition 4.2.21(i) we derive thatSpec Gn⊗ZFp is the union of the specialfibres of the affine schemesSpec Z[X−i

pk−1 ], for i = 0, ..., pk−1 − 1. Each of those contribute anaffine lineA1

Fp, soSpec Gn ⊗Z Fp consists of exactlypk−1 connected components. In total, we

have therefore exactlypk points inSpec Gn(Fp), so (i) follows.(ii): to give an endomorphism ofGm is the same as giving a compatible system of endomor-

phisms ofGm(n), one for eachn ∈ N. In caseFp ⊂ R, lemma 4.2.17 shows that this is alsoequivalent to the datum of a compatible system of morphismsφk : SpecR → Spec Gpk ⊗Z Fp,for everyk ≥ 0. From proposition 4.2.21(ii) we can further deduce that, under the morphismξk, each of thepk+1 connected components ofSpec Gpk+1 ⊗Z Fp gets mapped onto one of thepk+1 rational points ofSpec Gpk ⊗Z Fp. Sinceφk−1 = ξk φk, we see that the image ofφk−1

is contained inSpec Gpk−1(Fp), for everyk > 0. Taking (i) into account, we see that an en-domorphism ofGm,R is the same as the datum of a compatible system of continuous mapsSpecR→ Z/pkZ. Since thep-adic topology ofZp is the inverse limit of the discrete topologieson theZ/pkZ, the claim follows.

4.3. Modules of almost finite rank. Let A be aV a-algebra,P an almost finitely generatedprojectiveA-module andφ ∈ EndA(P ).

4.3.1. We say thatφ is Λ-nilpotentif there exists an integeri > 0 such thatΛiAφ = 0. Notice

that theΛ-nilpotent endomorphisms ofP form a bilateral ideal of the unitary ringEndA(P ).Notice also thatΛi

AP is an almost projectiveA-module for everyi ≥ 0; indeed, this is easilyshown by means of lemma 2.4.15. For aΛ-nilpotent endomorphismφwe introduce the notation

det(1P + φ) :=∑

i≥0

trΛiAP/A

(ΛiAφ).


Notice that the above sum consists of only finitely many non-zero terms, so thatdet(1P + φ) isa well defined element ofA∗.

Lemma 4.3.2. LetP be an almost finitely generated projectiveA-module.

(i) If φ is aΛ-nilpotent endomorphism ofP andα : A→ A′ is any morphism ofV a-algebras,setP ′ := P ⊗A A′. Then:det(1P ′ + φ⊗A 1A′) = α(det(1P + φ)).

(ii) Letφ, ψ ∈ EndA(P ) such thatφ ψ andψ φ areΛ-nilpotent. Then:det(1P + φ ψ) =det(1P + ψ φ).

Proof. (i) is a straightforward consequence of the definitions. As for (ii), it is clear thatψ φ isΛ-nilpotent and the stated identity follows directly from lemma 4.1.2(i).

4.3.3. Now, letφ, ψ ∈ EndA(P ) be two endomorphisms. SetB := A[X, Y ]/(Xn, Y n) andPB := P ⊗AB; φ andψ induce endomorphisms ofPB that we denote again by the same letters.ClearlyX ·φ andY ·ψ areΛ-nilpotent; hence we get elementsdet(1PB

+X ·φ), det(1PB+Y ·ψ)

anddet(1PB+X ·φ+Y ·ψ+XY ·ψ φ) in B∗. Notice that any element ofB∗ can be written

uniquely as anA∗-linear combination of the monomialsX iY j with 0 ≤ i, j < n. Moreover, itis clear thatdet(1PB

+X · φ) =∑

0≤i<n trΛiAP/A

(ΛiAφ) ·X i, and similarly forψ.

Proposition 4.3.4. With the above notation, the following identity holds :

det(1PB+X · φ) · det(1PB

+ Y · ψ) = det(1PB+X · φ+ Y · ψ +XY · ψ φ).(4.3.5)

Proof. First of all we remark that, whenP is a freeA-module of finite rank, the above identityis well-known, and easily verified by working with matrices with entries inA∗. Suppose nextthatP is arbitrary, butφ = ε · φ′, ψ = ε · ψ′ for someφ′, ψ′ ∈ EndA(P ) andε ∈ m. Pick a freeA-moduleF of finite rank, and morphismsu : P → F , v : F → P such thatv u = ε · 1P .Setφε := u φ′ v : F → F and define similarlyψε. Clearlydet(1PB

+X · φ) = det(1PB+

X · ε · φ′) = det(1PB+X · v u φ′) = det(1PB

+X · φε) (by lemma 4.3.2(ii)) and similarlyfor the other terms appearing in (4.3.5). Thus we have reduced this case to the case of a freeA-module. Finally, we deal with the general case. The foregoing shows that the sought identityis known at least whenφ andψ are replaced byε · φ, resp.ε · ψ, for anyε ∈ m. Equivalently,consider theA-algebra endomorphismα : B → B defined byX 7→ ε ·X, Y 7→ ε · Y and letCbe theB-algebra structure onB determined byα; by lemma 4.3.2(i) we have

det(1PB+X · ε · φ) = det(1PC

+ (X · φ)⊗B 1C) = α(det(1PB+X · φ))

and similarly for the other terms appearing in (4.3.5). Thus, the images underα of the twomembers of (4.3.5) coincide. But applyingα to a monomial of the forma ·X iY j has the effectof multiplying it by εi+j; by (B), the (i + j)-powers of elements ofm generatem, hence theclaim follows easily.

Corollary 4.3.6. If φ, ψ ∈ EndA(P ) are twoΛ-nilpotent endomorphisms, then

det(1P + φ) · det(1P + ψ) = det(1P + φ+ ψ + φ ψ).

Proof. For an arbitraryα ∈ EndA(P ), one can defineP ′ := P⊗AA[[X]] anddet(1P ′+X ·α) :=∑i≥0 trΛi

AP/A(Λi

Aα) ·X i ∈ A∗[[X]]. Then proposition 4.3.4 implies that the analogue of (4.3.5)holds inA∗[[X, Y ]]. But if α is Λ-nilpotent, the power seriesdet(1P ′ + X · α) is actually apolynomial inA∗[X]; the claim then follows by evaluating the polynomialsdet(1P ′ + X · φ),det(1P ′ + Y · ψ) anddet(1P ′ +X · φ+ Y · ψ +XY · φ ψ) for X = Y = 1.


4.3.7. Next, forP as above, set

χP (X) :=∑

i≥0 trΛiAP/A

(ΛiA1P ) ·X i ∈ A∗[[X]]

ψP (X) :=∑

i≥0 trSymiAP/A

(1SymiAP

) ·X i ∈ A∗[[X]].

Corollary 4.3.8. LetP be an almost finitely generated projectiveA-module. Then:

(i) the power seriesχP (X) defines an endomorphism of the formal groupGm,A∗.(ii) χP (X) · ψP (−X) = 1.(iii) χP (X) ∈ 1 + EP/A∗[[X]].

Proof. (i) is immediate. For (ii), recall that, for everyn > 0 there is an acyclic Koszul complex(cp. [17, Ch.X,§9, n.3, Prop.3])

0→ ΛnAQ→ (Λn−1

A Q)⊗A (Sym1AQ)→ ...→ (Λ1

AQ)⊗A (Symn−1A Q)→ Symn

AQ→ 0.

From proposition 4.1.4 we derive, by a standard argument, that the trace is an additive functionon arbitrary bounded acyclic complexes. Then, taking into account lemma 4.1.3 we obtain:∑n

i=0(−1)i · trΛn−iA P/A(1Λn−i

A P ) · trSymiAP/A

(1SymiAP

) = 0 for everyn > 0. This is equivalent tothe sought identity. To show (iii) we remark more precisely thatEΛr

AP/A⊂ EP/A for everyr > 0.

Indeed, setB := A/EP/A. Then, by proposition 2.4.28(i),(iii):EΛrAP/A

·B = EΛrB(P⊗AB)/B = 0,

whence the claim.

Definition 4.3.9. Let P be an almost finitely generated projectiveA-module.(i) The formal rankof P is the ring homomorphismf.rkA(P ) : G∞ := Z[α,

(α2

), ...] → A∗

associated toχP (X).(ii) We say thatP is of almost finite rankif, for everyε ∈ m, there exists an integeri ≥ 0 such

thatε · ΛiAP = 0.

(iii) We say thatP is of finite rankif there exists an integeri ≥ 0 such thatΛiAP = 0.

(iv) Let r ∈ N; we say thatP hasconstant rank equal tor if Λr+1A P = 0 andΛr

AP is aninvertibleA-module.

Remark 4.3.10. (i): It follows easily from lemma 2.3.7(vi) that every uniformly almost finitelygenerated projectiveA-module is of finite rank.

(ii): Notice that if P is of finite rank, thenχP (X) is a polynomial, whence it defines anendomorphism of the algebraic groupGm,A∗. In this case, it follows thatχP (X) is of the form(1 + X)α, whereα : SpecA∗ → Z is a continuous function (whereZ is seen as a discretetopological space). More precisely, there is an obvious injective ring homomorphism

C 0(SpecA∗,Z)→ EndA(Gm,A∗) β 7→ (1 +X)β(4.3.11)

which allows to identify the continuous functionα with the formal rank ofP . Moreover, ifΛiAP = 0, it is clear thatα(SpecA∗) ⊂ 0, ..., i− 1.The main result of this section is theorem 4.3.28, which describes general modules of al-

most finite rank as infinite products of modules of finite rank.The first step is lemma 4.3.12,concerned with the case of anA-module of rank one.

Lemma 4.3.12.LetP be an almost finitely generated projectiveA-module such thatΛ2AP = 0.

There existsV a-algebrasA0, A1 and an isomorphism ofV a-algebrasA ≃ A0 × A1 such thatP ⊗A A0 = 0 andP ⊗A A1 is an invertibleA1-module.

Proof. Since the natural mapP × P → Λ2AP is universal for alternatingA-bilinear maps on

P × P , we have

f(p) · q = f(q) · p for everyf ∈ (P ∗)∗ andp, q ∈ P∗.(4.3.13)


Using (4.3.13) we deriveωP/A(p ⊗ f)(q) = trP/A(ωP/A(p ⊗ f)) · q for everyf ∈ (P ∗)∗ andp, q ∈ P∗. In other words,ωP/A(p⊗f) = trP/A(ωP/A(p⊗f)) ·1P , for everyf ∈ (P ∗)∗, p ∈ P∗.By linearity we finally deduce

φ = trP/A(φ) · 1P for all φ ∈ EndA(P ).(4.3.14)

Now, by remark 4.3.10(ii), the hypothesisΛ2AP = 0 also implies thatχP (X) = (1 +X)α, for

a continuous functionα : SpecA∗ → 0, 1. We can decompose accordinglyA = A0 ×A1, sothatα(SpecAi∗) = i, which gives the sought decomposition. We can now treat separately thetwo casesA = A0 andA = A1. In casef.rkA(P ) = 0, thentrP/A(1P ) = 0, and then (4.3.14)implies thatP = 0. In casef.rkA(P ) = 1, thentrP/A(1P ) = 1 and (4.3.14) implies that thenatural mapA→ EndA(P ) : a 7→ a · 1P is an inverse fortrP/A, thusA ≃ P ⊗A P ∗.

The next step consists in analyzing the structure ofA-modules of finite rank. To this purposewe need some preliminaries of multi-linear algebra.

4.3.15. For everyn ≥ 0 let n := 1, ..., n; for a subsetI ⊂ n let |I| be the cardinality ofI;for a given partitionn = I ∪ J , let≺ denote the total ordering onn that restricts to the usualordering onI and onJ , and such thati ≺ j for everyi ∈ I, j ∈ J . Finally letεIJ be the signof the unique order-preserving bijection(n, <)→ (n,≺).

Let M be anyA-module. Given elementsm1, m2, ..., mn in M∗, andI ⊂ n a subset ofelementsi1 < i2 < ... < i|I|, letmI := mi1 ∧ ... ∧ mi|I| ∈ Λ

|I|AM∗ (with the convention that

m∅ = 1 ∈ A∗ = Λ0AM∗).

4.3.16. LetM,N be any twoA-modules. For everyi, j ≥ 0 there is a natural morphism

ΛiAM ⊗A Λj

AN → Λi+jA (M ⊕N)(4.3.17)

determined by the rule:

m1 ∧ ... ∧mi ⊗ n1 ∧ ... ∧ nj 7→ (m1, 0) ∧ (m2, 0) ∧ ... ∧ (0, n1) ∧ ... ∧ (0, nj)

for all m1, ..., mi ∈ M∗ andn1, ..., nj ∈ N . The morphisms (4.3.17) assemble to an isomor-phism ofA-modules

Λ•AM ⊗A Λ•AN → Λ•A(M ⊕N).(4.3.18)

Clearly, there is a unique gradedA-algebra structure onΛ•AM ⊗A Λ•AN such that (4.3.18) isan isomorphism of (graded-commutative)A-algebras. Explicitly, givenxi ∈ Λai

AM , yi ∈ ΛbiAN

(i = 1, 2) one verifies easily that the product onΛ•AM ⊗A Λ•AN is fixed by the rule

(x1 ⊗ y1) · (x2 ⊗ y2) = (−1)a2b1 · (x1 ∧ x2)⊗ (y1 ∧ y2).(4.3.19)

ThenΛ•AM ⊗A Λ•AN is even a bigradedA-algebra, if we letΛiAM ⊗A Λj

AN be the gradedcomponent of bidegree(i, j).

4.3.20. Next, letδ : M →M ⊕M be the diagonal morphismm 7→ (m,m) (for all m ∈M∗).It induces a morphismΛ•Aδ : Λ•AM → Λ•A(M ⊕ M) of A-algebras. We let∆ : Λ•AM →Λ•AM ⊗A Λ•AN be the composition of the morphismΛ•Aδ and the inverse of the isomorphism(4.3.18). For everya, b ≥ 0 we also let∆a,b : Λ•AM → Λa

AM ⊗A ΛbAN be the composition of

∆ and the projection onto the graded component of bidegree(a, b). The morphisms∆a,b areusually called ”co-multiplication morphisms”. An easy calculation shows that:

∆a,b(x1 ∧ x2 ∧ ... ∧ xa+b) =∑

I,J

εIJ · xI ⊗ xJ(4.3.21)


where the sum ranges over all the partitionsa + b = I ∪ J such that|I| = a. Let nowx1, ..., xa, y1, ..., yb ∈ M∗. Since∆ is a morphism ofA-algebras, we have∆(xa ∧ yb) =∆(xa) ·∆(yb). Hence, using (4.3.19) and (4.3.21) one deduces easily:

∆a,b(xa ∧ yb) =∑

I,J,K,L

εIJ · εKL · (−1)|J | · (xI ∧ yK)⊗ (xJ ∧ yL).(4.3.22)

where the sum runs over all partitionsI ∪ J = a,K ∪ L = b such that|J | = |K|.Lemma 4.3.23.Suppose thatΛa+1

A M = 0 for some integera ≥ 0. Let 0 < b ≤ a andx1, ..., xa, y1, ..., yb ∈M∗. Then the following identity holds inΛa

AM ⊗A ΛbAM :

xa ⊗ yb =∑

I,J

εJI · (xJ ∧ yb)⊗ xI

where the sum ranges over all the partitionsa = I ∪ J such that|I| = b.

Proof. For a given subsetB ⊂ b we let

γ(yB) :=∑

I,J

εIJ · (xI ∧ yB)⊗ xJ − xa ⊗ yB

where the sum is taken over all the partitionsI ∪ J = a such that|J | = |B|. Notice thatγ(y∅) = 0. We have to show thatγ(yb) = 0. To this purpose we show the following:

Claim4.3.24. If |B| > 0, then

∆a,|B|(xa ∧ yB) =∑

K,L

εKL · (−1)|K| · γ(yK) ∧ yL(4.3.25)

where the sum ranges over all the partitionsK ∪ L = B.

Proof of the claim:Using (4.3.22), the difference between the two sides of (4.3.25) is seen to beequal to

∑K,L εKL · (−1)|K| · xa ⊗ (yK ∧ yL) =

∑K,L ·(−1)|K| · xa ⊗ yB, where the sum runs

over all partitionsK ∪ L = B. A standard combinatorial argument shows that this expressioncan be rewritten asxa ⊗ yB ·

∑|B|k=0(−1)k ·

(|B|k

), which vanishes if|B| > 0.

To conclude the proof of the lemma, we remark that∆a,b vanishes ifb > 0 because byassumptionΛa+1

A M = 0; then the claim follows by induction on|B|, using claim 4.3.24.

Lemma 4.3.26.LetP be anA-module such thatΛn+1A P = 0 and assume that eitherP is flat

or 2 is invertible inA∗. ThenΛ2A(Λn

AP ) = 0.

Proof. For anyA-moduleM andr ≥ 0 there exists an antisymmetrizer operator (cp. [15, Ch.III, §7.4, Remarque])

ar : M⊗r →M⊗r m1 ⊗ ...⊗mr 7→∑

σ∈Sr

sgn(σ) ·mσ(1) ⊗ ...⊗mσ(r).

Clearlyar factors thorughΛrAM , and in caseM is free of finite rank, it is easy to check (just

by arguing with basis elements) that the induced mapar : ΛrAM → Im(ar) is an isomorphism.

This is still true also in caser! is invertible inA∗, since in that case one checks thatar/r! isidempotent (seeloc. cit.). More generally, ifM is flat then, by [51, Ch.I, Th.1.2],M! is thefiltered colimit of a direct system of freeA∗-modules of finite rank, so also in this casear is anisomorphism. Notice that, again by [51, Ch.I, Th.1.2], ifP is flat, thenΛk

AP is also flat, for everyk ≥ 0. Hence, to prove the lemma, it suffices to verify thatIm(a2 : (Λn

AP )⊗2 → (ΛnAP )⊗2) = 0

whenΛn+1A P = 0. However, this follows easily from lemma 4.3.23.

We are now ready to return toA-modules of finite rank.


Proposition 4.3.27.LetP be an almost projectiveA-module of finite rank; say thatΛrAP = 0.

There exists a natural decompositionA ≃ A0 ×A1 × ...×Ar−1 such thatPi := P ⊗A Ai is anAi-module of constant rank equal toi for everyi = 0, ..., r − 1.

Proof. We proceed by induction onr; the caser = 2 is covered by lemma 4.3.12. By lemma4.3.26 we haveΛ2

A(Λr−1A P ) = 0, so by lemma 4.3.12, there is a decompositionA ≃ A′r−1 ×

A′r−2, such that forPi := P ⊗AA′i (i = r− 2, r− 1) the following holds.Λr−1A′

r−2(Pr−2) = 0 and

Λr−1A′

r−1(Pr−1) is an invertibleAr−1-module. It follows in particular thatχPr−1(X) is a polynomial

of degreer−1, and its leading coefficient is invertible inAr−1. HenceχPr−1(X) = (1+X)r−1.By induction,A′r−2 admits a decompositionA′r−2 ≃ Ar−2 × ...×A0 with the stated properties;it suffices then to takeAr−1 := A′r−1.

Theorem 4.3.28.Let P be an almost projectiveA-module of almost finite rank. Then thereexists a natural decompositionA ≃∏∞i=0Ai such that:

(i) limi→∞

AnnV a(Ai) = V a (for the uniform structure of definition2.3.1);

(ii) for i ∈ N, let Pi := P ⊗A Ai; thenP ≃ ∏∞i=0 Pi and everyPi is anAi-module of finiteconstant rank equal toi.

Proof. Let mλλ∈I be the filtered family of finitely generated subideals ofm. For everyλ ∈ I,letAλ := A/AnnA(mλ). By hypothesis,Pλ := P⊗AAλ is anAλ-module of finite rank; say thatthe rank isr(λ). By proposition 4.3.27 we have natural decompositionsAλ ≃ Aλ,0×...×Aλ,r(λ)

such thatP ⊗A Aλ,i is anAλ,i-module of constant rank equal toi for every i ≤ r(λ). Thenaturality of the decomposition means that for everyλ, µ ∈ I such thatmλ ⊂ mµ, we haveAµ,i ⊗A Aλ ≃ Aλ,i. for everyi ≤ r(µ). In particular,mλAµ,i = 0 for everyi > r(λ). Byconsidering the short exact sequence of cofiltered systems of A-modules:

0→ (AnnA(mλ))λ∈I → (A)λ∈I → (Aλ)λ∈I → 0

we deduce easily thatA ≃ limλ∈I

Aλ and therefore we obtain a decompositionA ≃ ∏∞i=0Ai,

with Ai := limλ∈I

Aλ,i for every i ∈ N. Notice that, for everyi ∈ N and everyλ ∈ I, the

natural morphismAi → Aλ,i is surjective with kernel killed bymλ. It follows easily thatΛi+1Ai

(P ⊗A Ai) = 0 andΛiAi

(P ⊗A Ai) is an invertibleAi-module. Furthermore, for everyλ ∈ I, mλ · Ai = 0 for all i > r(λ), which implies (i). Finally, for everyλ ∈ I, mλ kills thekernel of the projectionP →∏r(λ)

i=0 Pi, soP is isomorphic to the infinite product of thePi.

4.4. Localisation in the flat site. Throughout this sectionP denotes an almost finitely gen-erated projectiveA-module. The following definition introduces the main tool used in thissection.

Definition 4.4.1. Thesplitting algebraof P is theA-algebra:

Split(A,P ) := Sym•A(P ⊕ P ∗)/(1− ζP ).

We endowSym•A(P ⊕ P ∗) with the structure of graded algebra such thatP is placed in degreeone andP ∗ in degree−1. ThenζP is a homogeneous element of degree zero, and consequentlySplit(A,P ) is also a gradedA-algebra.

4.4.2. We define a functorS : A-Alg→ Set by assigning to everyA-algebraB the setS(B)of all pairs(x, φ) wherex ∈ (P ⊗A B)∗, φ : P ⊗A B → B such thatφ(x) = 1.

Lemma 4.4.3. (i) Split(A,P ) is a flatA-algebra.

(ii) Split(A′, P ⊗A A′) ≃ Split(A,P )⊗A A′ for everyA-algebraA′.(iii) Split(A,P ) represents the functorS.


Proof. For everyk ≥ 0 we haveζkP ∈ (SymkAP )⊗A (Symk

AP∗) ⊂ gr0(Sym2k

A (P ⊕ P ∗)). It iseasy to verify the formula:

grkSplit(A,P ) ≃ colimj∈Z

(Symk+jA P )⊗A (Symj

AP∗)(4.4.4)

where the transition maps in the direct system are given by multiplication by ζP . In particular,it is clear thatgrkSplit(A,P ) is a flatA-module, so (i) holds. (ii) is immediate. To show (iii),let us introduce the functorT : A-Alg → Set that assigns to everyA-algebraB the set of allpairs(x, φ) wherex ∈ (P ⊗A B)∗ andφ : P ⊗A B → B. SoS is a subfunctor ofT .

Claim4.4.5. The functorT is represented by theA-algebraSym•A(P ⊕ P ∗).

Proof of the claim:Indeed, there are natural bijections:

(P ⊗A B)∗∼−→ HomB(P ∗ ⊗A B,B)

∼−→ HomA(P ∗, B)∼−→ HomA-Alg(Sym•AP

∗, B)

which show that the functorB 7→ (P⊗AB)∗ is represented by theA-algebraSymAP∗. Working

out the definitions, one finds that the composition of these bijections assigns to an elementx ∈ (P ⊗A B)∗ the uniqueA-algebra morphismfx : Sym•AP

∗ → B such thatfx(ψ) = ψ(x)for everyψ ∈ HomA(P,A). Similarly, the functorB 7→ HomB(P ⊗A B,B) is representedby Sym•AP , and again, one checks that the bijection assigns toφ : P ⊗A B → B the uniqueA-algebra morphismgφ : Sym•AP → B such thatgφ(p) = φ(p) for everyp ∈ P∗. It followsthatT is represented by(Sym•AP )⊗A (Sym•AP

∗) ≃ Sym•A(P ⊕ P ∗).For everyA-algebraB we have a natural map:αB : T (B) → B∗ given by(x, φ) 7→ φ(x).

This defines a natural transformation of functorsα : T → (−)∗. Moreover, let us consider thetrivial map βB : T (B) → B∗ that sends everything onto the element1 ∈ B∗. β is anothernatural transformation fromT to the almost elements functor. Clearly :

S(B) = Equal( T (B)αB //

βB

// B∗ ).

We remark that the functorB 7→ B∗ onA-algebras, is represented byA[X] := Sym•AA. There-fore there are morphismsα∗, β∗ : A[X] → Sym•A(P ⊕ P ∗) that represent these natural trans-formations. It follows thatS is represented by theA-algebra

Coequal( A[X]α∗

//

β∗// Sym•A(P ⊕ P ∗) ).

To determineα∗ andβ∗ it suffices to calculate them on the elementX ∈ A[X]∗. It is easy tosee thatα∗(X) = 1. To conclude the proof it suffices therefore to show:

Claim4.4.6. β∗(X) = ζP .

Proof of the claim: In view of the definitions, and using the notation of the proofof claim4.4.5, the claim amounts to the identity:(gφ ⊗ fx)(ζP ) = φ(x) for every(x, φ) ∈ T (B). Bynaturality, it suffices to show this forB = A. Now, for everyε ∈ m, we can writeε · ζP =∑

i qi ⊗ ψi for someqi ∈ P , ψi ∈ P ∗ and we have∑

i qi · ψi(b) = ε · b for all b ∈ P∗. Hence(gφ ⊗ fx)(

∑i qi ⊗ ψi) =

∑i φ(qi) · ψi(x) = φ(

∑i qi · ψi(x)) = φ(εx) and the claim follows.

Remark 4.4.7. The construction of the splitting algebra occurs already, in a tannakian context,in Deligne’s paper [23] : see the proof of lemma 7.15 inloc.cit.


4.4.8. We recall that for everyk ≥ 0 there are natural morphisms

SymkAP

αP−→ ΓkAPβP−→ Symk

AP

such thatβP αP = k! · 1SymkAP

andαP βP = k! · 1ΓkAP

. (to obtain the morphisms, one canconsider the flatA∗-moduleP!, thus one can assume thatP is a module over a usual ring; thenαP is obtained by extending multiplicatively the identity morphismSym1

AP = P → P = Γ1AP ,

andβP is deduced from the homogeneous degreek polynomial lawP ⊗A B → (P ⊗A B)⊗k

defined byx 7→ x⊗k). Moreover(ΓkAP )∗ ≃ SymkAP∗.

Lemma 4.4.9. With the notation of(4.4.8)we have:(αP ⊗A 1SymkAP

∗)(ζkP ) = k! · ζΓkAP

.

Proof. Suppose first thatP is a freeA-module, lete1, ..., en be a base ofP∗ ande∗1, ..., e∗n the

dual base ofP ∗. ThenΓkAP is the freeA-module generated by the basise[n1]1 · ... · e[nk]

k where0 ≤ ni ≤ k for i = 1, ..., k and

∑j nj = k. The dual of this basis is the basis ofSymk

AP∗

consisting of the elementse∗n11 · ... · e∗nk

k . Furthermore,ζP =∑

i ei ⊗ e∗i and thereforeζkP =∑n

(kn

)(en1

1 · ... · enkk )⊗ (e∗n1

1 · ... · e∗nkk ), wheren := (n1, ..., nk) ranges over the multi-indices

submitted to the above conditions and(kn

):= k!

n1!·...·nk!. Then the claim follows straightforwardly

from the identity:αP (en11 · ... · enk

k ) = n1! · ... ·nk! · e[n1]1 · ... · e[nk]

k . For the general case we shalluse the following

Claim 4.4.10. LetM be an almost finitely generated projectiveA-module and pick, for a givenε ∈ m, morphismsu : M → F andv : F →M with v u = ε ·1M . Thenv⊗u∗(ζF ) = ε · ζM .

Proof of the claim:We have a commutative diagram

F ⊗A F ∗ v⊗u∗ //

M ⊗AM∗

EndA(F ) // EndA(M)

where the vertical morphisms are the natural ones, and wherethe bottom morphism is given byφ 7→ v φ u. Then the claim follows by an easy diagram chase.

Pick morphismsu : P → F andv : F → P with vu = ε·1P . We consider the commutativediagram

SymkAF ⊗A Symk

AF∗

SymkAv⊗Symk

Au∗

//

αF⊗1Symk

AF∗

SymkAP ⊗A Symk

AP∗

αP⊗1Symk

AP∗

ΓkAF ⊗A SymkAF∗

ΓkAv⊗Symk

Au∗

// ΓkAP ⊗A SymkAP∗.

By claim 4.4.10, we havev ⊗ u∗(ζF ) = ε · ζP ; whenceSymkAv ⊗ Symk

Au∗(ζkF ) = εk · ζkP .

Moreover, we remark that(ΓkAv) (SymkAu∗)∗ = (ΓkAv) (ΓkAu) = εk · 1Γk

AP, therefore claim

4.4.10 (applied forM = ΓkAP ) yieldsΓkAv⊗SymkAu∗(ζΓk

AF) = εk ·ζΓk

AP. Since we already know

the lemma forF , a simple diagram chase shows that(α⊗A 1SymkAP

∗)(εk · ζkP ) = εk · k! · ζΓkAP

.Since thek-powers of elements ofm generatem, the claim follows.

Lemma 4.4.11.LetP be as above and suppose thatζP is nilpotent inSym•A(P ⊕ P ∗).(i) If Q ⊂ A∗, thenχP (X) = (1 +X)−α for some continuous functionα : SpecA∗ → N.(ii) If Fp ⊂ A∗, thenEP/A is a Frobenius-nilpotent ideal.


Proof. (i): Since Q ⊂ A∗, thenk! is invertible inA∗ for everyk ≥ 0; by lemma 4.4.9 itfollows thatSymk

AP = 0, thusψP (X) is a polynomial. By corollary 4.3.8,ψP (−X) definesan endomorphism ofGm,A∗ , thereforeψP (X) = (1 + X)α for some continuous functionα :SpecA∗ → N; then the claim follows by corollary 4.3.8(ii).

(ii): Let (f, p) ∈ (P ∗ ⊕ P )∗; in the notation of the proof of lemma 4.4.3, we can write(f, p) ∈ T (A). It follows that(f, p) corresponds to a morphism ofA-algebrasSym•A(f, p)∗ :Sym•A(P ⊕ P ∗)→ A. In particularSym•A(f, p)∗(ζkP ) = Sym•A(f, p)∗(ζP )k for everyk ≥ 0. Byinspecting the proof of claim 4.4.6, we deduceSym•A(f, p)∗(ζP ) = f(p) for everyf ∈ P ∗ andp ∈ P . By hypothesis,ζp

n

P = 0 for every sufficiently largen. It follows thatEP/A is Frobeniusnilpotent.

Lemma 4.4.12.LetR0 be a noetherian commutative ring,R anR0-algebra andM a flatR-module. ThenM = 0 if and only ifM ⊗R0 κ = 0 for every residue fieldκ ofR0 (i.e., for everyfieldκ of the formFrac(R0/p), wherep is some prime ideal ofR0).

Proof. Clearly we have only to show the direction⇐. It suffices to show thatMp = 0 for everyprime ideal ofR0. Hence we can assume thatR0 is local, in particular of finite Krull dimension.We proceed by induction on the dimension ofR0. If dimR0 = 0, thenR0 is a local artinianring, hence a power of its maximal idealm is equal to0. By assumption,M/mM = 0, i.e.M = mM . ThenM = mkM for everyk ≥ 0, soM = 0. Next, suppose thatdimR0 = dand the lemma already known for all rings of dimension strictly less thand. Assume first thatR0 is an integral domain and pickf ∈ S := R0 \ 0. ThenR0/fR0 has dimension strictlyless thand, so by induction we haveM/fM = 0, i.e. M = fM . Due to the flatness ofM ,we have:AnnM(f) = AnnR(f) ·M = AnnR(f) · fM = 0. This implies that the kernel ofthe natural mapM → S−1M is trivial. On the other hand, by hypothesisS−1M = 0, whenceM = 0 in this case. For a generalR0 of dimensiond, notice that the above argument impliesthat, for every minimal prime idealp of R0, we havepM = M . But the product of all (finitelymany) minimal prime ideals is contained in the nilpotent radical R of R0, whenceRM = M ,and finallyM = 0 as claimed.

4.4.13. Let nowp ∈ SpecA∗. By composingf.rkA(P ) with the mapA∗ → A∗/p, we obtaina ring homomorphism

f.rkA(P, p) : G∞ → A∗/p.

In case(A∗/p)a 6= 0, we can interpretf.rkA(P, p) as the formal rank ofP ⊗A (A/pa). Moreprecisely, letπ : A → A/pa be the natural projection; thenπ∗ factors through a mapπ′ :A∗/p→ (A/pa)∗ and we have:f.rkA/pa(P ⊗A (A/pa)) = π′ f.rkA(P, p).

Even if (A∗/p)a = 0, the morphismf.rkA(P, p) can still be interpreted as the map associatedto an endomorphism ofGm,A∗/p, so it still makes sense to ask whetherf.rkA(P, p) is an integer,as indicated in remark 4.3.10(ii).

Lemma 4.4.14.LetP be an almost finitely generated projectiveA-module andp ∈ SpecA∗.If B is anyAa∗p-algebra andq ∈ SpecB∗ such thatr(q) := f.rkB(P ⊗AB, q) is an integer, thenr(p) := f.rkA(P, p) is also an integer andr(p) = r(q).

Proof. Indeed, let us consider the natural mapsG∞ → A∗ → B∗; under the assumptions, thecontraction ofq in A∗ is contained inp. Since the image of

(αi

)in B∗/q is

(r(q)i

), it follows that

the same holds inA∗/p.

Definition 4.4.15. We say that anA-moduleP admits infinite splittingsif there is an infinitechains of decompositions of the form:P ≃ A⊕ P1, P1 ≃ A⊕ P2, P2 ≃ A⊕ P3, ...


Theorem 4.4.16.Let P be an almost finitely generated projectiveA-module. The followingconditions are equivalent:

(i) P is of almost finite rank.(ii) For all A-algebrasB 6= 0, we have:

⋂r>0 EΛr

B(P⊗AB)/B = 0.(iii) For all A-algebrasB 6= 0, PB := P ⊗AB does not admit infinite splittings, and moreover

if PB ≃ Bn ⊕ Q for someB-moduleQ andχQ(X) = (1 + X)−α for some continuousfunctionα : SpecB∗ → N, thenQ = 0.

(iv) For all A-algebrasB 6= 0, PB does not admit infinite splittings, and moreover ifPB ≃Bn ⊕Q for someB-moduleQ, then :(a) If Fp ⊂ B∗ andQ = IQ for a Frobenius-nilpotent idealI ⊂ B, thenQ = 0;(b) If Q ⊂ B∗ andSymr

BQ = 0 for somer ≥ 1, thenQ = 0.

Proof. (i) ⇒ (ii) : indeed, from proposition 2.4.28(iii) one sees that, for everyA-moduleof almost finite rank, everyA-algebraB and everyε ∈ m, there existsr ≥ 0 such thatε · EΛr

B(P⊗AB)/B = 0.(ii) ⇒ (iii) : let B 6= 0 be anA-algebra; by hypothesis, there existsr ≥ 0 such thatJr :=

EΛrB(P⊗AB)/B 6= B. Suppose thatP⊗AB admits infinite splittings. TheB/Jr-modulePB/JrPB

has rank< r, and at the same time it admits infinite splittings, a contradiction.Suppose next, that there is a decompositionP⊗AB ≃ Bn⊕Q; thenQ is obviously of almost

finite rank. Suppose thatχQ(X) = (1 +X)α has the shape described in (ii). We reduce easilyto the case whereα is a constant function. However,χQ/JrQ(X) is a polynomial of degree< r,thusα = 0, and thenQ = 0 by theorem 4.3.28(ii).

(iii) ⇒ (iv) : suppose thatFp ⊂ B∗ andQ = IQ for some Frobenius-nilpotent idealI. ThenχQ(X) ∈ 1 + I∗[[X]], which means that the image ofχQ(X) in End(Gm,B∗/I∗) is the trivialendomorphism. But we have a commutative diagram

C 0(SpecB∗,Zp) //

End(Gm,B∗)

C 0(SpecB∗/I∗,Zp) // End(Gm,B∗/I∗)

where the horizontal maps are those defined in remark 4.3.10(ii), and are bijective by corollary4.2.22. The left vertical map is induced by restriction to the closed subsetSpecB∗/I∗, andsinceI is Frobenius-nilpotent, it is a bijection as well. It follows that the right vertical map isbijective, whencef.rkB(Q) = 0, and finallyQ = 0 by (iii).

Next, consider the case whenQ ⊂ B∗ andSymrBQ = 0 for somer ≥ 1. It follows thatζQ

is nilpotent inSym•B(Q⊕Q∗). By lemma 4.4.11(i),χQ(X) = (1 +X)−α for some continuousα : SpecB∗ → N. Then (iii) implies thatQ = 0.

To show that (iv)⇒ (i), we will use the following:

Claim 4.4.17. Assume (iv). Then:Split(B,Q) = 0⇒ Q = 0.

Proof of the claim:SupposeQ 6= 0 andSplit(B,Q) = 0; then (4.4.4) implies that for everyε ∈ m there existsj ≥ 0 such thatε · ζjQ = 0. We haveεQ 6= 0 for someε ∈ m. Fromthe flatness ofQ, we deriveAnnSym•

B(Q⊕Q∗)(ε) = AnnB(ε) · Sym•B(Q ⊕ Q∗), hence we canreplaceB by B/AnnB(ε), Q by Q/AnnB(ε) · Q, thereby achieving thatζQ is nilpotent inSym•B(Q ⊕ Q∗) and stillQ 6= 0. Using lemma 4.4.12 (and the functoriality ofSplit(B,Q)for base extensionsB → B′) we can further assume thatB∗ contains eitherQ or one of thefinite fieldsFp. If Q ⊂ B∗, thenk! is invertible inB∗ for everyk ≥ 0; by lemma 4.4.9 itfollows thatSymk

BQ = 0, whenceQ = 0 by (iv), a contradiction. IfFp ⊂ B∗, then by lemma4.4.11(ii),EQ/B is Frobenius-nilpotent. However, from proposition 2.4.28(i) it follows easily


thatQ = EQ/B ·Q, whenceQ = 0, again by (iv), and again a contradiction. In either case, thisshows thatSplit(B,Q) 6= 0, as claimed.

Claim4.4.18. Assume (iv). ThenA/EP/A is a flatA-algebra.

Proof of the claim:It suffices to show thatAp/(EP/A)p is a flatAp-algebra for every prime idealp ⊂ A∗. If EP/A∗ is not contained inp, then(EP/A)p = Ap, so there is nothing to prove in thiscase. We assume therefore that

EP/A∗ ⊂ p.(4.4.19)

We will show thatPp = 0 in such case, whenceAp/(EP/A)p = Ap, so the claim will follow.From (4.4.19) and corollary 4.3.8(iii) we know already that

f.rkA(P, p) = 0.(4.4.20)

Suppose thatPp 6= 0; then there existsε ∈ m such thatεPp 6= 0. Define inductivelyA0 := Ap,Q0 := Pp, Ai+1 := Split(Ai, Qi) andQi+1 as anAi+1-module such thatQi ⊗Ai

Ai+1 ≃Ai+1 ⊕ Qi+1, for everyi ≥ 0 (the existence ofQi+1 is assured by lemma 4.4.3(iii)). Thencolimn∈N

An ≃ 0, since, after base change to thisV a-algebra,P admits infinite splittings. This

implies that there existsn ∈ N such thatεAn+1 = 0 andεAn 6= 0. However, sinceAn+1 isflat overAn, we have:An+1 = AnnAn+1(ε) = AnnAn(ε) · An+1. SetA′ := An/AnnAn(ε);thenSplit(A′, Qn ⊗An A

′) = 0, soQn ⊗An A′ = 0 by claim 4.4.17. By flatness ofQn, this

means thatεQn = 0; in particular,n > 0. By definition,Qn−1 ⊗An−1 An ≃ An ⊕ Qn;it follows thatQ′ := Qn−1 ⊗An−1 A

′ ≃ A′, in particularf.rkA′(Q′) = 1 and consequentlyf.rkA′(P ⊗A A′) = n > 0; in view of lemma 4.4.14, this contradicts (4.4.20), thereforePp = 0,as required.

Claim4.4.21. Assuming (iv), the natural morphismφ : A→ (A/EP/A)× Split(A,P ) is faith-fully flat.

Proof of the claim:The flatness is clear from claim 4.4.18. Hence, to prove the claim, it sufficesto show that((A/EP/A)× Split(A,P ))⊗A (A/J) 6= 0 for every proper idealJ ⊂ A,. But theconstruction ofφ commutes with arbitrary base changesA → A′, therefore we are reduced toverify that(A/EP/A)× Split(A,P ) 6= 0 whenA 6= 0. By claim 4.4.17, this can fail only whenP = 0; but in this caseEP/A = 0, so the claim follows.

We can now conclude the proof of the theorem: define inductively as in the proof of claim4.4.18:A0 := A,Q0 := P ,Ai+1 := Split(Ai, Qi) andQi+1 as anAi+1-module such thatQi⊗Ai

Ai+1 = Ai+1 ⊕ Qi+1, for everyi ≥ 0. The same argument as inloc. cit. shows that, for everyε ∈ m, there existsn ∈ N such thatεAn = 0. We may assume thatεAn−1 6= 0. Moreover, byclaim 4.4.21 (and an easy induction),B := A0/EQ0/A0×A1/EQ1/A1×...×An−1/EQn−1/An−1×Anis a faithfully flatA-algebra. However, one checks easily by induction thatP ⊗A (Ai/EQi/Ai

)is a freeAi/EQi/Ai

-module of ranki, for everyi < n. Hence,ΛnB(P ⊗A B) ≃ Λn

An(P ⊗A An),

which is therefore killed byε. By faithful flatness, so isΛnAP . The proof is concluded.

Proposition 4.4.22. If P is a faithfully flat almost projectiveA-module of almost finite rank,thenSplit(A,P ) is faithfully flat overA.

Proof. If A = 0 there is nothing to prove, so we assume thatA 6= 0. In this case, it suffices toshow thatSplit(A,P ) ⊗A A/I 6= 0 for every proper idealI of A. However,Split(A,P ) ⊗AA/I ≃ Split(A/I, P/IP ), and sinceP is faithfully flat, P/IP 6= 0; hence we are reduced toshowing thatSplit(A,P ) 6= 0 whenP is faithfully flat. Suppose thatSplit(A,P ) = 0; then(4.4.4) implies that for everyε ∈ m there existsj ≥ 0 such thatε · ζjP = 0. SinceP 6= 0,we haveεP 6= 0 for someε ∈ m. From the flatness ofP , we deriveAnnSym•

A(P⊕P ∗)(ε) =


AnnA(ε) ·Sym•A(P⊕P ∗), hence we can replaceA byA/AnnA(ε), P byP/AnnA(ε) ·P , whichallows us to assume thatζP is nilpotent inSplit(A,P ). Using lemmata 4.4.12 and 4.4.3(ii), wecan further assume thatA∗ contains eitherQ or one of the finite fieldsFp. If Q ⊂ A∗, thenk!is invertible inA∗ for everyk ≥ 0; by lemma 4.4.9 it follows thatSymk

AP = 0, whenceP = 0by theorem 4.4.16(iv), which contradicts our assumptions,so the proposition is proved in thiscase. Finally, suppose thatFp ⊂ A∗, then by lemma 4.4.11(ii),EP/A is Frobenius-nilpotent.However, sinceP is faithfully flat, proposition 2.4.28(iv) says thatEP/A = A, soA = 0, whichagain contradicts our assumptions.

4.4.23. For anyV a-algebraA we have a (large) fpqc site on the category(A-Alg)o (in somefixed universe!); as usual, this site is defined by the pretopology whose covering families arethe finite familiesSpecCi → SpecB | i = 1, ..., n such that the induced morphismB →C1 × ...× Cn is faithfully flat (notation of (3.3.3)).

Theorem 4.4.24.Every almost projectiveA-module of finite rank is locally free of finite rankin the fpqc topology of(A-Alg)o.

Proof. We iterate the construction ofSplit(A,P ) to split off successive free submodules of rankone. We use the previous characterization of modules of finite rank (proposition 4.3.27) to showthat this procedure stops after finitely many iterations. Byproposition 4.4.22, the output of thisprocedure is a faithfully flatA-algebra.

Theorem 4.4.24 allows to prove easily results on almost projective modules of finite rank, byreduction to the case of free modules. Here are a few examplesof this method.

Lemma 4.4.25.Let P be an almost projectiveA-module of constant rank equal tor ∈ N.Then, for every integer0 ≤ k ≤ r, the natural morphism

ΛkAP ⊗A Λr−k

A P → ΛrAP x⊗ y 7→ x ∧ y(4.4.26)

is a perfect pairing.

Proof. By theorem 4.4.24, there exists a faithfully flatA-algebraB such thatPB := P ⊗A Bis a freeB-module of rankr. It suffices to prove the assertion for theB-modulePB, in whichcase the claim is well known.

4.4.27. Keep the assumptions of lemma 4.4.25. Takingk = 1 in (4.4.26), we derive a naturalisomorphism

βP : (Λr−1A P )∗

∼→ P ⊗A (ΛrAP )∗.

Now, let us consider anA-linear morphismφ : P → Q of A-modules of constant rank equal tor. We set

ψ := βP (Λr−1A φ)∗ β−1

Q : Q⊗A (ΛrAQ)∗ → P ⊗A (Λr

AP )∗.

Proposition 4.4.28.With the notation of(4.4.27), we have:

ψ (φ⊗A 1(ΛrAQ)∗) = 1P ⊗A (Λr

Aφ)∗ and (φ⊗A 1(ΛrAP )∗) ψ = 1Q ⊗A (Λr

Aφ)∗.

Especially,φ is an isomorphism if and only if the same holds forΛrAφ.

Proof. After faithfully flat base change, we can assume thatP andQ are free modules of rankr. Then we recognize Cramer’s rule in the above identities.


4.4.29. Keep the assumption of lemma 4.4.25 and letφ be anA-linear endomorphism ofP .SetB := A[X,X−1] andPB := P ⊗A B. Obviouslyφ induces aΛ-nilpotent endomorphism ofPB, which we denote by the same letter. Hence we can define

χφ(X) := Xr · det(1PB−X−1 · φ) ∈ A∗[X]

(notation of (4.3.1)).

Proposition 4.4.30.With the notation of(4.4.29), we haveχφ(φ) = 0 in EndA(P ).

Proof. Again, we can reduce to the case of a freeA-module of constant finite rank, in whichcase we conclude by Cayley-Hamilton.

Corollary 4.4.31. Keep the assumptions of(4.4.29), and suppose thatφ is integral over a sub-ring S ⊂ A∗. Then the coefficients ofχφ are integral overS.

Proof. The assumption means thatφ satisfies an equation of the kindφn +∑n−1

i=0 aiφi = 0,

whereai ∈ S for every i = 0, ..., n − 1. We can assume thatP is free, in which case wereduce to the case of an endomorphism of a freeR-module of finite rank, whereR is a usualcommutative ring containingS; we can further suppose thatR is of finite type overZ, and itis easily seen that we can even replaceR by its associated reduced ringR/nil(R). ThenRinjects into a finite product of fields

∏iKi, and we can replaceS by its integral closure in∏

iKi, which allows us to reduce to the case whereR is a field. In this case the coefficientsof χφ are elementary symmetric polynomials in the eigenvalues ofφ, so it suffices to show thatthese eigenvaluese1, ..., er are integral overS. But this is clear, since we have more preciselyenj +

∑n−1i=0 aie

i = 0 for everyj ≤ r.

To conclude this section, we want to apply the previous results to analyze in some detail thestructure of invertible modules : it turns out that the notion of invertibility is rather more subtlethan for usual modules over rings.

Definition 4.4.32. Let M be an invertibleA-module. ClearlyM ⊗A M is invertible as well,consequently the mapA → EndA(M ⊗A M) → A : a 7→ a · 1M⊗AM is an isomorphism(by the proof of lemma 4.1.5(iii)). Especially, for the transposition endomorphismθM |M ofM ⊗A M : x ⊗ y 7→ y ⊗ x, there exists a unique elementuM ∈ A∗ such thatθM |M =uM · 1M⊗AM . Clearlyu2

M = 1. We say thatM is strictly invertibleif uM = 1.

Lemma 4.4.33.For an invertibleA-module the following are equivalent:

(i) M is strictly invertible;(ii) Λ2

AM = 0;(iii) M is of almost finite rank;(iv) there exists a faithfully flatA-algebraB such thatM ⊗A B ≃ B.

Proof. (i) ⇒ (ii): indeed, the conditionuM = 1 says that the antisymmetrizer operatora2 :M⊗2 → M⊗2 vanishes (cp. the proof of lemma 4.3.26); sinceM is flat, (ii) follows.

(ii) ⇒ (iii) and (iv)⇒ (i) are obvious. To show that (iii)⇒ (iv) let us setB := Split(A,M);by proposition 4.4.22B is faithfully flat overA, andB ⊗A M ≃ B ⊕ X for someB-moduleX. ClearlyB ⊗A M is an invertibleB-module, therefore, by lemma 4.1.5(ii), the evaluationmorphism gives an isomorphism(B⊕X)⊗A (B⊕X∗) ≃ B⊕X⊕X∗⊕ (X⊗AX∗) ≃ B. Byinspection, the restriction of the latter morphism to the direct summandB equals the identity ofB; henceX = 0 and (iv) follows.

Lemma 4.4.34. If M is invertible, thentrM/A(1M) = uM .


Proof. Pick arbitraryf ∈M∗∗ ,m,n ∈ M∗. Then, directly from the definition ofuM we deducethatf(m) · n = uM · f(n) ·m. In other words,ωM/A(n ⊗ f) = uM · evM/A(n⊗ f) · 1M . Bylinearity we deduce thatφ = uM · trM/A(φ) · 1M for everyφ ∈ EndA(M). By lettingφ := 1M ,and taking traces on both sides, we obtain:trM/A(1M) = uM · trM/A(1M)2. But sinceM isinvertible,trM/A(1M) is invertible inA∗, whenceuM · trM/A(1M) = 1, which is equivalent tothe sought identity.

Proposition 4.4.35.LetM be an invertibleA-module. Then:

(i) M ⊗AM is strictly invertible.(ii) There exists a natural decompositionA ≃ A1×A−1 whereM ⊗AA1 is strictly invertible,

A−1∗ is aQ-algebra andSym2A−1

(M ⊗A A−1) = 0.

Proof. (i): it is clear thatM⊗n is invertible for everyn. Let σ ∈ Sn be any permutation; it iseasy to verify that the morphismσM : M⊗n → M⊗n : x1 ⊗ x2 ⊗ ...⊗ xn 7→ xσ(1) ⊗ xσ(2) ⊗...⊗xσ(n) equalsusgn(σ)

M ·1M⊗n. Especially, the transposition operator on(M ⊗AM)⊗2 acts viathe permutation:x⊗ y ⊗ z ⊗w 7→ z ⊗w ⊗ x⊗ y whose sign is even. ThereforeuM⊗AM = 1,which is (i).

(ii): It follows from (i) that the antisymmetrizer operatora2 on (M ⊗A M)⊗2 vanishes;a fortiori it vanishes on the quotient(Λ2

AM)⊗2, thereforeΛ2A(Λ2

AM) ≃ Im(a2 : Λ2AM →

Λ2AM) = 0. Then lemma 4.3.12 says that there exists a natural decomposition A ≃ A1 × A−1

such that(Λ2AM)⊗AA1 = 0 and(Λ2

AM)⊗AA−1 is invertible. To show thatA−1∗ is aQ-algebra,it is enough to show thatA−1/pA−1 = 0 for every primep. Up to replacingA by A/pA, wereduce to verifying that, ifFp ⊂ A∗ andM is invertible, thenM is of almost finite rank. Tothis aim, it suffices to verify that the equivalent condition(iv) of theorem 4.4.16 is satisfied.If B 6= 0 andMB := M ⊗A B ≃ B ⊕ X, then the argument in the proof of lemma 4.4.33shows thatX = 0 and thereforeMB does not admit infinite splittings. Finally, it remains onlyto verify condition (a) ofloc. cit. So suppose thatMB ≃ Bn ⊕ Q. If n > 0, we have just seenthatQ = 0; if n = 0, andQ/IQ = 0 for some idealI, then by the faithfulness ofM (lemma4.1.5(iii)) we must haveI = B; if I is Frobenius nilpotent it follows thatB = 0. Finally,setM−1 := M ⊗A A−1; notice that, sinceA−1∗ is a Q-algebra, the endomorphism group ofGm,A−1∗ is isomorphic toA−1∗, and thereforeχM−1(X) = (1 +X)α, whereα is an element ofA−1∗ which can be determined by looking at the coefficient ofχM−1(X) in degree1. One findsα = trM−1/A−1

(1M−1). In view of lemma 4.4.34, we can rewriteα = uM−1; therefore

trΛ2A−1

M−1/A−1(1Λ2

A−1M−1

) =

(uM−1

2

).(4.4.36)

On the other hand, sinceΛ2A−1

M−1 is an invertibleA−1-module of finite rank, we know that theleft-hand side of (4.4.36) equals1; consequentlyuM−1 = −1. This means that, inM⊗2

−1 , theidentity x ⊗ y = −y ⊗ x holds for everyx, y ∈ M−1∗; therefore, the kernel of the projectionM⊗2−1 → Sym2

A−1M−1 contains all the elements of the form2 · x ⊗ y; in other words, multipli-

cation by2 is the zero morphism inSym2A−1

M−1; sinceA−1∗ is aQ-algebra, this at last showsthatSym2

A−1M−1 vanishes, and concludes the proof of the proposition.

4.5. Construction of quotients by flat equivalence relations.We will need to recall somegeneralities on groupoids, which we borrow from [25, Exp. V].

4.5.1. If C is any category admitting fibred products and a final object, aC -groupoid is thedatum of two objectsX0,X1 of C , together with ”source” and ”target” morphismss, t : X1 →X0, an ”identity” morphismι : X0 → X1 and a further ”composition” morphismc : X2 → X1,


whereX2 is the fibre product in the cartesian diagram:

X2t′ //

s′

X1

s

X1

t // X0.

The datum(X0, X1, s, t, c, ι) is subject to the following condition. For every objectS of C ,the setX0(S) := HomC (S,X0) is a groupoid, with set of morphisms given byX1(S), and foreveryφ ∈ X1(S), the source and target ofφ are respectivelys(φ) := φ s andt(s) := φ s;furthermore the composition law inX0(S) is given byc(S) : X1(S)×X0(S) X1(S) → X1(S).The above conditions amount to saying thats ι = t ι = 1X0 and the commutative diagrams

X2

c //

t′//

s′

X1

s

X2c //

t′

X1

t

X1

s //

t// X0 X1

t // X0

(4.5.2)

are cartesian inC both for the square made up from the upper arrows and for the square madeup from the lower arrows (cp. [25, Exp. V§1]).

One says that the groupoidG := (X0, X1, s, t, c, ι) has trivial automorphisms, if the mor-phism(s, t) : X1 → X0 ×X0 is a (categorical) monomorphism. (This translates in categoricalterms the requirement that for every objectS of C , and everyx ∈ X0(S), the automorphismgroup ofx in X0(S) is trivial).

It is sometimes convenient to denote byX ×(α,β) Z the fibre product of two morphismsα : X → Y andβ : Z → Y .

4.5.3. Given a groupoidG, and a morphismX0 → X ′0, we obtain a new groupoidG×X0 X′0

by taking the datum(X ′0, X1 ×X0 X′0, s×X0 1X′

0, t×X0 1X′

0, c×X0 1X′

0, ι×X0 1X′

0).

Moreover, suppose thatC admits finite coproducts and that all such coproducts are disjointuniversal (cp. [4, Exp.II, Def.4.5]). Denote byY ∐ Z the coproduct of two objectsY andZ ofC . LetG′ := (X ′0, X

′1, s′, t′, c′, ι′) be another groupoid; one can define a groupoidG ∐ G′ by

taking the datum(X0 ∐X ′0, X1 ∐X ′1, s∐ s′, t∐ t′, c∐ c′, ι∐ ι′).

4.5.4. In the following we will be concerned with groupoids in the categoryA-Algo of A-schemes, whereA is anyV a-algebra. We will use the general terminology for almost schemesintroduced in (3.3.3), and complemented by the following:

Definition 4.5.5. Let φ : X → Y be a morphism ofA-schemes,G := (X0, X1, s, t, c, ι) agroupoid in the category ofA-schemes.

(i) We say thatφ is aclosed imbedding(resp. isalmost finite, resp. isetale, resp. isflat, resp.is almost projective) if the corresponding morphismφ♯ : OY → OX is an epimorphismof OY -modules (resp. enjoys the same property). We say thatφ is anopen and closedimbeddingif it induces an isomorphismX

∼→ Y1 onto one of the factors of a decompositionY = Y1 ∐ Y2.

(ii) We say thatG is aclosed equivalence relationif the morphism(s, t) : X1 → X0 ×X0 isa a closed imbedding. We say thatG is flat (resp. etale, resp.almost finite, resp.almostprojective) if the morphisms : X1 → X0 enjoys the same property. We say thatG is offinite rankif OX1 is an almost projectiveOX0-module of finite rank. Furthermore, we setX0/G := Spec OG

X0, whereOG

X0⊂ OX0 is the equalizer of the morphismss♯ andt♯.


4.5.6. LetG := (X0, X1, s, t, c, ι) be a groupoid of finite rank inA-Algo. by assumptionOX1 is an almost projectiveOX0-module of finite rank, hence, by proposition 4.3.27, there is adecompositionOX0 ≃

∏ri=0Bi such thatCi := OX1 ⊗OX0

Bi is of constant rank equal toi, fori = 0, ..., r. SetX0,i := SpecBi.

Lemma 4.5.7. In the situation of(4.5.6), there is a natural isomorphism of groupoids:

G ≃ (G×X0 X0,1)∐ ...∐ (G×X0 X0,r).

Proof. For everyi ≤ r, letαi : X0,i → X0 be the open and closed imbedding defined by (4.5.6).SetX1,i := X0,i ×(αi,s) X1 (soX1,i = SpecCi). Moreover, letX ′1,i := X0,i ×(αi,t) X1, βi :X ′1,i → X1 the open and closed imbedding (obtained by pulling backαi),X2,i := X1,i×(βi,s′)X2

andX2,i := X ′1,i ×(βi,s′) X2. There follow natural decompositionsX1 ≃ X ′1,1 ∐ ... ∐X ′1,r andX2 ≃ X ′2,1 ∐ ... ∐ X ′2,r, such thats′ decomposes as a coproduct of morphismsX ′2,i → X ′1,i.By the construction ofX0,i, it is clear thatX ′o2,i has rank equal toi as anX ′o1,i-module, for everyi ≤ r. In other words, the above decompositions fulfill the conditions of proposition 4.3.27.Similarly, we obtain decompositionsX1 ≃ X1,1 ∐ ... ∐ X1,r andX2 ≃ X2,1 ∐ ... ∐ X2,r

which fulfill the same conditions. However, these conditions characterize uniquely the factorsoccuring in it, thusX1,i = X ′1,i for i ≤ r. The claim follows easily.

Lemma 4.5.8. LetG := (X0, X1, s, t, c, ι) be a groupoid of finite rank inA-Algo. If G hastrivial automorphisms, then it is a closed equivalence relation.

Proof. Using lemma 4.5.7 we reduce easily to the case whereOX1 is of constant rank, say equalto r ∈ N. LetY := X0×X0; sinceG has trivial automorphisms, the morphism(s, t) : X1 → Yis a monomorphism; equivalently, the natural projectionspr1, pr2 : X1 ×Y X1 → X1 areisomorphisms. LetD := Im((s, t)♯ : OY → OX1); it follows that the natural morphismspr♯1, pr♯2 : OX1 → OX1 ⊗D OX1 are isomorphisms and consequently,

(OX1/D)⊗D OX1 = 0.(4.5.9)

We need to show thatOX1 = D, or equivalently, thatOX1/D = 0. However, by theorem4.4.24, we can find a faithfully flatOX0-algebraB such thatC := B ⊗OX0

OX1 ≃ Br. LetD′ := B ⊗B D; it follows thatC is a faithful finitely generatedD′-module. It suffices to showthatC/D′ = 0, and we know already from (4.5.9) that(C/D′)⊗D′C = 0. By proposition 3.4.4it follows thatC/D′ is a flatD′-module; consequentlyC/D′ ⊂ (C/D′) ⊗D′ C, and the claimfollows.

4.5.10. LetB be anA-algebra,P an almost finitely generated projectiveB-module. For everyintegeri ≥ 0, we define aB-linear morphism

ΓiB(EndB(P )a)→ B(4.5.11)

as follows (see (8.1.13) for the definition of the functorΓiB : B-Mod → B-Mod). LetR beanyB∗-algebra; we remark that the natural mapβP : EndB(P )a! ⊗B∗ R → EndRa(Ra ⊗A P )a!is an isomorphism. Hence, for everyi ≥ 0, we can define a map of sets

λiR : EndB(P )a! ⊗B∗ R→ EndB(ΛiBP )a! ⊗B∗ R

by lettingφ 7→ β−1Λi

BP(Λi

RaβP (φ)a). In the terminology of [61], the system of mapsλiR forms

a homogeneous polynomial law of degreei from EndB(P )a! to EndB(ΛiBP )a! , so it induces a

B∗-linear mapλi : ΓiB∗(EndB(P )a! ) → EndB(Λi

BP )a! . After passing to almost modules, weobtain aB-linear morphism

ΓiB(EndB(P )a)→ EndB(ΛiBP )a.(4.5.12)

Then (4.5.11) is defined as the composition of (4.5.12) and the trace morphismtrΛiBP/B

.


4.5.13. LetC be an almost finite projectiveB-algebra. Defineµ : C → EndB(C)a as in(4.1.7). By composition ofΓiBµ and (4.5.11) we obtain aB-linear morphism

ΓiBC → B

characterized by the condition:c[i] 7→ σi(c) := trC/B(ΛiCµ(c)).

4.5.14. The construction of (4.5.13) applies especially toan almost finite projective groupoidG := (X0, X1, s, t, c, ι). In such case one verifies, using the cartesian diagrams (4.5.2), thatσi(t

♯(f)) ∈ OGX0

for everyf ∈ OX0∗ and everyi ≤ r: the argument is the same as in the proofof [25, Exp.V, Th.4.1]. In this way one obtainsOG

X0-linear morphisms

TG,i : ΓiOGX0

OX0

Γit♯−→ ΓiOX0OX1 → OG

X0f [i] 7→ σi(t

♯(f)).(4.5.15)

Theorem 4.5.16.LetG := (X0, X1, s, t, c, ι) be anetale almost finite and closed equivalencerelation inA-Algo. ThenG is effective and the natural morphismX0 → X0/G is etale andalmost finite projective.

Proof. See [25, Exp.IV,§3.3] for the definition of effective equivalence relation. By (4.5.2),we have an identificationX2 ≃ X1 ×(s,s) X1.; therefore, the natural diagonal morphismX1 →X1 ×(s,s) X1 gives a sectionδ : X1 → X2 of the morphisms′ : X2 → X1. Furthermore, sinceX2 = X1 ×(s,t) X1, the pair of morphisms(1X1 , ι s) : X1 → X1 induces another morphismψ0 : X1 → X2; similarly, letψ1 : X1 → X2 be the morphism induced by the pair(ι t, 1X1)(these are the degeneracy maps of the simplicial complex associated toG: cp. [25, Exp. V,§1]).By arguing withT -points (and exploiting the interpretation (4.5.1) ofX0(T ), X1(T ), etc.) onechecks easily, first thatψ1 = δ, and second, that the two commutative diagrams

X0ι //

ι

X1

ψ1

X1t //

ψ1

X0

ι

X1

ψ0 // X2 X2t′ // X1

(4.5.17)

are cartesian. Since by assumptions is etale, corollary 3.1.9 implies thatδ is an open andclosed imbedding; consequently the same holds forι. Let e0 ∈ OX1∗ (resp. e1 ∈ OX2∗)be the idempotent corresponding to the open and closed imbedding ι (resp. δ); sinceG is aclosed equivalence relation, for everyε ∈ m we can writeε · e0 =

∑ni s

♯(bi) · t♯(b′i) for somebi, b

′i ∈ OX1∗ . In view of (4.5.17) we deduce thatε ·e1 =

∑ni (t′♯s♯(bi)) ·(t′♯t♯(b′i)). However,

s t′ = t s′ andt t′ = t c, consequently

ε · e1 =

n∑

i

(s′♯ t♯(bi)) · (c♯ t♯(b′i)).

Finally, thanks to remark 4.1.17, and again (4.5.2), we can write:

ε · f =

n∑

i

s♯ TrX1/X0(f · t♯(bi)) · t♯(b′i) for everyf ∈ OX1∗.(4.5.18)

If we now let f := t♯(g) in (4.5.18) we deduce:ε · t♯(g) =∑

i s♯(TG,1(g · bi)) · t♯(b′i) =∑

i t♯(TG,1(g · bi)) · t♯(b′i) for everyg ∈ OX0∗. Sincet♯ is injective, this means that:

ε · g =

n∑

i

TG,1(g · bi) · b′i for everyg ∈ OX0∗.(4.5.19)

It follows easily thatOX0 is an almost finitely generated projectiveOGX0

-module. Furthermore,let us introduce the bilinear pairingtG := TG µOX0

/OGX0

: OX0 ⊗OGX0

OX0 → OGX0

.


Claim 4.5.20. tG is a perfect pairing.

Proof of the claim:We have to show that the associatedOX0-linear morphism

τG : OX0 → O∗X0:= alHomOG

X0(OX0 ,O

GX0

)

is an isomorphism. From (4.5.19) it follows easily thatτG is a monomorphism. Letφ : OX0 →OGX0

be aOGX0

-linear morphism; it remains only to show that, for everyε ∈ m, there existsb ∈ OX0∗ such thatτG(b) = ε · φ. Let α : OX1 → On

X0, β : On

X0→ OX1 be defined by the

rules:c 7→ (TrX1/X0(c · b1), ...,TrX1/X0

(c · bn)) and(x1, ..., xn) 7→∑n

i xi · b′i for all c ∈ OX1∗,x1, ..., xn ∈ OX0∗. We remark thatIm(α t♯) ⊂ (OG

X0)n and Im(β (t♯)n) ⊂ OX0, so that

we deduce, by restriction, morphismsα0 : OX0 → (OGX0

)n andβ0 : (OGX0

)n → OX0. Letψ := ((φ β0) ⊗OG

X01OX0

) α : OX1 → OX0. By theorem 4.1.14 we can find, for every

ε ∈ m, an elementc ∈ OX1∗ such thatε · ψ = τX1/X0(c). Using (4.5.18) we derive easily

ε · c = ε ·∑ni ψ(t♯(bi)) · t♯(b′i) = ε ·∑n

i φ β0 α0(bi) · t♯(b′i) = ε ·∑ni φ(ε · bi) · t♯(bi). In

particular,ε · c = t♯(b) for someb ∈ B, so the claim follows.By assumption, the morphismπ : C := OX0 ⊗OG

X0OX0 → OX1 induced by the pair(s♯, t♯) is

an epimorphism. Moreover, by construction, we have the identity:

TrX1/X0 π = 1OX0

⊗OGX0TG,1.

By claim 4.5.20 we see that1OX0⊗OG

X0TG,1 induces a perfect pairingC ⊗OX0

C → OX0 ; onthe other hand,TrX1/X0 is already a perfect pairing, by theorem 4.1.14. It then follows thatπmust be a monomorphism, henceC ≃ OX1, which shows thatG is effective; then it is easy toverify thatTG,1 is actually the trace of theOG

X0-algebraOX0, which is consequently etale over

OGX0

.

Proposition 4.5.21.Let G = (X0, X1, s, t, c, ι) be a groupoid of finite rank. ThenOX0∗ isintegral overOG

X0∗.

Proof. By lemma 4.5.7 we can reduce to the case where the rank ofOX1 is constant, say equalto r. The assertion is then a direct consequence of the following:

Claim 4.5.22. Let f ∈ OX0∗. With the notation of (4.5.14) we have:

(t♯(f))r + TG,1(f) · (t♯(f))r−1 + TG,2(f) · (t♯(f))r−2 + ...+ TG,r(f) = 0.

Proof of the claim:We apply proposition 4.4.30 (i.e. Cayley-Hamilton’s theorem) to the endo-morphismt♯(f) · 1OX1

: OX1 → OX1 .

Proposition 4.5.23.Let G be anetale closed equivalence relation of finite rank. ThenG isuniversally effective and the morphismX0 → X0/G is etale, faithfully flat and almost finiteprojective.

Proof. Everything is known by theorem 4.5.16, except for the faithfulness, which follows fromthe following:

Claim 4.5.24. Under the assumptions of proposition 4.5.21, letI ⊂ OGX0

be an ideal such thatI · OX0 = OX0. ThenI = OG

X0.

Proof of the claim:First of all, letOX0 ≃∏r

i=0Bi be the decomposition as in (4.5.6); onederives easily a corresponding decompositionOG

X0≃∏r

i=0BGi , so we can assume that the rank

of OX1 is constant, equal tor. Let J ⊂ OGX0

be any ideal, and setC := OGX0/J . We have a

natural isomorphismΓiC(OX0 ⊗OG

X0C) ≃ ΓiOG

X0

(OX0)⊗OGX0C.


Composing with (4.5.15)⊗OGX0C, we derive aC-linear morphism:ψi : ΓiC(OX0 ⊗OG

X0C)→ C.

By inspecting the construction, one shows easily thatψi(1[i]) =

(ri

)(indeed, by flat base change

one reduces easily to the case whereOX1 is a freeB-module of rankr, in which case the resultis obvious). Let us now takeJ = I. ThenΓiC(OX0 ⊗OG

X0C) = 0 for everyi > 0, whence

1 = ψr(1[r]) = 0 in C, and the claim follows.

Proposition 4.5.25.Keep the assumptions of proposition4.5.23. If X0 is an almost finite (resp.almost finitely presented, resp. flat, resp. almost projective, resp. weakly unramified, resp.unramified, resp. weaklyetale, resp.etale)SpecA-scheme, then the same holds for theSpecA-schemeX0/G.

Proof. By proposition 4.5.23,OX0 is a faithful almost finitely generated projectiveOGX0

-module,henceEOX0

/OGX0

= OGX0

by proposition 2.4.28(iv). It follows easily that, for every ε ∈ m there

existsn ∈ N such thatε · 1OGX0

factors as a composition ofOGX0

-linear morphisms:

OGX0→ On

X0→ OG

X0.(4.5.26)

The assertions for “almost finite” and for “almost projective” are immediate consequences. Toprove the assertion for “almost finitely presented” we use the criterion of proposition 2.3.15(ii).Indeed, let(Nλ, φλµ | λ) be a filtered system ofA-modules; we apply the natural transformation(2.3.16) to the sequence of morphisms (4.5.26) : sinceOX0 is almost finitely presented, so isOnX0

, hence the claim follows by a little diagram chase. The assertions for “flat” and “weaklyunramified” are easy and shall be left to the reader. To conclude, it suffices to consider theassertion for “unramified”. Now, by proposition 4.5.23 it follows thatOX0 ⊗A OX0 is an al-most finitely generated projectiveOG

X0⊗A OG

X0-module; since by assumptionOX0 is an almost

projectiveOX0 ⊗A OX0-module, we deduce from lemma 2.4.7 thatOX0 is an almost projectiveOGX0⊗AOG

X0-module. Using (4.5.26) we deduce thatOG

X0is almost projective overOG

X0⊗AOG

X0

as well.


5. HENSELIZATION AND COMPLETION OF ALMOST ALGEBRAS

This chapter deals with more advanced aspects of almost commutative algebras : we beginwith the definitions ofJacobson radicalrad(A) of an almost algebraA, of henselian pairandhenselizationof a pair(A, I), whereI ⊂ A is an ideal contained inrad(A). These notionsare especially well behaved whenI is atight ideal (definition 5.1.5), in which case we can alsoprove a version of Nakayama’s lemma (lemma 5.1.6).

In section 5.3 we explain what is alinear topologyon anA-module and anA-algebra; as usualone is most interested in the case ofI-adic topologies. In caseA is I-adically complete andI istight, we show that the functorB 7→ B/IB from almost finitely presented etaleA-algebras toalmost finitely presented etaleA/I-algebras is an equivalence (theorem 5.3.27). For the proofwe need some criteria to ensure that anA-algebra is unramified under various conditions : suchresults are collected in section 5.2, especially in theorem5.2.12 and its corollary 5.2.15.

In section 5.5, theorem 5.3.27 is further generalized to thecase where the pair(A, I) istight henselian (see theorem 5.5.6, that also contains an analogous statement concerning almostfinitely generated projectiveA-modules). The proof is a formal patching argument, which canbe outlined as follows. First one reduces to the case whereI is principal, say generated byf ,and sinceI is tight, one can assume thatf ∈ m; hence, we want to show that a given etalealmost finitely presentedA/fA-algebraB0 lifts uniquely to anA-algebraB of the same type;in view of section 5.3 one can liftB0 to an etale algebraB∧ over thef -adic completionA∧

of A; on the other hand, the almost spectrumSpecA is a usual scheme away from the closedsubscheme defined byI, so we can use standard algebraic geometry to liftB0[f

−1] to an etalealgebraB′ overA[f−1]. Finally we need to show thatB∧ andB′ can be patched in a uniqueway overSpecA; this amounts to showing that certain commutative diagramsof functors are2-cartesian (proposition 5.5.5).

The techniques needed to constructB′ are borrowed from Elkik’s paper [31]; for our purposewe need to extend and refine slightly Elkik’s results, to dealwith non-noetherian rings. Thismaterial is presented in section 5.4; its usefulness transcends the modest applications to almostring theory presented here.

The second main thread of the chapter is the study of thesmooth locusof an almost scheme;first we consider the affine case: as usual, an affine schemeX over S := SpecA can beidentified with the fpqc sheaf that it represents; then the smooth locusXsm of X is a certainnatural subsheaf, defined in terms of the cotangent complexLX/S . To proceed beyond sim-ple generalities one needs to impose some finiteness conditions onX, whence the definitionof almost finitely presentedscheme overS. For such affineS-schemes we can show that thesmooth locus enjoys a property which we could callalmost formal smoothness. Namely, sup-pose thatI ⊂ A is a tight ideal such that the pair(A, I) is henselian; suppose furthermore thatσ0 : S0 := SpecA/I → X is a section lying in the smooth locus ofX; in this situation it doesnot necessarily follow thatσ0 extends to a full sectionσ : S → X, howeverσ always existsif σ0 extends to a section over some closed subscheme of the formSpecA/m0I (for a finitelygenerated subidealm0 ⊂ m).

Next we consider quasi-projective almost schemes; ifX is such a scheme, the invertible sheafOX(1) defines a quasi-affineGm-torsorY → X, and we define the smooth locusXsm just asthe projection of the smooth locus ofY . This is presumably not the best way to defineXsm,but anyway it suffices for the applications of section 5.8. Inthe latter we consider again a tighthenselian pair(A, I), and we study some deformation problems forG-torsors, whereG is aclosed subgroup scheme ofGLn defined overSpecA and fulfilling certain general assumptions(see (5.8.4)). For instance, theorem 5.8.21 says that everyG-torsor over the closed subschemeSpecA/I extends to aG-torsor over the whole ofSpecA; the extension is however not unique,


but any two such extensions are close in a precise sense (theorem 5.8.19) : here the almost for-mal smoothness of the quasi-projective almost scheme(GLn/G)a comes into play and accountsfor the special quirks of the situation.

5.1. Henselian pairs.

Definition 5.1.1. Let A be aV a-algebra. TheJacobson radicalof A is the idealrad(A) :=rad(A∗)

a (where, for a ringR, we have denoted byrad(R) the usual Jacobson ideal ofR).

Lemma 5.1.2. Let R be aV -algebra,I ⊂ R an ideal. ThenIa ⊂ rad(Ra) if and only ifmI ⊂ rad(R).

Proof. Let us remark the following:

Claim5.1.3. If S is any ring,J ⊂ S an ideal, thenJ ⊂ rad(S) if and only if, for everyx ∈ Jthere existsy ∈ J such that(1 + x) · (1 + y) = 1.

Proof of the claim:Suppose thatJ ⊂ rad(S) and letx ∈ J ; then1 + x is not contained in anymaximal ideal ofS, so it is invertible. Find someu ∈ S with u · (1+x) = 1; settingy := u−1,we derivey = −x − xy ∈ J . Conversely, suppose that the condition of the claim holds for allx ∈ J . Letx ∈ J ; we have to show thatx ∈ rad(S). If this were not the case, there would be amaximal idealq ⊂ S such thatx /∈ q; then we could finda ∈ S such thatx · a ≡ −1 (mod q),so1 + x · a ∈ q, especially1 + a · x is not invertible, which contradicts the assumption.

Let φ : mI → R→ Ra∗ be the natural composed map.

Claim5.1.4. Imφ is an ideal ofRa∗ andmI ⊂ rad(R) if and only if Imφ ⊂ rad(Ra

∗).

Proof of the claim:The first assertion is easy to check, and clearly we only have to verify the”if” direction of the second assertion, so suppose thatImφ ⊂ rad(Ra

∗). Notice thatKerφ is asquare-zero ideal ofR. Then, using claim 5.1.3, we deduce that for everyx ∈ m · I there existsz ∈ m · I such that(1 + x) · (1 + z) = 1 + a, wherea ∈ Kerφ, soa2 = 0. Consequently(1 + x) · (1 + z) · (1− a) = 1, so the elementy := z − a− z · a fulfills the condition of claim5.1.3.

On the other hand, it is clear thatIa ⊂ rad(Ra) := rad(Ra∗)a if and only if Imφ ⊂ rad(Ra

∗),so the lemma follows from claim 5.1.4.

Given aV a-algebraA and an idealI ⊂ rad(A), one can ask whether the obvious analogue ofNakayama’s lemma holds for almost finitely generatedA-modules. As stated in lemma 5.1.6,this is indeed the case, at least if the idealI has the property singled out by the followingdefinition, which will play a constant role in the sequel.

Definition 5.1.5. Let I be an ideal of aV a-algebraA. We say thatI is tight if there exists afinitely generated subidealm0 ⊂ m and an integern ∈ N such thatIn ⊂ m0A.

Lemma 5.1.6. Let A be aV a-algebra,I ⊂ rad(A) a tight ideal. IfM is an almost finitelygeneratedA-module withIM = M , we haveM = 0.

Proof. Under the assumptions of the lemma we can find a finitely generatedA-moduleQ suchthatm0M ⊂ Q ⊂ M . It follows thatM = InM ⊂ m0M ⊂ Q, soM = Q and actuallyM isfinitely generated. LetM0 ⊂ M∗ be a finitely generatedA∗-submodule withMa

0 = M ; clearlymM0 ⊂ I∗M0 andm(I∗)

n ⊂ m0A∗. Therefore:

m0M0 ⊂ mn+2M0 ⊂ I∗mn+1M0 ⊂ ... ⊂ In+1

∗ mM0 ⊂ I∗m ·m0M0 ⊂ m0M0

and consequentlymM0 = m0M0 = mI∗ · m0M0. By lemma 5.1.2 we havemI∗ ⊂ rad(A∗),hencem0M0 = 0 by Nakayama’s lemma, thus finallymM0 = 0, i.e.M = 0, as claimed.


Corollary 5.1.7. LetA, I be as in lemma5.1.6; suppose thatφ : N → M is anA-linear mor-phism of almost finitely generated projectiveA-modules such thatφ⊗A1A/I is an isomorphism.Thenφ is an isomorphism.

Proof. From the assumptions we derive thatCoker(φ) ⊗A (A/I) = 0, henceCoker φ = 0,in view of lemmata 2.3.17(i) and 5.1.6. By lemma 2.3.17(ii) it follows thatKerφ is almostfinitely generated; moreover, sinceM is flat,Ker(φ)⊗A (A/I) = Ker(φ⊗A 1A/I) = 0, whenceKerφ = 0, again by lemma 5.1.6.

Definition 5.1.8. LetA be aV a-algebra,I ⊂ rad(A) an ideal. We say that(A, I) is ahenselianpair if (A∗,m · I∗) is a henselian pair. If in addition,I is tight, we say that(A, I) is a tighthenselian pair.

Remark 5.1.9. For the convenience of the reader, we recall without proofs afew facts abouthenselian pairs.

(i) For a ringR, letRred := R/nil(R) and denote by√I the radical of the idealI. Then the

pair (R, I) is henselian if and only if the same holds for the pair(Rred,√I · Rred).

(ii) Suppose thatI ⊂ rad(R); then(R, I) is a henselian pair if and only if the same holds forthe pair(Z ⊕ I, I), whereZ ⊕ I is endowed with the ring structure such that(a, x) · (b, y) :=(ab, ay + bx + xy) for everya, b ∈ Z, x, y ∈ I. Indeed, this follows easily from the followingcriterion (iii), which is shown in [60, Ch.XI,§2, Prop.1].

(iii) AssumeI ⊂ rad(R); then the pair(R, I) is henselian if and only if every monic poly-nomial p(X) ∈ R[X] such thatp(X) ≡ (X2 − X)m (mod I[X]) decomposes as a productp(X) = q(X) · r(X) whereq(X), r(X) are monic polynomials inR[X] and q(X) ≡ Xm

(mod I[X]), r(X) ≡ (X − 1)m (mod I[X]).(iv) Let J ⊂ I be a subideal. If the pair(R, I) is henselian, the same holds for(R, J).(v) If (R, I) is a henselian pair, andR → S is an integral ring homomorphism, then the pair

(S, IS) is henselian ([60, Ch.XI,§2, Prop. 2]).

Lemma 5.1.10.Let R be a V -algebra, I ⊂ rad(R) an ideal. Then the pair(Ra, Ia) ishenselian if and only the same holds for the pair(R,mI).

Proof. It comes down to checking that(R,mI) is henselian if and only if(Ra∗,mI

a∗ ) is. To this

aim, letφ : R → Ra∗ be the natural map. LetS := Imφ andJ := φ(mI) ⊂ S. SinceKerφ is

a square-zero ideal inR, it follows from remark 5.1.9(i) that(R,mI) is henselian if and only if(S, J) is. However, it is clear that the induced mapJ → mIa∗ is bijective, hence remark 5.1.9(ii)and lemma 5.1.2 say that(S, J) is henselian if and only if(Ra

∗,mIa∗ ) is.

5.1.11. Suppose that(R, I) is a henselian pair, withR a V -algebra. Then, in view of lemma5.1.10 and remark 5.1.9(iv) we see that(Ra, Ia) is also henselian. This gives a way of producingplenty of henselian pairs.

Lemma 5.1.12.LetB be an almost finiteA-algebra,I ⊂ A an ideal.

(i) The induced ring homomorphismA∗ → φ(A∗) + mB∗ is integral.(ii) If I ⊂ rad(A), thenIB ⊂ rad(B).

(iii) If (A, I) is a henselian pair, the same holds for the pair(B, IB).

Proof. (i): for a given finitely generated subidealm0 ⊂ m, pick a finitely generated submoduleQ ⊂ B∗ with m0B∗ ⊂ Q; notice that(m0Q)2 ⊂ m0Q, henceφ(A∗) + m0Q is a subring ofB∗,finite overA∗. As φ(A∗) + mB∗ is a filtered union of such subrings, the assertion follows. (ii)follows from (i) and from lemma 5.1.2. (iii) is a direct consequence of (i), of remark 5.1.9(v)and of (5.1.11).


5.1.13. Given aV a-algebraA and an idealI ⊂ A, we define thehenselizationof the pair(A, I) as the unique pair (up to unique isomorphism)(Ah, Ih) which satisfies the (almost versionof the) usual universal property (cp. [60, Ch.XI,§2, Def.4]). SupposeA = Ra andI = Ja fora V -algebraR and an idealJ ⊂ R, and let(R′, J ′) be a henselization of the pair(R,mJ);one can easily check that(R′a, J ′a) is a henselization of(A, I). It follows in particular that, ifI ⊂ rad(A) and(Ah, Ih) is a henselization of(A, I), then the morphismA → Ah is faithfullyflat ([60, Ch.XI,§2]).

Lemma 5.1.14.Let m0 ⊂ m be a finitely generated subideal. Then there exists an integern := n(m0) > 0 such that((m0A)∗)

n ⊂ m0A∗ for anyV a-algebraA.

Proof. We proceed by induction on the numberk of generators ofm0. To start out, letB be anyV a-algebra andf ∈ B∗ any almost element. The endomorphismB → B : x 7→ f · x inducesan isomorphismα : B := B/AnnB(f)

∼→ fB, and a commutative diagram:

B∗ ⊗V B∗µB∗ //

α∗⊗α∗

B∗

f ·α∗

(fB)∗ ⊗V (fB)∗

µB∗ // (fB)∗.

It follows that

f · (fB)∗ = ((fB)∗)2(5.1.15)

as ideals ofB∗, which takes care of the casek = 1. Suppose now thatk > 1; let us writem0 = x1·V +m1, wherem1 is generated byk−1 elements. We apply (5.1.15) withB := A/m1Aandf = x1 to deduce:

(m0A/m1A)∗ · (m0A/m1A)∗ = x1 · (m0A/m1A)∗ ⊂ Im((x1A)∗ → (A/m1A)∗).

Therefore((m0A)∗)2 ⊂ (x1A)∗+(m1A)∗. After raising the latter inclusion to some high power,

the inductive assumption onm1 allows to conclude.

Corollary 5.1.16. LetA be aV a-algebra,I ⊂ rad(A) a tight ideal. Then:

(i) I∗ ⊂ rad(A∗).(ii) If (A, I) is a henselian pair, then the same holds for the pair(A∗, I∗).

Proof. For integersn,m large enough we can write:(I∗)nm ⊂ (In∗ )m ⊂ (m0A)m∗ ⊂ m0A∗,

thanks to lemma 5.1.14. Then by lemma 5.1.2 we deduce(I∗)nm+1 ⊂ rad(A∗), which implies

(i). Similarly, we deduce from lemma 5.1.14 that√

mI =√I, so (ii) follows from remark

5.1.9(i).

Proposition 5.1.17.Let (A, I) be a tight henselian pair. The natural morphismA → A/Iinduces a bijection from the set of idempotents ofA∗ to the set of idempotents of(A/I)∗.

Proof. Pick an integerm > 0 and a finitely generated subidealm0 ⊂ m such thatIm ⊂ m0 ·A.We suppose first thatm has homological dimension≤ 1. By corollary 5.1.16(ii), the quotientmapA∗ → A∗/I∗ induces a bijection on idempotents. So we are reduced to showing that thenatural injective mapA∗/I∗ → (A/I)∗ induces a surjection on idempotents. To this aim, itsuffices to show that an idempotent almost elemente : V a → A/I always lifts to an almostelemente : V a → A; indeed, the image ofe insideA∗/I∗ will then necessarily agree withe.Now, the obstruction to the existence ofe is a classω1 ∈ Ext1

V a(V a, I). On the other hand,proposition 2.5.13(i) ensures that, for every integern > 0, e admits a unique idempotent lifting


en : V a → A/In. Let more generallyωn ∈ Ext1V a(V a, In) be the obstruction to the existence

of a lifting of en to an almost element ofA; the imbeddingIn ⊂ I induces a map

Ext1V a(V a, In)→ Ext1

V a(V a, I)(5.1.18)

and clearly the image ofωn under (5.1.18) agrees withω1. Thus, the proposition will follow inthis case from the following:

Claim 5.1.19. Forn sufficiently large, the map (5.1.18) vanishes identically.

Proof of the claim:We will prove more precisely that the natural map

Ext1V a(V a,m0I)→ Ext1

V a(V a, I)(5.1.20)

vanishes; the claim will follow forn := m + 1. A choice of generatorsε1, ..., εk for m0

determines an epimorphismφ : I⊕k → m0I; notice thatExt2V a(V a,Kerφ) = 0 due to lemma

2.4.14(i),(ii) and the assumption on the homological dimension ofm. Thus, the induced map

Ext1V a(V a, I⊕k)→ Ext1

V a(V a,m0I)(5.1.21)

is surjective. To prove the claim, it suffices therefore to show that the composition of (5.1.21)and (5.1.20) vanishes, which is obvious, since the modules in question are almost zero.

Finally suppose thatm is arbitrary; by theorem 2.1.12(i.b),(ii.b),m is the colimit of a filteredfamily of subideals(mα ⊂ m | λ ∈ Λ), such thatm0 ⊂ mλ = m2

λ and mλ := mλ ⊗V mλ

is V -flat for everyλ ∈ Λ, and moreover eachmλ has homological dimension≤ 1. We maysuppose that(A, I) = (R, J)a for a henselian pair(R, I), whereR is aV -algebra andJm ⊂m0R. Each pair(V,mλ) is a basic setup, and we denote by(Aλ, Iλ) the tight henselian paircorresponding to(R, J) in the almost category associated to(V,mλ) (soAλ is the image ofR under the localization functorV -Alg → (V,mλ)

a-Alg). Let e be an idempotent almostelement ofA/I (which is an object of(V,m)a-Alg); e is represented by a uniqueV -linearmapf : m → R/J and by the foregoing, for everyλ ∈ Λ the restrictionfλ : mλ → R/Jlifts to a unique idempotent mapgλ : mλ → R. By uniqueness, the mapsgλ glue to a mapcolimλ∈Λ

gλ : m→ R which is the sought lifting ofe.

5.2. Criteria for unramified morphisms. The following lemma generalizes a case of [42,Partie II, lemma 1.4.2.1].

Lemma 5.2.1. Let A → C be a morphism ofV a-algebras,f ∈ A∗ any almost element andM aC-module. Suppose thatM [f−1] is a flatC[f−1]-module,M/fM is a flatC/fC-module,TorA1 (C,A/fA) = 0 andTorAi (M,A/fA) = 0 for i = 1, 2. ThenM is a flatC-module.

Proof. Let N be anyC-module; We need to show thatTorC1 (M,N) = 0. To this aim we

consider the short exact sequence0 → K → Nj→ N [f−1] → L → 0 wherej is the natural

morphism. We haveTorC1 (M,N [f−1]) = TorC[f−1]1 (M [f−1], N [f−1]) = 0, therefore we are

reduced to showing :

Claim 5.2.2. TorC1 (M,K) = TorC2 (M,L) = 0.

Proof of the claim:Notice thatK =⋃k>0 AnnK(fk), and similarly forL, so it suffices to show

thatTorCi (M,Q) = 0 for i = 1, 2 and everyC-moduleQ such thatfkQ = 0 for some integerk > 0. By considering the short exact sequence0 → AnnQ(fk−1) → Q → Q′ → 0, an easyinduction onk further reduces to showing thatTorCi (M,Q) = 0 for i = 1, 2, in casefQ = 0.However, the morphismsC → C/fC andA → C determine base change spectral sequences(cp. [67, Th.5.6.6])

E2pq :=TorC/fCp (TorCq (M,C/fC), Q)⇒ TorCp+q(M,Q)F 2pq :=TorCp (TorAq (C,A/fA),M)⇒ TorAp+q(M,A/fA).


The spectral sequenceF yields an exact sequence:

TorA2 (M,A/fA) // TorC2 (C/fC,M) // TorA1 (C,A/fA)⊗C M EDBC

GF@A// TorA1 (M,A/fA) // TorC1 (C/fC,M) // 0

which impliesTorCi (M,C/fC) = 0 for i = 1, 2. ThusE2pq = 0 for q = 1, 2 and we get

TorCi (M,Q) ≃ TorC/fCi (M/fM,Q) for i = 1, 2.

SinceM/fM is a flatC/fC-module, the claim follows.

Lemma 5.2.3. Let M be anA-module,f ∈ A∗ an almost element, and denote byA∧ :=limn∈N

(A/fnA) thef -adic completion ofA. Then we have:

(i) AnnA(M [f−1]) · AnnA(M ⊗A A∧) ⊂ AnnA(M).(ii) If M is almost finitely generated andM/fM = M [f−1] = 0 thenM = 0.

Proof. (i): let a ∈ AnnA(M [f−1])∗, b ∈ AnnA(M⊗AA∧)∗ and denote byµa, µb : M →M thescalar multiplication morphisms. Froma·M [f−1] = 0 we deduce thataM ⊂ ⋃n>0 AnnM(fn);it follow that the natural morphismaM → (aM) ⊗A A∧ is an isomorphism. Now the claimfollows by inspecting the commutative diagram:

M //

µa

M ⊗A A∧0=µb⊗1A∧ // M ⊗A A∧

µa⊗1A∧

aMµb // (aM)⊗A A∧ ∼ // (aM)⊗A A∧.

(ii): for a given finitely generated subidealm0 ⊂ m, pick a finitely generatedA-moduleQ ⊂ M such thatm0M ⊂ Q. By assumption,Q[f−1] = 0; hence there exists an integern ≥ 0such thatm0 · fnQ = 0, whencem2

0 · fnM = 0. HoweverM = fM by assumption, thusm2

0M = 0, and finallyM = 0, sincem0 is arbitrary.

Theorem 5.2.4.Let A be aV a-algebra,f ∈ A∗ any almost element andI ⊂ A an almostfinitely generated ideal withI2 = I. Suppose that bothI[f−1] ⊂ A[f−1] and(I + fA)/fA ⊂A/fA are generated by idempotents inA[f−1]∗ and respectively(A/fA)∗. ThenI is generatedby an idempotent ofA∗.

Proof. We start out with the following:

Claim5.2.5. Let I, J ⊂ A be two ideals such thatJ is nilpotent andI2 = I. If I := Im(I →A/J) is generated by an idempotent of(A/J)∗, thenI is generated by an idempotent ofA∗.

Proof of the claim:We apply proposition 2.5.13 to the non-unitary extension0 → J ∩ I →I → I → 0 to derive that the idempotente that generatesI lifts uniquely to an idempotente ∈ I∗. Thene induces decompositionsA ≃ A0 × A1 andI = (IA0)× (IA1) such thate = 0(resp.e = 1) onA0 (resp. onA1). We can therefore reduce to the casesA = A0 or A = A1.If A = A0, thenI ⊂ J , henceI = In = 0 for n sufficiently large. IfA = A1 then1 ∈ I∗, soI = A; in either case, the claim holds.

Claim5.2.6. In the situation of (3.4.7), suppose thatI ⊂ A0 is an ideal such thatIA1 andIA2

are generated by idempotents. ThenI is generated by an idempotent ofA0∗.


Proof of the claim:By applying termwise the functorM 7→ M∗ to (3.4.8), we obtain a com-mutative diagram (3.4.8)∗ which is still cartesian. By assumption, we can find idempotentse1 ∈ A1∗, e2 ∈ A2∗ with eiAi = IAi (i = 1, 2). It is then easy to see that the images ofe1ande2 agree inA3∗; consequently there exists a unique elemente0 ∈ A0∗ whose image inAiagrees withei for i = 1, 2. Such element is necessarily an idempotent; moreover, since thenaturalA0-linear morphismI → (IA1) × (IA2) is a monomorphism, we deduce easily that(1 − e0) · I = 0, whenceI ⊂ e0A0. LetM := (e0A0)/I; clearlyM ⊗A0 Ai = 0 for i = 1, 2.Using lemma 3.4.18(i) we deduce thatM = 0, whencee0 ∈ I, as stated.

Suppose next thatf is a regular element ofA∗. Denote byA∧ thef -adic completion ofA.One verifies easily that the natural commutative diagram

A //

A[f−1]

A∧ // A∧[f−1]

(5.2.7)

is cartesian.

Claim 5.2.8. In the situation of (5.2.7), letI ⊂ A be an ideal such that bothI[f−1] ⊂ A[f−1]andI · A∧ ⊂ A∧ are generated by idempotents. ThenI is generated by an idempotent.

Proof of the claim:As in the proof of claim 5.2.6, we find inA∗ an idempotente such thateA∧ = IA∧ andeA[f−1] = I[f−1], and deduce thatI ⊂ eA. LetM := (eA)/I; on one handwe haveM ⊗A A∧ = 0. On the other hand,M [f−1] = 0, i.e. M =

⋃n∈N AnnM(fn), which

implies thatM ⊗A A∧ = M . HenceM = 0, and the claim follows.

Claim 5.2.9. The theorem holds in caseA is f -adically complete.

Proof of the claim:Say thatIm(I → A/fA) = e1 · (A/fA), for an idempotente1 ∈ (A/fA)∗.For everyn > 0 setIn := Im(I → A/fnA); applying proposition 2.5.13 to the non-unitaryextensions

0→ (fnA/fn+1A) ∩ In+1 → In+1 → In → 0

we construct recursively a compatible system of idempotents en ∈ In∗, and by claim 5.2.5,engeneratesIn for everyn > 0; whence an elemente ∈ lim

n∈N(A/fnA)∗ ≃ (lim

n∈NA/fnA)∗ ≃ A∗.

Clearlye is again an idempotent, and it induces decompositionsA ≃ A0 × A1, I ≃ (IA0) ×(IA1), such thate = 0 (resp.e = 1) inA0 (resp. inA1). We can thus assume that eitherA = A0

or A = A1. If A = A0, thenen = 0 for everyn > 0, soI ⊂ ⋂n>0 fnA = 0. If A = A1, then

en = 1 for everyn > 0, i.e. In = A/fnA for everyn > 0, that is,I is dense inA. It followseasily thatAnnA(I) = 0. Using lemma 2.4.6 we deduceAnnA[f−1](I[f

−1]) = 0; sinceI[f−1]is generated by an idempotent, this means thatI[f−1] = A[f−1]. SetM := A/I; the foregoingshows thatM fulfills the hypotheses of lemma 5.2.3(ii), whenceM = 0, which implies theclaim.

Claim 5.2.10. The theorem holds in casef is a non-zero-divisor ofA∗.

Proof of the claim:According to claim 5.2.8, it suffices to prove that the idealIA∧ ⊂ A∧ isgenerated by an idempotent. However, sinceA∧/fA∧ ≃ A/fA, this follows directly from ourassumptions and from claim 5.2.9.

After these preparations, we are ready to prove the theorem.Let T :=⋃n>0 AnnA(fn);

by claim 5.2.10, we know already that the idealI · (A/T ) is generated by an idempotent, andthe same holds for the idealIA∧. SinceT ⊗A A∧ ≃ T , it follows that the natural morphism


φ : T → TA∧ is an epimorphism; one verifies easily thatJ := Kerφ = T ∩ (⋂n>0 f

nA), andthen it is clear thatJ2 = 0. Furthermore, it follows that the natural commutative diagram

A/J //

A/T

A∧ // A∧/TA∧

is cartesian. Taking claim 5.2.6 into account, we deduce that the image ofI in A/J is generatedby an idempotent. Lastly, we invoke claim 5.2.5 to show thatI is indeed generated by anidempotent.

Remark 5.2.11. One may wonder whether every almost finitely generated idealI ⊂ A suchthatI = I2, is generated by an idempotent. We do not know the answer to this question.

Theorem 5.2.12.Letφ : A→ B be a morphism ofV a-algebras,I, J ⊂ A any two ideals andf ∈ A∗ any almost element.

(i) If φ⊗A 1A/I andφ⊗A 1A/J are weakly unramified, then the same holds forφ⊗A 1A/IJ .(ii) If φ⊗A 1A/I andφ⊗A 1A/J are unramified, then the same holds forφ⊗A 1A/IJ .(iii) If φ is flat and bothφ⊗A 1A/fA andφ⊗A 1A[f−1] are weakly unramified, thenφ is weakly

unramified.(iv) If φ is almost finite and bothφ ⊗A 1A/fA and φ ⊗A 1A[f−1] are unramified, thenφ is

unramified.

Proof. To start out, we remark the following:

Claim5.2.13. LetA′ be anyA-algebra and setB′ := B⊗AA′; then for everyB′⊗A′B′-moduleM we have a natural isomorphism:TorB⊗AB

1 (B,M) ≃ TorB′⊗A′B′

1 (B′,M).

Proof of the claim:Notice that the short exact sequence ofB ⊗A B-modules

0→ IB/A → B ⊗A B → B → 0.(5.2.14)

is split exact as a sequence ofB-modules (anda fortiori as a sequence ofA-modules); hence(5.2.14)⊗AN is again exact for everyA-moduleN . We deduce easily thatTorB⊗AB

1 (B,N ⊗AB ⊗A B) = 0 for everyA-moduleN . On the other hand, the morphismB ⊗A B → B′ ⊗A′ B′

determines a base change spectral sequence (cp. [67, Th.5.6.6])

E2pq := TorB

′⊗A′B′

p (TorB⊗ABq (B,B′ ⊗A′ B′),M)⇒ TorB⊗AB

p+q (B,M)

for everyB′ ⊗A B′-moduleM . The foregoing yieldsE2p1 = 0 for everyp ∈ N, so the claim

follows.

In order to show (i) we may replaceA byA/IJ and therefore assume thatIJ = 0; then wehave to prove thatTorB⊗AB

1 (B,N) = 0 for everyB ⊗A B-moduleN . By a simple devissagewe can further reduce to the case where eitherIN = 0 or JN = 0. SayIN = 0; we applyclaim 5.2.13 withA′ := A/I, B′ := B/IB to deduceTorB⊗AB

1 (B,N) ≃ TorB′⊗A′B′

1 (B′, N);however the latter module vanishes by our assumption onφ⊗A 1B′ .

For (ii) we can again assume thatIJ = 0. We need to show thatB is an almost projectiveB ⊗A B-module, and we know already that it is flat by (i). SetI ′ := I(B ⊗A B), J ′ :=J(B ⊗A B); by assumptionB/IB (resp.B/JB) is almost projective overB ⊗A B/I ′ (resp.B ⊗A B/J ′); in light of proposition 3.4.21 we deduce thatB/(IB ∩ JB) is almost projectiveoverB ⊗A B/(I ′ ∩ J ′). Then, since(I ′ ∩ J ′)2 ⊂ I ′J ′ = 0, the assertion follows from lemma3.2.25(i).


Next we remark that, under the assumptions of (iii), the hypotheses of lemma 5.2.1 are ful-filled by C := B ⊗A B andM := B, so assertion (iii) holds. Finally, suppose thatφ isalmost finite; henceΩB/A is an almost finitely generatedA-module, and by assumption wehaveΩB/A ⊗A (A/fA) = ΩB/A[f−1] = 0, thereforeΩB/A = 0, in view of lemma 5.2.3(ii).Set I := Ker(B ⊗A B → B); it follows that I2 = I, and I is almost finitely generatedas anA-module, soa fortiori as aB ⊗A B-module. Moreover, from proposition 3.1.4 andour assumption onφ ⊗A 1A/fA andφ[f−1], it follows that bothI[f−1] ⊂ B ⊗A B[f−1] andIm(I → B ⊗A (B/fB)) are generated by idempotents. Then theorem 5.2.4 says that also I isgenerated by an idempotent of(B ⊗A B)∗, so thatB is almost projective as aB ⊗A B-module,by remark 3.1.8, which proves (iv).

Corollary 5.2.15. Let I ⊂ rad(A) be a tight ideal. IfB is an almost finiteA-algebra andB/IB is unramified overA/I, thenB is unramified overA.

Proof. Under the stated assumptions,ΩB/A is an almost finitely generatedA-module such thatΩB/A ⊗A A/I = 0; by lemma 5.1.6 it follows that

ΩB/A = 0.(5.2.16)

SetIB/A := Ker(µB/A : B ⊗A B → B); we derive thatIB/A = I2B/A. Pickn > 0 and a finitely

generated subidealm0 ⊂ m with In ⊂ m0A; let ε1, ..., εk be a set of generators form0.

Claim 5.2.17. IB/A[ε−1i ] is generated by an idempotent ofB ⊗A B[ε−1

i ], for everyi ≤ k.

Proof of the claim:Notice thatA[ε−1i ] is a (usual)V -algebra (that is, the localization functor

A[ε−1i ]∗-Mod → A[ε−1

i ]-Mod is an equivalence) andB[ε−1i ] is a finiteA[ε−1

i ]-algebra; from(5.2.16) and [40, Exp.I, Prop.3.1] we deduce thatB[ε−1

i ] is unramified overA[ε−1i ], which

implies the claim.

Claim 5.2.18. IB/A/m0IB/A is generated by an idempotent as well.

Proof of the claim:We apply theorem 5.2.12(ii) withJ = Im to deduce, by induction onm,thatB/Im+1B is an unramifiedA/Im+1-algebra for everym ∈ N; it follows thatB/m0B isunramified overA/m0A, whence the claim.

Using theorem 5.2.12(iv) and claims 5.2.17, 5.2.18, the assertion is now easily verified byinduction on the numberk of generators ofm0,

The following lemma shows that morphisms of unramifiedA-algebras can be read off fromtheir ”graphs”. This result will be useful in section 5.5.

5.2.19. LetA → B be an unramified morphism ofV a-algebras, andC anyA-algebra. Sup-pose thate ∈ (B ⊗A C)∗ is an idempotent; toe we can attach a morphism ofA-algebras:

Γ(e) : C → B ⊗A C → e · (B ⊗A C) c 7→ 1⊗ c 7→ e · (1⊗ c) (c ∈ C∗)and, in caseΓ(e) is an isomorphism, we obtain a morphismfe : B → C after composingΓ(e)−1 and the natural morphism

∆(e) : B → B ⊗A C → e · (B ⊗A C) b 7→ b⊗ 1 7→ e · (b⊗ 1) (b ∈ B∗)Conversely, To a given a morphismφ : B → C of A-algebras, we can associate an idempotenteφ ∈ (B ⊗A C)∗, by settingeφ := (1B ⊗A φ)(eB/A), whereeB/A is the idempotent provided byproposition 3.1.4. Then, since the commutative diagram

B ⊗A B1B⊗φ //

µB/A

B ⊗A CµC/A(φ⊗1C)

B

φ // C


is cocartesian, we deduce from proposition 3.1.4 that the corresponding morphismΓ(eφ) is anisomorphism. In fact, the following holds:

Lemma 5.2.20.The ruleφ 7→ eφ of (5.2.19)induces a natural bijection fromHomA-Alg(B ,C)onto the set of idempotentse ∈ (B ⊗A C)∗ such thatΓ(e) is an isomorphism. Its inverse is therule e 7→ φe.

Proof. Let φ : B → C be given and sete := eφ. We need to show thatfe = φ, or equivalently∆(e) = Γ(e) φ. This translates into the equality:

e · (b⊗ 1) = e · (1⊗ φ(b)) for everyb ∈ B∗.(5.2.21)

The latter can be rewritten as(1⊗ φ)(eB/A · (1⊗ b− b ⊗ 1)) = 0 which follows from identity(iii) in proposition 3.1.4. Conversely, fore given such thatΓ(e) is an isomorphism, setφ := φe;we have to check thateφ = e. By construction ofφ we have∆(e) = Γ(e) φ, which meansthat (5.2.21) holds for the pair(e, φ). Let J ⊂ B ⊗A C be the ideal generated by the elementsof the formb⊗ 1− 1⊗φ(b), for all b ∈ B∗; we know thateJ = 0, which means that the naturalmorphismB⊗AC → e(B⊗AC) factors through the natural morphismB⊗AC → eφ(B⊗AC).Since bothΓ(e) andΓ(eφ) are isomorphisms, it follows thate(B ⊗A C) = eφ(B ⊗A C), soe = eφ.

5.3. Topological algebras and modules.

Definition 5.3.1. LetA be aV a-algebra,M anA-module.(i) A linear topologyonM is a non-empty familyL of submodules ofM that satisfies the

following conditions. IfI, J ∈ L , thenI ∩ L ∈ L ; if I ∈ L andI ⊂ J , thenJ ∈ L .We say that a submoduleI is openif I ∈ L .

(ii) Let F be a family of submodules ofM . The topology generated byF is the smallestlinear topology containingF .

(iii) Let L be a linear topology onM and I ⊂ M a submodule. Theclosureof I is thesubmoduleI :=

⋂J∈L (I + J).

(iv) We say that the topologyL is completeif the natural morphismM → limI∈L

M/I is an

isomorphism.(v) Let LM , LA be topologies onM , resp. onA, andI an open ideal ofA. We say that the

topologyLM is I-preadic(resp.I-c-preadic) if the family of submodules(InM | n ∈ N)

(resp. (In ·M | n ∈ N)) is a cofinal subfamily inLM . In either case, we say thatI isan ideal of definitionfor LM . Furthermore, we say that the topologyLM is I-adic if it isI-preadic and complete. We similarly define anI-c-adictopology.

5.3.2. One defines as usual the notions ofcontinuous, resp. openmorphism ofA-modules,of induced topologyon submodules of a linearly topologized module and ofcompletionof atopological module. Notice that, ifA is anI-c-preadic topologicalV a-algebra andM anI-c-preadicA-module (for some open idealI ⊂ A) then we haveInM = In ·M for everyn ∈ N.

Lemma 5.3.3. Let M , N be twoA-modules endowed with descending filtrationsFM :=(Mn | n ∈ N), FN := (Nn | n ∈ N) so thatM0 = M , N0 = N andM (resp.N) is completefor the topology generated byFM (resp. byFN ). Suppose furthermore thatφ : M → N isa morphism ofA-modules which respect the filtrations, and such that the induced morphismsgriφ : griM → griN are epimorphisms for everyi ∈ N. Thenφ is an epimorphism.

Proof. Under the assumptions, the induced morphismφn : M/Mn → N/Nn is an epimor-phism for everyn ∈ N. Clearly it suffices to show then thatlim

n∈N

1 Ker(φn) vanishes. However,

from Coker(griφ) = 0 one deduces by snake lemma that the induced morphismKer(φi+1) →Ker(φi) is an epimorphism, so the claim follows.


5.3.4. Let(An; φn : An+1 → An | n ∈ N) be a projective system ofV a-algebras. ThenA∞ := lim

n∈NAn is naturally endowed with a linear topology, namely the topology L generated

by the family(Ker(A∞ → An) | n ∈ N). We callL theprojective topologydefiningA∞.

Lemma 5.3.5. Let (An; φn : An+1 → An | n ∈ N) be a projective system ofV a-algebras,A∞its projective limit, and suppose that:

(a) φn is an epimorphism for everyn ∈ N.(b) Kerφn = Ker(φ0 ... φn : An+1 → A0)

n+1 for everyn ∈ N.Then we have:

(i) Let I := Ker(A∞ → A0). The projective topology ofA∞ is I-c-preadic. This topology iscomplete and moreoverIn+1 = Ker(A∞ → An) for everyn ∈ N.

(ii) Conversely, suppose thatA is any complete linearly topologizedV a-algebra andI ⊂ Aan open ideal such that the topology ofA is I-c-preadic. SetAn := A/In+1 for everyn ∈ N. Then the projective system(An | n ∈ N) satisfies conditions(a)and(b).

Proof. To show (i) it suffices to show thatIn+1 = Ker(A∞ → An) for everyn ∈ N. WeendowIn+1 (resp.Ker(A∞ → An)) with the descending filtrationFi := In+1+j (resp.Gi :=Ker(A∞ → An+j)) (i ∈ N). Notice first that the natural morphismA∞ → Ak+1 induces anisomorphismI/Gk+1

∼→ Ker(Ak+1 → A0). By (b) we deduce isomorphisms

(Ik+1 +Gk+1)/Gk+1∼→ Kerφk for everyk ∈ N.(5.3.6)

Especially,Ik+1 ⊂ Ker(A∞ → Ak). Since the latter is a closed ideal, we deduceIk+1 ⊂Ker(A∞ → Ak) for everyk ∈ N. Therefore, we obtain an imbeddingIn+1 ⊂ Ker(A∞ → An)that respects the filtrationsF• andG•. SincegriG• ≃ Ker φi+n, we derive easily from (5.3.6)that the induced morphismgriF• → griG• is an epimorphism for everyi ∈ N, so (i) follows bylemma 5.3.3. Under the assumptions of (ii) it is obvious thatcondition (a) holds. To show (b)comes down to verifying the identityIn+1 + In+2 = In+1, which is obvious sinceIn+2 is anopen ideal.

Remark 5.3.7. Notice that, in the situation of lemma 5.3.5, the pair(A∞, I) is henselian. Onesees this as follows, using the criterion of remark 5.1.9(iii). Let Rn := A∗/In+1

∗, Jn :=In∗/In+1

∗ for everyn ∈ N; let alsoR := (limn∈N

Rn) and denote byJ the image oflimn∈N

Jn in

R. SinceJn is a nilpotent ideal ofRn, the pair(Rn, Jn) is henselian. Let nowp(X) ∈ R[X]and denote bypn(X) the image ofp(X) in Rn(X), for everyn ∈ N; suppose thatp0(X)decomposes inR0[X] asp0(X) = q0(X) · r0(X); by the cited criterion we derive a compatiblefamily of decompositionspn(X) = qn(X)·rn(X) inRn[X] for everyn ∈ N; hencep(X) admitsa similar decomposition inR[X], so(R, J) is a henselian pair, which implies the contention, inview of lemma 5.1.10.

Lemma 5.3.8. LetA∞ be the projective limit of a system ofV a-algebras satisfying conditions(a)and(b) of lemma5.3.5. LetJ ⊂ A∞ be a finitely generated ideal such thatJ is open. Then:

(i) J = J .(ii) Every finite system of generatorsx1, ..., xr of J defines an open morphismAr∞ → J (for

the topologies onAr∞ andJ induced fromA∞).(iii) Jk is open for everyk ∈ N.

Proof. By lemma 5.3.5 the topology onA∞ is c-preadic for some defining idealI. The assump-tion means thatIn ⊂ J for a sufficiently large integern.

Claim 5.3.9. The natural morphism:(J ∩ Im)/(J ∩ Im+1)→ Im/Im+1 is an isomorphism foreverym ≥ n.


Proof of the claim:It is an easy verification that shall be left to the reader.

Let x1, ..., xr ∈ J∗ be a system of generators forJ , andf : Ar∞ → J the correspondingepimorphism. SetMk := f−1(J ∩ Ik). To start out,Mk contains(Ik)⊕r, whenceMk is open inAr∞ for everyk ∈ N. This already shows thatf is continuous. We define a descending filtrationonMn by settingFk := IkMn for everyk ∈ N. In view of claim 5.3.9, the morphismf inducesan epimorphismMn → In/In+1, hence for everyk ∈ N, the morphisms:

Ik ·Mn → (Ik · In)/(Ik · In+1)→ In+k/In+k+1(5.3.10)

are epimorphisms as well. The composition of the morphisms (5.3.10) extends to an epimor-phismFk → In+k/In+k+1, for everyk ∈ N. In other words, if we endowIn with its I-c-preadicfiltration, thenf induces a morphismφ : Mn → In of filtered modules, such thatgr•φ is anepimorphism of graded modules. By assumptionIn is complete for its filtration. Similarly,Mn

is complete for its filtrationF•; indeed, this follows by remarking that(In+k)⊕r ⊂ Fk ⊂ (Ik)⊕r

for everyk ∈ N. Then lemma 5.3.3 says thatφ is an epimorphism,i.e. f(Mn) = In; it followsthatJ is an open ideal, hence equal to its closure. More generally,the same argument provesthat for everyk ∈ N, f(Fk) = In+k. Since by the foregoing,(Fk | k ∈ N) is a cofinal system ofopen submodules ofAr∞, we deduce thatf is an open morphism. Finally the identity

f((Jn)⊕r) = Jn+1 for everyn ∈ N

together with (ii) and an easy induction, yields (iii).

5.3.11. LetI be an ideal of definition for the projective topology on a ringA∞ as in lemma5.3.5(i). We wish now to give a criterion to ensure that the projective topology onA∞ is ac-tually I-adic. This can be achieved in caseI is tight, as shown by proposition 5.3.12, whichgeneralizes [26, Ch.0, Prop.7.2.7].

Proposition 5.3.12.LetA∞ be as in lemma5.3.8andI an ideal of definition for the projectivetopology ofA∞. Assume thatI is tight and moreover thatI/I2 is an almost finitely generatedA∞-module. Then :

(i) There existsn ∈ N and a finitely generated idealJ ⊂ A∞ such thatIn ⊂ J ⊂ I.(ii) The topology ofA∞ is I-adic.

Proof. Using the natural epimorphism(I/I2)⊗k → Ik/Ik+1 we deduce thatIk/Ik+1 is almostfinitely generated for everyk ≥ 0; then the same holds forI/Ik+1. Let m0 ⊂ m be a finitelygenerated subideal andn ∈ N such thatIn ⊂ m0; pick a finitely generated idealm1 ⊂ m withm0 ⊂ m2

1; we can find a finitely generatedA∞-moduleQ ⊂ I/In+2 such thatm1 · (I/In+2) ⊂Q; up to replacingQ by m1 ·Q, we can then achieve thatQ is generated by the images of finitelymany almost elementsx1, ..., xt of I and moreoverm0 ·(I/In+2) ⊂ Q. Then, sinceIn+2 is openwe deduce

In+1/In+2 ⊂ Q.(5.3.13)

Let J ⊂ A∞ be the ideal generated byx1, ..., xt. From (5.3.13) we deduce that the naturalmorphism(J ∩ In+1)/(J ∩ In+2) → In+1/In+2 is an isomorphism. SinceIk(J ∩ In+1) ⊂J ∩ In+1+k, the same holds more generally for the morphisms(J ∩ In+1+k)/(J ∩ In+2+k) →In+1+k/In+2+k, for everyk ∈ N. This easily implies thatJ ′ := J ∩ In+1 is densein In+1, i.e.J ′ = In+1; especially,J is open inA∞ and thereforeJ = J by lemma 5.3.8(i), which provesassertion (i). Assertion (ii) follows easily from (i) and lemma 5.3.8(iii).


5.3.14. Let now(An | n ∈ N) be an inverse system ofV a-algebras satisfying conditions (a)and (b) of lemma 5.3.5 and letI := Ker(A∞ → A0). The induced functorsAn+1-Mod →An-Mod : M 7→ An ⊗An+1 M define an inverse system of categories(An-Mod | n ∈ N). Insuch situation one can define a natural functor

A∞-Modtop → 2-limn∈N

An-Mod(5.3.15)

from the category of topologicalA∞-modules whose topology isI-c-adic (and of continuousA∞-linear morphisms) to the 2-limit of the foregoing inverse system of categories (see [43,Ch.I] for generalities on 2-categories). Namely, letI := Ker(A∞ → A0); then to anA∞-moduleM one associates the compatible system(M/In+1M | n ∈ N).

Lemma 5.3.16.The functor(5.3.15)is an equivalence of categories.

Proof. We claim that a quasi-inverse to (5.3.15) can be given by associating to any compatiblesystem(Mn | n ∈ N) theA∞-moduleM∞ := lim

n∈NMn, with the linear topology generated by the

submodulesKn := Ker(M∞ → Mn), for all n ∈ N. In order to show that this topology onM∞is I-c-adic (whereI := Ker(A∞ → A0)), it suffices to verify thatKn = In+1M∞ for everyn ∈ N. ClearlyIn+1M∞ ⊂ Kn, hence we are reduced to showing thatKn ⊂ Km + In+1M∞for everym > n or equivalently, thatKn/Km equals the image ofIn+1M∞ insideMm, whichis obvious.

In the following we will seek conditions under which (5.3.15) can be refined to equivalencesbetween interesting subcategories, for instance to almostfinitely generated, or almost projectivemodules.

Lemma 5.3.17.Let M∞ be a topologicalA∞-module whose topology isI-c-adic (for someideal of definitionI ⊂ A∞). Suppose thatN ⊂M∞ is a finitely generated submodule such thatN is open inM∞. Then :

(i) N = N .(ii) Every finite set of generatorsx1, ..., xr ofN determines an open morphismAr∞ → N .

(iii) For every open idealJ ⊂ A∞, the submoduleJN is open inM∞.

Proof. Mutatis mutandis, this is the same as the proof of lemma 5.3.8; the details can thus besafely entrusted to the reader.

Lemma 5.3.18.Keep the notation of(5.3.14)and assume thatI is a tight ideal. LetM be atopologicalA∞-module whose topology isI-c-adic and let(Mn | n ∈ N) be the image ofMunder(5.3.15). Suppose also thatM0 is an almost finitely generatedA0-module. Then:

(i) M admits an open finitely generated submodule.(ii) M is almost finitely generated.

(iii) M ⊗A∞ An ≃Mn for everyn ∈ N.

Proof. Using lemma 3.2.25(ii) and the hypothesis onM0 we deduce thatMn is almost finitelygenerated for everyn ∈ N. Choose finitely generatedm0 ⊂ m andn ≥ 0 such thatIn ⊂ m0 ·A;it follows easily that there are almost elementsx1, ..., xt of M whose images inMn+1 generatea submoduleQ such thatm0 ·Mn+1 ⊂ Q. This implies thatIn+1M/In+2M ⊂ Q. LetN ⊂Mbe the submodule generated byx1, ..., xt. Then, arguing as in the proof of proposition 5.3.12we see thatIn+1M ⊂ N . HenceN = N in view of lemma 5.3.17(i), so (i) holds. Furthermore,by lemma 5.3.17(iii),In ·M is open for everyn ∈ N (since it containsIn ·N), so (iii) followseasily. Finally,M is almost finitely generated because the natural morphismMn → M/N is anepimorphism.


5.3.19. In the situation of (5.3.14), letM be anA∞-module with anI-c-adic topology, andlet (Mn | n ∈ N) be its image under (5.3.15). We suppose now thatMn is an almost projectiveAn-module for everyn ∈ N. Let us choose an epimorphismf : A

(S)∞ →M . We endow the free

A∞-moduleA(S)∞ with the topology generated by the family of submodules(In ·A(S)

∞ | n ∈ N).Notice thatf is continuous for this topology, hence it extends to a continuous morphismf∧ :

F∞ → M on the completionF∞ of A(S)∞ .

Lemma 5.3.20.With the notation of(5.3.19), we have:

(i) f∧ is topologically almost split, i.e., for everyε ∈ m there exists a continuous morphismg : M → F∞ such thatf∧ g = ε · 1M .

(ii) If M is almost finitely generated, thenM is almost projective.

Proof. By lemma 5.3.16 the set of continuous morphismsM → F∞ is in natural bijection withlimn∈N

HomAn(Mn, Fn), whereFn := F∞/In+1F∞ for everyn ∈ N. On the other hand, under the

assumptions of the lemma,f induces an epimorphism:

limn∈N

alHomAn(Mn, Fn)→ limn∈N

alHomAn(Mn,Mn).

Assertion (i) follows easily. Suppose now thatM is almost finitely generated; then for everyfinitely generated subidealm0 ⊂ m we can findm ≥ 0 and a morphismh : Am∞ → M suchthatm0 · Coker h = 0. Let ε1, ..., εt be a system of generators form0; then for everyi, j ≤ t

we can pick a morphismφ : A(S)∞ → Am∞ such thath φ = εi · εj · f ; after taking completions,

this relation becomesh φ∧ = εi · εj · f∧. Chooseg : M → F∞ as in (i); we deduceεi · εj · ε · 1M = h φ∧ g, which as usual shows thatεi · εj · ε annihilatesalExt1

A∞(M,N) for

everyA∞-moduleN , so (ii) holds.

5.3.21. For aV a-algebraA, let us denote byA-Modafpr the full subcategory ofA-Mod

consisting of all almost finitely generated projectiveA-modules. Let now(An | n ∈ N) andI ⊂ A∞ be as in (5.3.14). We define natural functors

A∞-Modafpr → 2-limn∈N

An-Modafpr(5.3.22)

respectively (see (3.2.27)),

A∞-Etafp → A0-Etafp(5.3.23)

by assigning to a given objectM of A∞-Modafpr, the compatible system(M ⊗A∞ An | n ∈ N)

(resp. to an objectB of A∞-Etafp, theA0-algebraB ⊗A∞ A0).

Theorem 5.3.24.In the situation of(5.3.21), suppose thatI is tight. Then(5.3.22)is an equiv-alence of categories.

Proof. We claim that a quasi-inverse to (5.3.22) is obtained by assigning to any compatiblesystem(Mn | n ∈ N) theA∞-moduleM := lim

n∈NMn. Indeed, by the proof of lemma 5.3.16

there results thatM is endowed with a naturalI-c-adic topology; then by lemma 5.3.18(ii) wededuce thatM is almost finitely generated, and finally lemma 5.3.20(ii) says thatM is almostprojective. From lemma 5.3.18(iii) it follows that the functor thus defined is a right quasi-inverse to (5.3.22), so the latter is essentially surjective. Full faithfulness is a consequence ofthe following:

Claim5.3.25. Let P be an almost finitely generated projectiveA∞-module. Then the naturalmorphism

P → limn∈N

(P ⊗A∞ An)(5.3.26)

is an isomorphism.


Proof of the claim:This is clear ifP = Ar∞ for somer ≥ 0. For a generalP one chooses,for everyε ∈ m, a sequenceP → An∞ → P as in lemma 2.4.15 and applies to it the naturaltransformation (5.3.26); the claim follows by the usual diagram chase.

Theorem 5.3.27.In the situation of(5.3.21), suppose thatI is tight. Then(5.3.23)is an equiv-alence of categories.

Proof. It follows easily from theorem 3.2.28(ii) that the natural functor 2-limn∈N

An-Etafp →A0-Etafp is an equivalence. Hence we are reduced to showing that the natural functor

A∞-Etafp → 2-limn∈N

An-Etafp(5.3.28)

is an equivalence. To this aim, we claim that the rule(Bn | n ∈ N) 7→ limn∈N

Bn defines a quasi-

inverse to (5.3.28). Taking into account theorem 5.3.24, this will follow once we have shown:

Claim 5.3.29. Let B be an almost finitely generated projectiveA∞-algebra such thatB0 :=B ⊗A∞ A0 is unramified. ThenB is unramified.

Proof of the claim:To start out, we apply theorem 5.2.12(ii) withJ = In to deduce, by induc-tion onn, thatB/In+1B is unramified overA∞/In+1 for everyn ∈ N. SetBn := B ⊗A∞ An;it follows thatBn is unramified overAn for everyn ∈ N. By proposition 3.1.4 we deducethat there exists a compatible system of idempotent almost elementsen of Bn ⊗An Bn, suchthat en · IBn/An = 0 andµBn/An(en) = 1 for everyn ∈ N. Hence we obtain an idempotentin (lim

n∈NBn ⊗An Bn)∗ ≃ lim

n∈N(Bn ⊗An Bn)∗; however, the latter is isomorphic to(B ⊗A∞ B)∗,

in view of claim 5.3.25. Then conditions (ii) and (iii) of proposition 3.1.4 follow easily byremarking thatµB/A∞ = lim

n∈NµBn/An andIB/A∞ = lim

n∈NIBn/An .

5.4. Henselian approximation of structures over adically complete rings. This section re-views and complements some results of Elkik’s article [31].Especially, we wish to show howthe main theorems ofloc.cit. generalize to the case of not necessarily noetherian rings.In prin-ciple, this is known ([31, Ch.III,§4, Rem.2, p.587] explains briefly how to adapt the proofs tomake them work in some non-noetherian situations), but we feel that it is worthwhile to givemore details.

Definition 5.4.1. LetR be a ring,F := R[X1, ..., XN ] a freeR-algebra of finite type,J ⊂ F afinitely generated ideal, and letS := F/J . We define an ideal ofF by setting:

HR(F, J) := AnnF Ext1S(LS/R, J/J2).

Lemma 5.4.2. In the situation of definition5.4.1, we have:

(i) For anyR-algebraR′ let F ′ := R′ ⊗R F . ThenHR(F, J) · F ′ ⊂ HR′(F ′, JF ′).(ii) The open subsetSpecS \ V (HR(F, J) · S) is the smooth locus ofS overR.

(iii) HR(F, J) annihilatesExt1S(LS/R, N) for everyS-moduleN .

Proof. According to [46, Ch.III, Cor.1.2.9.1] there is a natural isomorphism inD(S-Mod):

τ[−1LS/R ≃ (0→ J/J2 ∂→ S ⊗F ΩF/R → 0)(5.4.3)

where∂ is induced by the universal derivationd : F → ΩF/R. SinceΩF/R is a freeF -module,we derive a natural isomorphism:

Ext1S(LS/R, J/J2) ≃ EndS(J/J

2)/∂∗HomF (ΩF/R, J/J2).(5.4.4)

Let nowR′ be anR-algebra andh ∈ F ; in view of (5.4.4), the conditionh ∈ HR(F, J) meansprecisely that the scalar multiplicationh : J/J2 → J/J2 factors through∂. It follows that


h : (J/J2)⊗R R′ → (J/J2)⊗R R′ factors through∂ ⊗S 1S′. However, the latter map admits afactorization:

∂ ⊗S 1S′ : (J/J2)⊗R R′ α→ J ′/J ′2∂′→ S ′ ⊗F ′ ΩF ′/R′

whereα is a surjective map. It follows easily that the scalar multiplicationh : J ′/J ′2 → J ′/J ′2

factors through∂′, which shows (i).To show (ii), let us first pick anyh ∈ HR(F, J); we have to prove thatSh is smooth overR.

However, the foregoing shows that multiplication byh onJ/J2 factors throughS ⊗F ΩF/R; ifnowh is invertible, this means that∂ is a split imbedding, thereforeSh is formally smooth overR by the Jacobian criterion [28, Ch.0, Th.22.6.1]. SinceS is of finite presentation overR, theassertion follows. Conversely, leth ∈ F be an element such thatSh is smooth overR; we wishto show thathn ∈ HR(F, J) for n large enough. One can either prove this directly using thedefinition ofHR(F, J), or else by using the following lemma 5.4.6 and the well knownfact thathn ∈ HS for n large enough.

Finally we recall the natural isomorphism:

HomD(S-Mod)(LS/R, N [1]) ≃ HomD(S-Mod)(τ[−1LS/R, N [1])

which, in view of (5.4.3), shows that every morphismτ[−1LS/R → N [1] factors through a mapτ[−1LS/R → J/J2[1], whence (iii).

5.4.5. In the situation of definition 5.4.1, choose a finite system of generatorsf1, ..., fq forJ ; it is shown in [31, Ch.0,§2] how to construct an ideal, calledHS in loc.cit. with the sameproperties as in lemma 5.4.2(i),(ii). However, the definition ofHS depends explicitly on thechoice of the generatorsf1, ..., fq. This goes as follows. For every integerp and every multi-index(α) = (α1, ..., αp) ∈ Np with 1 ≤ α1 < α2... < αp ≤ q, set|α| := p and letJα ⊂ J bethe subideal generated byfα1 , ..., fαp and∆α ⊂ F the ideal generated by the determinants ofthe minors of orderp of the Jacobian matrix(∂fαi

/∂Xj | 1 ≤ i ≤ p, 1 ≤ j ≤ N). We set

HS :=∑

p≥0

∑

|α|=p

∆α · (Jα : J).

Though we won’t be using the idealHS, we want to explain how it relates to the more intrinsicHR(F, J). This is the purpose of the following:

Lemma 5.4.6. With the notation of(5.4.5), we have:HS ⊂ HR(F, J).

Proof. Let p ∈ N, α a multi-index with|α| = p, δ ∈ ∆p andx ∈ F such thatxJ ⊂ Jα. Wecan suppose thatδ = det(M), whereM = (∂fαj

/∂Xβi| 1 ≤ i, j ≤ p) for a certain multi-

indexβ with |β| = p. We consider the mapsφα : Rp → J/J2 : ei 7→ fαi(mod J2) and

πβ : S ⊗F ΩF/R → Rp : dXβj7→ ej for j = 1, ..., p anddXk 7→ 0 if k /∈ β1, ..., βp. Then the

matrix of the composedS-linear mapπβ ∂ φα : Sp → Sp is none else thanM . LetM ′ bethe adjoint matrix ofM , so thatM ·M ′ = δ · 1Sp, and we can compute:

φα M ′ πβ ∂ φα = φα (δ · 1Sp) = (δ · 1J/J2) φα.In other words, the mapsφαM ′πβ∂ andδ·1J/J2 agree onImφα = (Jα+J2)/J2. Therefore:(x ·φα)M ′ πβ ∂ = x · δ ·1J/J2, i.e the scalar multiplication byx · δ onJ/J2 factors through∂; as in the proof of lemma 5.4.2 this implies thatx · δ ∈ HR(F, J), as claimed.

5.4.7. LetR be a (not necessarily noetherian) ring,t ∈ R a non-zero-divisor,I ⊂ R an ideal,S anR-algebra of finite presentation, which we write in the formS = F/J for some finitelygenerated idealJ ⊂ F := R[X1, ..., XN ]. For givena := (a1, ..., aN) ∈ RN , let pa ⊂ F be theideal generated by(X1 − a1, ..., XN − aN ).


Lemma 5.4.8. In the situation of(5.4.7), letn, h ≥ 0 be two integers anda ∈ RN such that

th ∈ HR(F, J) + pa J ⊂ pa + tnIF n > 2h.(5.4.9)

Then there existsb ∈ RN such that

b− a ∈ tn−hIRN and J ⊂ pb + (tn−hI)2F.

Proof. We are given a morphismσ : SpecR/tnI → SpecS whose restriction to the closedsubschemeSpecR/tn−hI we denote byσ0. The claim amounts to saying that there existsa lifting σ : SpecR/(tn−hI)2 → SpecS of σ0. By proposition 3.2.16, the obstruction tothe existence of an extension ofσ to a morphismSpecR/(tnI)2 → SpecS is a classω ∈Ext1S(LS/R, t

nI/(tnI)2). We have a commutative diagram:

tnI/(tnI)2 α //

β

tn−hI/(tn−hI)2

γ

tnI/(tnI)2 δ // tnI/t2n−hI2

whereα is induced by the inclusiontnI ⊂ tn−hI, β is the scalar multiplication byth, δ isthe restriction totnI of the natural projectionR/(tnI)2 → R/t2n−hI2, andγ is an isomor-phism induced by scalar multiplication byth : tn−hI

∼→ tnI. Since theS-module structure ontnI/(tnI)2 is induced by extension of scalars viaσ, it is clear thatpa ·ω = 0; on the other hand,by lemma 5.4.2(iii) we know thatHR(F, J) · ω = 0, henceth · ω = Ext1S(LS/R, β)(ω) = 0, andconsequentlyExt1S(LS/R, α)(ω) = 0. Since the latter class is the obstruction to the existence ofσ, the assertion follows.

Lemma 5.4.8 is the basis of an inductive procedure that allows to construct actual sectionsof X := SpecS, starting from approximate solutions of the system of equations defined by theidealJ . The section thus obtained live inX(R∧), whereR∧ is thetI-adic completion ofR.In caseI is finitely generated,R∧ is (separated) complete for the(tI)∧-adic topology, where(tI)∧ is the topological closure oftI in R∧; however, in later sections we will find situationswhere the relevant idealI is not finitely generated; in such case,R∧ is complete only for the(tI)∧-c-adic topology (see definition 5.3.1(v)). In this sectionwe carry out a preliminary studyof some topologies onR-modules and on sets of sections ofR-schemes; one of the main themesis to compare the topologies of, sayZ(R) whereZ is anR-scheme, and ofZ(R∧), whereR∧

is an adic completion ofR. For this reason, it is somewhat annoying that, in passing fromR toits completion, one is forced to replace a preadic topology by a c-preadic one. That is why weprefer to use a slightly coarser topology onR, described in the following:

Definition 5.4.10. LetR be any ring,t ∈ R a non-zero-divisor,I ⊂ R an ideal,M0 (resp.N)a finitely generatedR-module (resp.R[t−1]-module).

(i) The(t, I)-preadic topologyofM0 is the linear topology that admits the family of submod-ules(tnIM0 | n ∈ N0) as a cofinal system of open neighborhoods.

(ii) The (t, I)-preadic topologyof N is the linear topologyLN defined as follows. Choosea finitely generatedR-moduleN0 with anR-linear mapφ : N0 → N such thatφ[t−1] :N0[t

−1] → N is onto; thenLN is the finest topology such thatφ0 becomes an openmap when we endowN0 with the (t, I)-preadic topology. Therefore, the family ofR-submodules(φ(tkIN0) | k ∈ N) is a cofinal system of open neighborhoods of0 ∈ N forthe topologyLN .

The advantage of the(t, I)-adic topology is that the(t, I)-adic completionR∧ of R is com-plete for a topology of the same type, namely for the(t, I∧)-topology (whereI∧ ⊂ R∧ is thetopological closure ofI). Hence, for everyR-schemeZ, the topologies onZ(R) andZ(R∧)


admit a uniform description. After this caveat, let us stress nevertheless that all of the results ofthis section hold as well for the c-adic topologies.

5.4.11. In the situation of definition 5.4.10, it is easy to verify that the(t, I)-preadic topologyonN does not depend on the choice ofN0 andφ. Indeed, ifφ′ : N1 → N is another choice,then one checks thatφ(tkN0) ⊂ φ′(N1) for k ∈ N large enough, and symmetricallyφ1(t

kN1) ⊂φ0(N0), which implies the assertion.

Moreover, it is easy to check that everyR[t−1]-linear mapM → N of finitely generatedR[t−1]-modules is continuous for the respective(t, I)-preadic topologies.

Lemma 5.4.12.LetR be a noetherian ring,I ⊂ R an ideal, and suppose that the pair(R,I )is henselian. LetR be theI -adic completion ofR and setS := R ⊗R S. LetU ⊂ SpecSbe an open subset smooth overSpecR, andU ⊂ SpecS the preimage ofU . Then, for everyintegern and everyR-sectionσ : SpecR → SpecS whose restriction toSpecR \ V (IR)factors throughU , there exists anR-sectionσ : SpecR→ SpecS congruent toσ moduloI n,and whose restriction toSpecR \ V (I ) factors throughU .

Proof. It is [31, Ch.II, Th.2 bis].

Proposition 5.4.13.Keep the notation of(5.4.7)and letI ⊂ R be an ideal such that(R, tI) isa henselian pair. Leta ∈ RN andn, h ≥ 0 such that(5.4.9)holds. Then there existsb ∈ RN

such thatb− a ∈ tn−hIRN andJ ⊂ pb.

Proof. We consider first the following special case:

Claim5.4.14. The proposition holds ifR is complete for thetI-adic topology.

Proof of the claim:We apply repeatedly lemma 5.4.8 to obtain atI-adically convergent se-quence of elements(am ∈ RN | m ∈ N), with a0 := a and such thatam ≡ a (mod tn−hIRN)for everym ∈ N. The limit b of the sequence(am |m ∈ N) will do.

Let nextR be a general ring; letH ⊂ HR(F, J) be a finitely generated subideal such thatth ∈ H. We can writeR =

⋃λ∈ΛRλ for a filtered family of noetherian subrings(Rλ | λ ∈ Λ)

such thatt ∈ Rλ anda ∈ RNλ for everyλ ∈ Λ, and we setIλ := I ∩ Rλ for everyλ ∈ Λ.

We can also assume that the pair(Rλ, tIλ) is henselian for everyλ ∈ Λ. Let f1, ..., fq ∈ Fbe a finite set of generators forJ and g1, ..., gr ∈ F a finite set of generators forH; up torestricting to a cofinal family, we can then assume thatfi, gj ∈ Fλ := Rλ[X1, ..., XN ] for everyλ ∈ Λ and everyi ≤ q, j ≤ r. Let Jλ ⊂ Fλ be the ideal generated byf1, ..., fq and setSλ := Fλ/Jλ; let alsoHλ ⊂ Fλ be the ideal generated byg1, ..., gr. Again after replacingΛby a cofinal subfamily, we can achieve thatHλ ⊂ HRλ

(Fλ, Jλ) for everyλ ∈ Λ. Finally, letpλ,a ⊂ Fλ be the ideal generated byX1 − a1, ..., XN − aN ; we can assume thatth ∈ Hλ + pλ,aandJλ ⊂ pλ,a + tnIλFλ for everyλ ∈ Λ. With this setup, letRλ be thetIλ-adic completion ofRλ; we can apply claim 5.4.14 to deduce that there existsc ∈ RN

λ such thatc− a ∈ tn−hIλRNλ

and such thatJλ ⊂ pλ,c (wherepλ,c denotes the ideal generated byX1 − c1, ..., X − cn inRλ[X1, ..., XN ]). LetUλ := SpecSλ \ V (Hλ · Sλ) andSλ := Rλ ⊗Rλ

S; from lemma 5.4.2(ii)we know thatUλ is smooth overSpecRλ, and by constructionc determines aRλ-section ofSpecSλ whose restriction toSpecRλ \ V (tIλ) factors through the preimage ofUλ. Thereforelemma 5.4.12 ensures that there exists anRλ-sectionσ : SpecRλ → SpecSλ that agrees withcmodulo(tIλ)

n−h and whose restriction toSpecRλ\V (tIλ) factors throughU . Letπλ : Fλ → Sλbe the natural projection; there is a uniqueb ∈ RN

λ such thatpλ,b = π−1λ (Ker σ♯ : Sλ → Rλ);

this pointb has the sought properties.


5.4.15. Let nowX be an affine scheme of finite type overSpecR[t−1]; for everyn ∈ N letAnR[t−1] be then-dimensional affine space overSpecR[t−1]; for n large enough we can find

a closed imbeddingj : X → AnR[t−1] of SpecR[t−1]-schemes. The choice of coordinates

on AnR[t−1], yields a bijectionAn

R[t−1](R[t−1]) ≃ R[t−1]n, and thenj induces an injective mapj∗ : X(R[t−1]) ⊂ R[t−1]n. The (t, I)-adic topologyof X(R[t−1]) is defined as the subspacetopology induced byj∗ (whereR[t−1]n is endowed with its(t, I)-preadic topology).

Lemma 5.4.16.LetX be as in(5.4.15); we have:

(i) The(t, I)-adic topology ofX(R[t−1]) is independent of the choice of closed imbeddingjand of coordinates onAn

R[t−1].(ii) If Y is another affineR[t−1]-scheme, then the natural map(X ×R[t−1] Y )(R[t−1]) →

X(R[t−1]) × Y (R[t−1]) is a homeomorphism for the(t, I)-adic topologies of the corre-sponding schemes.

(iii) If U ⊂ X is any open subset, andσ ∈ U(R[t−1]), there existsf ∈ OX(X) such thatD(f) := X \ V (f) ⊂ U and such thatσ factors throughD(f).

(iv) If U ⊂ X is an affine open subset, then the(t, I)-adic topology onU(R[t−1]) agrees withthe topology induced from the(t, I)-adic topology onX(R[t−1]), andU(R[t−1]) is anopen subset ofX(R[t−1]).

Proof. (i): supposej1 : X → AnR[t−1] and j2 : X → Am

R[t−1] are two closed imbeddings,τ1 and τ2 the respective topologies onX(R[t−1]); by symmetry it suffices to show that theidentity map(X(R[t−1]), τ1) → (X(R[t−1]), τ2) is continuous. However,j2 can be extendedto some morphismφ : An

R[t−1] → AmR[t−1], and we come down to showing that the induced map

φ∗ : R[t−1]n → R[t−1]m is continuous for the(t, I)-adic topology. We can further reduce to thecase ofm = 1, in which caseφ is given by a single polynomialf ∈ R[t−1, T1, ..., Tn], andφ∗is the map(a1, ..., an) 7→ f(a1, ..., an). Let x0 ∈ R[t−1]n and sety0 := f(x0); we have to showthat, for everyk ∈ N there existsh ∈ N such thatf(x0 + thI) ⊂ y + tkI. The Taylor formulagives an identity of the form:f(T ) = y0 +

∑r∈Nn\0 ar · (T − x0)

r, wherear ∈ R[t−1] foreveryr ∈ Nn andar = 0 for all but finitely manyr. Let s ∈ N be an integer large enoughso thattsar ∈ R for everyr ∈ Nn; obviouslyh := k + s will do. (ii) is easily reduced to thecorresponding statement for(t, I)-adic topologies on direct sums ofR[t−1]-modules; we leavethe details to the reader.

(iii): we can writeU = X \ V (J), whereJ ⊂ S := OX(X), and a sectionσ ∈ U(R[t−1])induces a mapφ : S → R[t−1] such thatφ(J)R = R. Let ai ∈ R andfi ∈ J , i = 1, ..., n with∑

i ai · φ(fi) = 1 and setf :=∑

i fiai; it follows that the image ofσ is contained in the affineopen subsetSpecS[f−1] ⊂ U . To show (iv), we can suppose, thanks to (iii), thatU = X \V (f)for somef ∈ OX(X). Let φf : X → A1

R[t−1] be the morphism defined byf , andΓf : X →X×R[t−1] A1

R[t−1] its graph; choose a closed imbeddingX ⊂ AnR[t−1]. The compositionj : X →

X ×R[t−1] A1R[t−1] → An+1

R[t−1] is another closed imbedding, which induces the same topology onX(R[t−1]) in view of (i). We havej−1(An

R[t−1] ×R[t−1] Gm,R[t−1]) = U , consequently we are

reduced to showing the assertion for the open imbeddingAnR[t−1] ×R[t−1] Gm,R[t−1] ⊂ An+1

R[t−1].Using (ii) we further reduce to considering the imbeddingGm,R[t−1] ⊂ A1

R[t−1]. This case canbe dealt with by explicit calculations.

Lemma 5.4.17.Let R be any ring,X a quasi-projectiveR-scheme. Then everyR-sectionSpecR→ X ofX factors through an open imbeddingU ⊂ X, whereU is an affineR-scheme.

Proof. By assumptionX is a locally closed subset ofPnR, for somen ∈ N .

Claim 5.4.18. The lemma holds ifX is a closed subscheme ofPnR.


Proof of the claim:Indeed, in this case we can assumeX = PnR. Therefore, letσ ∈ PnR(R); by[27, Ch.II, Th.4.2.4]σ corresponds to a rank one locally free quotientL of Rn+1. We choosea section of the projectionπ1 : Rn+1 → L, hence a decompositionRn+1 ≃ L ⊕ Ker π1; theinduced projectionπ2 : Rn+1 → Kerπ1 determines a closed imbeddingP(Kerπ1) → PnRrepresenting the transformation of functors that assigns to every rank one locally free quotientof Ker π1 the same module, seen as a quotient ofRn+1 via the projectionπ2. It is clear that theimage ofσ does not intersect the image ofP(Ker π1), and the complementPnR \ P(Kerπ1) isaffine (π1 is a section inΓ(PnR,OPn

R(1))).

Thanks to claim 5.4.18 we can assume thatX is an open subscheme of an affine schemeYof finite type overSpecR. In this case, the claim reduces to lemma 5.4.16(iii) (applied witht = 1).

5.4.19. Let nowX be a quasi-projectiveR[t−1]-scheme; each affine open subset ofX comeswith a natural(t, I)-adic topology. By lemma 5.4.16(iv) these topologies agreeon the intersec-tions of any two such affine open subsets, and according to lemma 5.4.17 we haveX(R[t−1]) =⋃U U(R[t−1]), whereU ranges on the family of all affine open subsetsU ⊂ X, soX(R[t−1])

can be endowed with a well defined(t, I)-adic topology, independent of all choices.

Lemma 5.4.20.The closed subschemeUn of Spec Z[x11, ..., xnn] that classifies then×n idem-potent matrices is smooth overSpec Z.

Proof. ClearlyUn is of finite type overSpec Z, hence it suffices to show thatUn is formallysmooth. Therefore, letR0 be a ring andI ⊂ R0 an ideal withI2 = 0; we need to show thatthe induced mapUn(R0) → Un(R0/I) is surjective,i.e. that everyn × n idempotent matrixM with entries inR0/I lifts to an idempotent matrix with entries inR0. Pick an arbitrarymatrixM ∈ Mn(R0) that liftsM ; letE := R0[M ] ⊂ Mn(R0) be the commutativeR0-algebragenerated byM , E ⊂ Mn(R0/I) be the image ofE, J ⊂ E the kernel of the induced mapE → E. We haveJ2 = 0, so we can apply proposition 2.5.13(i) to liftM to some idempotentmatrix inE.

Proposition 5.4.21.Let t ∈ R be a non-zero-divisor,I ⊂ R an ideal,R∧ := limn∈N

R/tnI the

(t, I)-adic completion ofR, I∧ the topological closure ofI in R∧, and suppose that the pair(R, tI) is henselian. LetX be a smooth quasi-projectiveR[t−1]-scheme, and endowX(R[t−1])(resp. X(R∧[t−1])) with its (t, I)-adic (resp. (t, I∧)-adic) topology. Then the natural mapX(R[t−1])→ X(R∧[t−1]) has dense image.

Proof. We begin with the following special case:

Claim5.4.22. The proposition holds ifX is affine.

Proof of the claim:Say thatX = SpecS, whereS is some finitely presented smoothR[t−1]-algebra, and letσ : S → R∧[t−1] be any element ofX(R∧[t−1]). We have to show that there areelements ofX(R[t−1]) in every(t, I)-adic neighborhood ofσ. To this aim, we pick a finitelypresentedR-algebraS0 such thatS0[t

−1] ≃ S; after clearing some denominators we can assumethatσ extends to a mapS0 → R∧. Let

S0 = R[X1, ..., XN ]/J(5.4.23)

be a finite presentation ofS0, and setH := HR(R[X1, ..., XN ], J) (notation of definition 5.4.1).By assumption we have

th ∈ H + J(5.4.24)

for h ∈ N large enough. The presentation (5.4.23) defines a closed imbeddingX ⊂ ANR , and

we can then find a sectionσ0 : SpecR → ANR that is (t, I)-adically close toσ, so that the


restrictions ofσ andσ0 agree onSpecR/tnI. Let p := Ker σ♯0 ⊂ R[X1, ..., XN ] be the idealcorresponding toσ0; it then follows that

J ⊂ p + tnIR[X1, ..., XN ].(5.4.25)

From (5.4.24) and (5.4.25) we deduce thatth ∈ H + tnIR[X1, ..., XN ] + p; up to enlargingn (which is harmless) we can assume thatn > 2h. Let H := (H + p)/p ⊂ R; we derive:th ∈ H + tnI, whenceth(1 + atk) ∈ H for somea ∈ I andk > 0; sincetI ⊂ rad(R), itfollows:

th ∈ H + p.(5.4.26)

By assumption we haveAnnR(t) = 0, hence from (5.4.24), (5.4.26) and proposition 5.4.13 wededuce the contention.

Claim 5.4.27. Suppose thatX = PrR[t−1]. Then every sectionσ : SpecR∧[t−1] → X factorsthrough an open imbeddingU ⊂ X, whereU is an affineR-scheme.

Proof of the claim:By [27, Ch.II, Th.4.2.4],σ corresponds to a rank one projective quotientL of(R∧[t−1])r+1; we can then find an idempotent(r+1)×(r+1) matrixe such thatCoker(e) = L.By lemma 5.4.20, the schemeU that represents the(r + 1) × (r + 1) idempotent matricesis smooth; clearlyU decomposes as a disjoint union of open and closed subschemesU =⋃r+1n=0 Un, whereUn represents the subfunctor that classifies all(r + 1) × (r + 1) idempotent

matrices of rankn, for everyn = 0, ..., r + 1. Therefore eachUn is smooth and affine overSpec Z and then by claim 5.4.22 it follows thate can be approximated closely by an idempotentmatrixe0 ∈Mr+1(R[t−1]) whose rank equals the rank ofe. We havee0 · e = e0 · (Ir+1−e0 + e)and ife0 is sufficiently(t, I)-adically close toe, the matrixIr+1−e0+e is invertible inMr+1(R),hence we can assume that:

Im(e0 · e) = Im(e0) ⊂ R∧[t−1]r+1.(5.4.28)

The projectionπ : R[t−1]r+1 → Im e0 : x 7→ e0(x)

determines a closed imbeddingP(Im e0)→ PrR[t−1] representing the transformation of functorsthat, to everyR[t−1]-algebraS and every rank one projective quotient ofS⊗R[t−1]Im(e0) assignsthe same module seen as a quotient ofS ⊗R[t−1] R[t−1]r+1 via π. By (5.4.28), the restriction of1R∧ ⊗R π to R∧ ⊗R Im(e) is an isomorphism, hence it remains such after every base changeR∧[t−1] → S; this means that the image ofσ lands in the complement ofP(Im e0), whichimplies the contention.

Claim 5.4.29. Let X be any quasi-projectiveR[t−1]-scheme,σ : SpecR∧[t−1] → X any sec-tion. Thenσ factors through an open imbeddingU ⊂ X, whereU is an affineR-scheme.

Proof of the claim:Due to claim 5.4.27 we can assume thatX is an open subscheme of an affineR[t−1]-scheme of finite type. Thus, we can writeX = Y \ V (J), whereJ ⊂ S := OY (Y ), anda sectionσ ∈ X(R∧[t−1]) induces a mapφ : S → R∧[t−1] such thatφ(J)R∧[t−1] = R∧[t−1].Let ai ∈ R∧[t−1], fi ∈ J such that

∑i ai · φ(fi) = 1; we choosebi ∈ R[t−1], (t, I)-adically

close toai (i = 1, ..., n), so that∑

i φ(fi) ·(ai−bi) ∈ tIR∧. SincetIR∧ ⊂ rad(R∧), we deduceeasily that

∑i φ(fi)bi ∈ R∧ is invertible inR∧. Setf :=

∑i fibi; it follows that the image ofσ

is contained in the affine open subsetSpecS[f−1] ⊂ X.The proposition follows easily from claims (5.4.22) and (5.4.29).

Proposition 5.4.30.Resume the assumptions of proposition5.4.21and letφ : X → Y be amorphism of quasi-projectiveR[t−1]-schemes; then we have:

(i) The mapφ∗ : X(R[t−1])→ Y (R[t−1]) is continuous for the(t, I)-adic topologies.


(ii) If φ is smooth,φ∗ is an open map.

Proof. (i): we can factorφ as a closed imbeddingX → X × Y followed by a projectionX × Y → Y , so it suffices to prove the claim for the latter maps. The caseof a projection isimmediate, and the case of a closed imbedding follows straightforwardly from lemma 5.4.16(i).

(ii): due to lemmata 5.4.16(iv) and 5.4.17 we can assume thatbothX andY are affineschemes, sayX = SpecS, Y = SpecT whereS andT are finitely generatedR[t−1]-algebrasandS is smooth overT , especiallyS is a finitely presentedT -algebra. Letσ : SpecR[t−1] →X be an element ofX(R[t−1]); after choosing presentations forS andT and clearing somedenominators, we can find morphisms of finitely presentedR-algebrasf : T0 → S0 andg :S0 → R such thatf [t−1] = φ♯ : T → S andg[t−1] = σ♯ : S → R[t−1]. LetX0 := SpecS0,Y0 := Spec T0, σ0 := Spec(g). We have to show that, for every sectionξ : SpecR[t−1] → Ysufficiently close toσ, there isξ′ ∈ X(R[t−1]) such thatφ∗(ξ′) = ξ.

Claim5.4.31. There is an open neighborhoodU ⊂ Y (R[t−1]) of φ∗(σ) such that everyξ ∈ Uextends to a sectionξ0 : SpecR→ Y0.

Proof of the claim:By construction,φ∗(σ) is induced by the mapg f : S0 → R; any othersectionξ ∈ Y (R[t−1]) is determined by a mapξ♯ : S → R[t−1]; we haveS = S0[t

−1] andS0 isof finite type overR, sayS0 = R[x1, ..., xN ]/J . Thereforeai := g f(xi) ∈ R for everyi ≤ N .The claim follows by observing that the set of sectionsξ ∈ Y (R[t−1]) such thatξ♯(xi)−ai ∈ Rfor everyi ≤ N forms an open neighborhood ofφ∗(σ).

Next, choose a presentation

S0 ≃ T0[x1, ..., xN ]/J(5.4.32)

and letH := HT0(T0[x1, ..., xN ], J) (notation of definition 5.4.1). SinceS is smooth overT , wehave

th ∈ H + J(5.4.33)

for h ∈ N large enough. Letξ ∈ Y (R[t−1]); by claim 5.4.31 we can suppose thatξ extends toa sectionξ0 : SpecR→ Y0. DefineX0(ξ0) as the fibre product in the cartesian diagram

X0(ξ0) //

X0

φ0

SpecR

ξ0 // Y0.

The presentation (5.4.32) induces a closed imbeddingX0(ξ0) ⊂ ANR whose defining ideal

J(ξ0) ⊂ R[x1, ..., xN ] is the image ofJ ; define similarlyH(ξ0) ⊂ R[x1, ..., xN ] as the im-age ofH. From (5.4.33) we deduce that

th ∈ H(ξ0) + J(ξ0).(5.4.34)

Suppose now thatξ is sufficiently close toσ; this means that the restrictions ofξ0 andφ0∗(σ0)agree on some closed subsetSpecR/tnI ⊂ SpecR. Hence, letσ0 : SpecR/tnI → X0(ξ) bethe restriction ofσ0, and choose any extension ofσ0 to a morphismσ1 : SpecR→ AN

R . Finally,let p := Ker σ♯1 ⊂ R[x1, ..., xN ] be the ideal corresponding toσ1. By construction we have

J(ξ0) ⊂ p + tnIR[x1, ..., xN ].(5.4.35)

Now, arguing as in the proof of claim 5.4.22 we see that (5.4.34) and (5.4.35) implyth ∈H(ξ0) + p, at least ifn > 2h, which can always be arranged. Finally, proposition 5.4.13showsthatσ0 can be extended to a sectionξ′ : SpecR→ X0(ξ0), as required.


5.4.36. Resume the assumptions of proposition 5.4.21 and let (X0, X1, s, t, c, ι) be a groupoidof quasi-projectiveR[t−1]-schemes (see (4.5.1) for our general notations concerninggroupoidsin a category). We have a natural map

π0(X0(R[t−1]))→ π0(X0(R∧[t−1]))(5.4.37)

where, for any groupoid of setsG := (G0, G1, sG, tG, cG, ιG), we denote byπ0(G) the set ofisomorphism classes of elements ofG0; this is the same as the set of connected components ofthe geometric realization of the simplicial set associatedto the groupoidG.

Theorem 5.4.38.Keep the notation of(5.4.36), and suppose thatX0 is smooth overR[t−1] andthat the morphism(s, t) : X1 → X0 ×R[t−1] X0 is smooth. Then(5.4.37)is a bijection.

Proof. Let σ ∈ X0(R∧[t−1]); sinceX0 is smooth overR[t−1] we can find sectionsσ0 ∈

X0(R∧[t−1]) arbitrarily (t, I)-adically close toσ (proposition 5.4.21). Then(σ, σ0) can be

made arbitrarily close to(σ, σ) ∈ X0 ×R[t−1] X0(R∧[t−1]); since the latter lies in the image

of X1(R∧[t−1]) under the morphism(s, t), it follows that the same holds for(σ, σ0), provided

σ0 is sufficiently close toσ (proposition 5.4.30(ii)). This shows that (5.4.37) is onto. Next,suppose thatσ, τ ∈ X0(R[t−1]) and that their imagesσ∧, τ∧ in X0(R

∧[t−1]) lie in the samehomotopy class. By definition, this means that(σ∧, τ∧) = (s, t)(α) for someα ∈ X1(R

∧[t−1]);sinceX1 is smooth overR[t−1] it follows that there exist sectionsα0 ∈ X1(R[t−1]) arbitrarilyclose toα (again proposition 5.4.21). Since(s, t)∗ is continuous (proposition 5.4.30(i)), it fol-lows that(σ0, τ0) := (s, t)∗(α0) can be made arbitrarily close to(σ, τ) inX0×R[t−1]X0(R[t−1]).This means that the pair(σ, σ0) can be made arbitrarily close to(σ, σ), hence(σ, σ0) is in theimage of(s, t)∗ providedσ andσ0 are sufficiently close. Soσ is in the same homotopy class asσ0. Likewise we can arrange thatτ andτ0 are in the same homotopy class, which shows that(5.4.37) is injective as well.

5.4.39. Letn ∈ N and defineUn as in lemma 5.4.20; sometimes we identifyUn to the functorwhich it represents. We define a functor fromFn : Z-Alg → Set as follows. Given a ringR, we letFn(R) be the set of all data of the form(S, T, φ, ψ), whereS, T ∈ Mn(R) are twoidempotent matrices andφ, ψ ∈ Mn(R) are two other matrices submitted to the followingconditions:

(a) φ · S = 0 = T · φ.(b) ψ · T = 0 = S · ψ.(c) (In − S) · (ψ · φ− In) = 0 = (In − T ) · (φ · ψ − In).

The meaning of (a) is thatφ induces a mapφ : Coker(S) → Coker(T ); likewise, (b) meansthatψ induces a mapψ : Coker(T ) → Coker(S). Finally (c) means thatφ ψ is the identityof Coker(T ), and likewise forψ φ. It is easy to see from this description that the functorFnis representable by an affineZ-scheme of finite type, which we shall denote by the same name.Moreover, the rule(S, T, φ, ψ) 7→ (S, T ) defines a transformation of functorsFn → Un × Un,whence a natural morphism of schemes:

Fn → Un ×Z Un.(5.4.40)

Lemma 5.4.41.The morphism(5.4.40)is smooth.

Proof. Since (5.4.40) is clearly of finite presentation, it sufficesto verify that it is formallysmooth. Hence, letR → R0 be a surjective ring homomorphism with nilpotent kernelJ ,let (S, T ) ∈ Un ×Z Un(R) and(S0, T0, φ0, ψ0) ∈ Fn(R0) such that(S0, T0) coincides with theimage of(S, T ) in Un×ZUn(R0). We need to show that there existφ, ψ ∈Mn(R) lifting φ0 andψ0, such that(S, T, φ, ψ) ∈ Fn(R). However, sinceS andT are idempotent,P := Coker(S)andQ := Coker(T ) are finitely generated projectiveR-modules. According to (5.4.39), the


induced mapφ0 : P0 =: P ⊗R R0 → Q0 := Q⊗R R0 is an isomorphism with inverseψ0. LetπP : P → P0, πQ : Q → Q0 be the projections; sinceψ0 πQ : Q → P0 is surjective, we canfind a mapφ : P → Q such thatψ0 πQ φ = πP . Using Nakayama’s lemma one checkseasily thatφ is an isomorphism that liftsφ0. Let α : Rn → Q andβ : Rn → P be the naturalprojections. We setφ := (In − T ) φ β andψ := (In − S) φ−1 α and leave to the readerthe verification of the identities (a)-(c) of (5.4.39).

Corollary 5.4.42. Resume the assumptions of proposition5.4.21. Then the base change func-tor R[t−1]-Mod → R∧[t−1]-Mod : M 7→ M ⊗R R∧ induces a bijection from the set ofisomorphism classes of finitely generated projectiveR[t−1]-modules to the set of isomorphismclasses of finitely generated projectiveR∧[t−1]-modules.

Proof. Resume the notation of (5.4.39) and lets, t : Fn → Un be the morphisms obtained bycomposing (5.4.40) with the two projections ontoUn. The datum(Un, Fn, s, t) can be com-pleted to a groupoid of schemes, by lettingι : Un → Fn be the morphism representing thetransformation of functors:S 7→ (S, S, In, In), andc : Fn ×Un Fn → Fn the morphism repre-senting the transformation:((S, T, φ1, ψ1), (S, T, φ2, ψ2)) 7→ (S, T, φ2 ·φ1, ψ2 ·ψ1). Since everyfinitely generated projective module can be realized as the cokernel of an idempotent endomor-phism of a free module of finite rank, the assertion is a straightforward consequence of theorem5.4.38 and lemmata 5.4.20, 5.4.41.

5.4.43. LetS be anR[t−1]-scheme,P a coherentOS-module. AnS-algebra structureonPis a datum(µ, 1) consisting of a mapµ : P⊗OS

P →P and a global section1 ∈P(S), suchthat(P, µ, 1) is anOS-algebra,i.e. such thatµ and1 satisfy the following conditions for everyopen subsetU ⊂ S and every local sectionsx, y, z ∈P(U) :(a) µ(µ(x⊗ y)⊗ z) = µ(x⊗ µ(y ⊗ z)).(b) µ(x⊗ y) = µ(y ⊗ x).(c) µ(1⊗ x) = x.

We say that anS-algebra structure onP is etale if the datum(P, µ, 1) is an etaleOS-algebra.We denote byAlgS(P) (resp.EtS(P)) the set of allS-algebra structures (resp. etaleS-algebrastructures) onP. If now P is a finitely presentedR[t−1]-module, we obtain a functor:

R[t−1]-Scheme→ Set S 7→ AlgS(OS ⊗R[t−1] P ).(5.4.44)

Lemma 5.4.45.Suppose thatP is a finitely generated projectiveR[t−1]-module. Then thefunctor(5.4.44)is representable by a finitely presentedR[t−1]-algebra.

Proof. Clearly (5.4.44) is a sheaf on the fppf topology ofSpecR[t−1], hence it suffices to showthat we can coverSpecR[t−1] by finitely many affine Zariski open subsetsUi, such that therestriction of (5.4.44) toUi is representable and finitely presented. However,P is locally free offinite rank on the Zariski topology ofSpecR[t−1], hence we can assume thatP is a freeR[t−1]-module. Lete1, ..., en be a basis ofP ; then a multiplication lawµ is determined by its valuesaij ∈ P on ei ⊗ ej; by writing aij =

∑k aijkek we obtainn3 elements ofR[t−1]; likewise,1 is

represented by elementsb1, ..., bn ∈ R[t−1], and conditions (a),(b) and (c) of (5.4.43) translateas a finite system of polynomial identities for theaijk and thebl; in other words, our functor isrepresented by a quotient of the free polynomial algebraR[t−1, Xijk, Yl | i, j, k, l = 1, ..., n] bya finitely generated ideal, which is the contention.

Lemma 5.4.46.Keep the assumptions of lemma5.4.45. LetXP be an affineR[t−1]-schemerepresenting the functor(5.4.44). Then the functor

R[t−1]-Scheme→ Set S 7→ EtS(OS ⊗R[t−1] P )(5.4.47)

is represented by an affine open subsetUP ⊂ XP . Moreover,UP is smooth overSpecR[t−1].


Proof. Let us show first that the functor (5.4.47) is formally smooth. Indeed, suppose thatZis an affineR[t−1]-scheme andZ0 ⊂ Z is a closed subset defined by a nilpotent ideal. Let(OZ ⊗R[t−1] P, µ0) be an etaleZ0-algebra structure; we can lift it to some etaleOZ-algebra(Q, µ) and thenQ is necessarily a locally free sheaf of finite rank (for instance by lemma3.2.25(i),(iii)); from proposition 3.2.30 we deduce thatQ ≃ OZ ⊗R[t−1] P , whence the claim.Next, let (P, µ, 1) be the universalOXP

-algebra structure onP ⊗R[t−1] OXP; let alsoδ ∈

OXP(XP ) be the discriminant of the trace form ofP. By theorem 4.1.14, a pointx ∈ XP is in

the support ofδ if and only if Px is not etale overOXP ,x; therefore, the subsetUP over whichP is etale is indeed open and affine. To conclude, it suffices toshow thatUP represents thefunctor (5.4.47). This amounts to showing that, for every morphismf : S → XP of R[t−1]-schemes, the algebraf ∗P is etale overOS if and only if the image off lands inUP . However,the latter statement follows easily from [30, Ch.IV, Cor.17.6.2].

5.4.48. LetS be anR[t−1]-scheme,P a coherentOS-module. We denote byAutOS(P) the

group ofOS-linear automorphisms ofP. Then, for a given finitely presentedR[t−1]-modulePwe obtain a group-valued functor

R[t−1]-Scheme→ Grp S 7→ AutOS(OS ⊗R[t−1] P ).(5.4.49)

Lemma 5.4.50.Keep the assumptions of lemma5.4.45. Then the functor(5.4.49) is repre-sentable by a finitely presentedR[t−1]-group scheme.

Proof. It is analogous to the proof of lemma 5.4.45 : up to restricting to a Zariski open subset,we can assume thatP is free of some rankn. Then the group scheme representing our functoris justGLn,R[t−1].

5.4.51. Let nowP be a finitely generated projectiveR[t−1]-module. For anyR[t−1]-scheme,let PS := OS ⊗R[t−1] P ; we wish to define an action ofAutOS

(PS) on the setEtS(PS).Indeed, ifµ : PS ⊗OS

PS →PS is any etaleS-algebra structure andg ∈ AutOS(PS), letµg

be the uniqueS-algebra structure onPS such thatg is an isomorphism of etaleOS-algebras:g : (PS, µ)

∼→ (PS, µg). It is obvious that the rule(g, µ) 7→ µg is a functorial group action.

LetUP be as in lemma 5.4.46, and letAutP be aR[t−1]-scheme representing the functor 5.4.49;the functorial map(g, µ) 7→ (µg, µ) is represented by a morphism of schemes:

AutP ×R[t−1] UP → UP ×R[t−1] UP .(5.4.52)

Lemma 5.4.53.The morphism(5.4.52)is etale.

Proof. The map is clearly of finite presentation, hence it suffices toshow that it is formallyetale. Therefore, letφ : Z → UP ×R[t−1] UP be a morphism ofR[t−1]-schemes,Z0 ⊂ Z aclosed subscheme defined by a nilpotent ideal, and suppose that the restriction ofφ toZ0 lifts toa morphismψ0 : Z0 → AutP ×R[t−1] UP . We need to show thatφ lifts uniquely to a morphismψ that extendsψ0. However, the datum ofφ is equivalent to the datum consisting of a pair ofZ-algebra structures(PZ , µ1) and(PZ , µ2). The datum ofψ0 is equivalent to the datum of aZ0-algebra structureµ0 onPZ0 , and of an automorphismg0 of PZ0 . Finally, the fact thatψ0 liftsthe restriction ofφmeans thatµ0 = µ2⊗Z 1Z0 , andg0 : (PZ0, µ1⊗Z 1Z0)

∼→ (PZ0 , µ2⊗Z 1Z0)is an isomorphism of etaleOZ0-algebras. By theorem 3.2.18(iii), such an isomorphism extendsuniquely to an isomorphism of etaleOZ-algebrasg : (PZ , µ1)

∼→ (PZ , µ2). The datum(g, µ2)is equivalent to the sought mapψ.

Proposition 5.4.54.Resume the assumptions of proposition5.4.21. Then the base change func-torR[t−1]-Alg→ R∧[t−1]-Alg induces an equivalence of categories from the category of finiteetaleR[t−1]-algebras to the category of finiteetaleR∧[t−1]-algebras.


Proof. Let (P ∧, µ∧) be a finite etaleR∧[t−1]-algebra; in particular,P ∧ is a finitely generatedprojectiveR∧[t−1]-module, hence by corollary 5.4.42 we can find a finitely generated projectiveR[t−1]-moduleP such thatR∧[t−1]⊗R[t−1] P ≃ P ∧. The functorial action (5.4.52) ofAutP onthe set of etale algebra structures onP defines a groupoid of quasi-projective schemes. It thenfollows from theorem 5.4.38 and lemmata 5.4.46, 5.4.53 thatthe base change map from theset of isomorphism classes ofR[t−1]-algebra structures onP to the set of isomorphism classesof R∧[t−1]-algebra structures onP ∧ is bijective. This shows that the base change functor isessentially surjective on finite etaleR∧[t−1]-algebras. To prove full faithfulness, letY1, Y2 beany two finite etale schemes overX := R[t−1]; we letH be the functor that assigns to everyR[t−1]-schemeZ the setHomZ(Z×XY1, Z×XY2). Since the functors represented byY1 andY2

are locally constant sheaves, say in the fppf topology ofX := SpecR[t−1], the same holds forthe functorH, hence the latter is represented by a finite etaleX-scheme (proposition 8.2.23),which we denote by the same name. We can viewH as a trivial groupoid (i.e. such that foreveryX-schemeZ, the only morphisms of the groupoidH(Z) are the identity morphisms ofits objects). In this case, the associated morphism(s, t) : H → H ×X H is none other than thediagonal morphism; especially, the latter is an open imbedding, hence theorem 5.4.38 appliesand yields the sought assertion.

5.5. Lifting theorems for henselian pairs. For the considerations that follow, it will be usefulto generalize a little our usual setup : we wish to work with sheaves of almost modules (oralmost algebras) on a scheme. This is just a matter of introducing the relevant language, so wewill proceed somewhat briskily.

5.5.1. LetX be a scheme overSpec(V ). For every open subsetU ⊂ X, Γ(U,OX) is aV -algebra, hence we obtain a sheaf ofV a-algebrasOa

X onX by settingΓ(U,OaX) := Γ(U,OX)a

for every openU ⊂ X. We refer to [26, Ch.0,§3.1] for generalities on sheaves with valuesin arbitrary categories; in particular the sheaves ofV a-modules onX form an abelian tensorcategory, and hence we can define a sheafF of Oa

X-modules onX (briefly: aOaX-module) as a

sheaf of almost modules endowed with a scalar multiplicationOaX⊗V a F → F . Those gadgets

form a category that we denote byOaX-Mod. There is a functor

Γ(X,OaU)-Mod→ Oa

X-Mod M 7→M∼(5.5.2)

defined as one expects. We say thatF is quasi-coherentif we can coverX by affine open sub-setsUi, such thatF|Ui

is in the essential image of a functor (5.5.2). We denote byOaX-Modqcoh

the full subcategory of quasi-coherentOaX-modules. Similarly, we denote byOa

X-Alg (resp.OaX-Algqcoh) the category ofOa

X-algebras (resp. quasi-coherentOaX-algebras) defined as one

expects.

5.5.3. Since the functorsM → M! andM 7→ M∗ from A-modules toA∗-modules are rightexact, we can globalize them to the situation of (5.5.1). Thus, for everyV -schemeX there arefunctors

OaX-Mod→ OX-Mod F 7→ F! (resp.F 7→ F∗)

which are left (resp. right) adjoint to the localization functor OX-Mod → OaX-Mod. The

functor F 7→ F! is exact and preserves quasi-coherence (as can be easily deduced fromproposition 2.4.33), hence it provides a left adjoint to thelocalization functorOX-Modqcoh →OaX-Modqcoh. The functorF 7→ F∗ does not preserve quasi-coherence, in general.

5.5.4. LetR be aV -algebra and setX := Spec(R). Using the full faithfulness of the functorF 7→ F! one can easily verify that the functorM 7→ M∼ fromRa-modules to quasi-coherentOaX-modules is an equivalence, whose quasi-inverse is given bythe global section functor.After these preliminaries, we are ready to state the following descent result which will be

crucial for the proof of theorem 5.5.6.


Proposition 5.5.5. (i) LetR be aV -algebra,J ⊂ R a finitely generated ideal andR → R′ aflat morphism inducing an isomorphismR/J → R′/JR′. LetX := Spec(R),X ′ := Spec(R′),U := X \ V (J) andU ′ := U ×X X ′. Then the natural commutative diagrams of functors:

OaX-Modqcoh

//

OaX′-Modqcoh

OaU -Modqcoh

// OaU ′-Modqcoh

OaX-Algqcoh

//

OaX′-Algqcoh

OaU -Algqcoh

// OaU ′-Algqcoh

are2-cartesian (that is, cartesian in the category of2-categories).

(ii) LetA be aV a-algebra,f ∈ A∗ a non-zero-divisor,A∧ thef -adic completion ofA. DenotebyA-Modf (resp. A-Algf ) the full subcategory off -torsion freeA-modules (resp.A-algebras), and similarly defineA∧-Modf (resp.A∧-Algf ). Then the natural commutativediagrams of functors

A-Modf //

A∧-Modf

A[f−1]-Mod // A∧[f−1]-Mod

A-Algf //

A∧-Algf

A[f−1]-Alg // A∧[f−1]-Alg

are2-cartesian.

Proof. (i): for the functors onOaX-modules, one applies the functorF 7→ F!, thereby reducing

to the corresponding assertion for quasi-coherentOX-modules. Under assumption (a), the latteris proved in [36, Prop.4.2] (actually, inloc.cit. one assumes thatX ′ is faithfully flat overX, butone can reduce to such case after replacingX ′ byX ′ ∐ (U1 ∐ ... ∐ Un), where(Ui | i ≤ n) isa finite cover ofU by affine open subsets; notice also thatloc.cit. omits the assumption thatJis finitely generated, but the proof works only under such assumption). Since all the functorsinvolved commute with tensor products, the assertion aboutOa

X-algebras follows formally.(ii): for modules one argues as in the proof of (i), except that instead of invoking [36], one

uses [9, Theorem]. For algebras, one has to proceed a little more carefully, since the tensor prod-uct of twof -torsion free modules may fail to bef -torsion free. Hence, let(B1, B2, β) the datumconsisting of anA[f−1]-algebraB1, anA∧-algebraB2 and an isomorphismβ : B1 ⊗A A∧ ∼→B2[f

−1] ofA∧[f−1]-algebras. LetI :=⋃n>0 AnnB2⊗A∧B2(f

n) and setC := B2⊗A∧B2/I; β in-duces an isomorphismγ : B1⊗AB1⊗AA∧ ∼→ C ofA∧-modules, so by the foregoing there existsanA-moduleD such thatD[f−1] ≃ B1 ⊗A B1 andD ⊗A A∧ ≃ C. Furthermore, the multipli-cation morphismµB2/A∧ factors through a morphismµ : D → B2, and consequently the datum(µB1/A[f−1], µ, γ, β) determines a unique morphismD → B. Let I ′ :=

⋃n>0 AnnB⊗AB(fn);

one verifies easily that(B ⊗A B/J) ⊗A A∧ ≃ C, so again the same sort of arguments showthatD ≃ B ⊗A B/J , hence we obtain a morphismµB/A : B ⊗A B → B that liftsµB1/A[f−1]

andµB2/A∧ . Arguing along the same lines one can now verify easily that(B, µB/A) is really aB-algebra : we leave the details to the reader.

Theorem 5.5.6.Let (A, I) be a tight henselian pair,P an almost finitely generated projectiveA/I-module,m1 ⊂ m a finitely generated subideal. We have:

(i) If m has homological dimension≤ 1, then there exists an almost finitely generated projec-tiveA-module such thatP ⊗A (A/I) ≃ P .

(ii) If P1 andP2 are two liftings ofP as in (i) and if there exists an isomorphismβ : P1 ⊗A(A/m1I)

∼→ P2 ⊗A (A/m1I), then there exists an isomorphismβ : P1 → P2 such thatβ ⊗A 1A/I = β ⊗A 1A/I .


(iii) With the notation of(5.3.21), the natural functorA-Etafp → (A/I)-Etafp is an equiva-lence of categories.

Proof. We begin by showing (ii): indeed, the obstruction to the existence of a morphismα :P1 → P2 such thatα⊗A1(A/m1I) = β is a classω ∈ Ext1

A(P1,m1IP2). The same argument usedin the proof of claim 5.1.19 shows that the natural mapExt1

A(P1,m1IP2) → Ext1A(P1, IP2)

vanishes identically, and proves the assertion.Next we wish to show that the functor of (iii) is fully faithful. Therefore, letB,C be two

almost finitely presented etaleA-algebras, andφ : B/IB → C/IC a morphism. Accordingto lemma 5.2.20,φ is characterized by its associated idempotent, call ite ∈ (B ⊗A C/IC)∗.SetD := B ⊗A C; according to lemma 5.1.12(iii), the pair(D, ID) is tight henselian. Thenproposition 5.1.17 says thate lifts uniquely to an idempotente ∈ D∗.

Claim5.5.7. The associated morphismΓ(e) is an isomorphism (notation of (5.2.19)).

Proof of the claim:Indeed, by naturality ofΓ, we haveΓ(e) ⊗A 1A/I = Γ(e), so the assertionfollows from corollary 5.1.7.

By claim 5.5.7 and lemma 5.2.20,e corresponds to a unique morphismφ : B → C whichis the sought lifting ofφ. The remaining steps to complete the proof of (iii) will apply as wellto the proof of (i). Pick an integern > 0 and a finitely generated subidealm0 ⊂ m such thatIn ⊂ m0A; we notice that assertions (i) and (iii) also hold whenI is nilpotent, since in this casethey reduce to theorem 3.2.28(i.b),(ii). It follows easilythat it suffices to prove the assertionsfor the pair(A, In), hence we can and do assume throughout thatI ⊂ m0A.

Claim5.5.8. Assertions (i) and (iii) hold ifI is generated by a non-zero-divisor ofA∗.

Proof of the claim:Say thatI = fA, for some non-zero-divisorf ∈ m0A∗, and letA∧ be thef -adic completion ofA. By theorem 3.2.28(i),P lifts to a compatible system(Pn | n ∈ N) ofalmost finitely generated projectiveA/In+1-modules; by theorem 5.3.24, the latter compatiblesystem gives rise to a unique almost finitely generated projectiveA∧-moduleP ∧. Notice thatfis regular onA∧, hence also onP ∧. SinceA∧[f−1] is a (usual)V -algebra, theA∧[f−1]-moduleP ∧[f−1] is finitely generated projective; it follows from corollary5.4.42 that there exists afinitely generated projectiveA[f−1]-moduleQ with an isomorphismβ : Q ⊗A A∧ ≃ P ∧[f−1].By proposition 5.5.5(ii) the datum(P ∧, Q, β) determines a uniquef -torsion-freeA-modulePwhich lifts P . Sincef is regular on bothP andA, we haveTorAi (P,A/fA) = 0 for i = 1, 2,henceP isA-flat, by virtue of lemma 5.2.1. Next, setC := A[f−1]×A∧; from lemma 5.2.3(i)we deduce thatAnnC(P ⊗A C)2 ⊂ AnnA(P ), and sinceTorA1 (C, P ) = 0, remark 3.2.26(i)implies thatP is almost finitely presented, therefore almost projective over A, which showsthat (i) holds. Likewise, letB be an almost finitely presented etaleA/I-algebra; by theorem5.3.27,B admits a unique lifting to an almost finitely presented etale A∧-algebraB∧. ThenB∧[f−1] is a finite etaleA∧[f−1]-algebra, hence by proposition 5.4.54 there exists a uniquefinite etaleA[f−1]-algebraB0 with an isomorphismβ : B0 ⊗B B∧ ∼→ B∧[f−1]. By proposition5.5.5(ii), the datum(B∧, B0, β) determines a uniquef -torsion-freeA-algebraB; the foregoingproof of assertion (i) applies to theA-module underlyingB and shows thatB is an almostfinitely generated projectiveA-algebra. By constructionB/fB is unramified overA/fA, sotheorem 5.2.12(iv) applies and shows thatB is unramified overA. Thus, we have shown thatthe functor of (iii) is essentially surjective under the present assumptions; since it is alreadyknown in general that this functor is fully faithful, assertion (iii) is completely proved in thiscase.

Claim5.5.9. Assertions (i) and (iii) hold ifI is a principal ideal.


Proof of the claim:Say thatI = fA for somef ∈ m0A∗. Let J :=⋃n>0 AnnA(fn); we have

a cartesian diagram

A/(J ∩ I) //

A/I

A/J // A/(I + J).

LetP be as in (i) and letB be a finitely presented etaleA/I-algebra; by claim 5.5.8 we can findan almost finitely generated projectiveA/J-moduleP 1 with an isomorphismβ : P 1/IP 1

∼→P/JP and a finitely presented etaleA/J-algebraB1 that liftsB/JB. By proposition 3.4.21, thedatum(P, P 1, β) determines a unique almost finitely generated projectiveA/(I ∩ J)-moduleP 2; likewise, using corollary 3.4.22 we obtain an etale almost finitely presentedA/(I ∩ J)-algebraB2 that liftsB. Next, letK :=

⋂n>0 f

nA and setN := K ∩ J ∩ I; we have a cartesiandiagram

A/N //

A/(I ∩ J)

A∧ // A∧/(I ∩ J)A∧.

Due to theorem 3.2.28(i.b) we can liftP 2 ⊗A A∧ to an almost finitely generated projectiveA∧/(I ∩ J)2A∧-moduleP ∧2 . For the same reason,P∧2 ⊗A A/f 2A can be lifted to a compatiblefamily (Qn | n ∈ N), whereQn is almost finitely generated projective overA/fn+2A for everyn ∈ N. Finally, by theorem 5.3.24, the projective limitQ of the system(Qn | n ∈ N) isan almost finitely generated projectiveA∧-module. By construction, there is an isomorphismβ : Q/f 2Q

∼→ P∧2 ⊗A A/f 2A; by assertion (ii) it follows that there exists an isomorphismβ : Q/(I ∩ J)2Q

∼→ P ∧2 that liftsβ ⊗A 1A/fA. By proposition 3.4.21, the datum(P 2, Q, β ⊗A1A/(I∩J)) determines a unique almost finitely generated projectiveA/N-moduleP1; sinceN2 =0, P1 can be further lifted to an almost finitely generated projectiveA-moduleP , so assertion(i) holds in this case. The proof of assertion (iii) is analogous, but easier : we need to showthatB2 lifts to an almost finitely presented etaleA-algebraB; to this aim, it suffices to showthatB2 lifts to an almost finitely presented etaleA/N-algebraB1, since in that caseB1 canbe lifted to an almost finitely presented etaleA-algebraB, by theorem 3.2.28(ii). To obtainB1 it suffices to find an almost finitely presented etaleA∧-algebraB∧ with an isomorphismβ : B∧ ⊗A A/(I ∩ J)

∼→ B2 ⊗A A∧; indeed, in this case the datum(B∧, B2, β) determinesa unique etale almost finitely presentedA/N-algebra in view of corollary 3.4.22. Finally, weconsider the natural functors

A∧-Etafp → A∧/(I ∩ J)A∧-Etafp → A/I-Etafp.

By theorem 5.3.27, the composition of these two functors is an equivalences of categories andthe rightmost functor is fully faithful by (ii), so the leftmost functor is an equivalence, thusB1

as sought can be found, which concludes the proof of (iii) in this case.

Claim 5.5.10. Assertions (i) and (iii) hold ifI is a finitely generated ideal.

Proof of the claim: We proceed by induction on the numbern of generators ofI, the casen = 1 being covered by claim 5.5.8. So supposen > 1 and letf1, ..., fn ∈ I∗ be a finite setof generators ofI. Let P be as in (i) andB any almost finitely presented etaleA/I-algebra.We letA′ := A/f1A andJ := Im(I → A′). By lemma 5.1.12(iii) the pair(A′, J) is againtight henselian, so by inductive assumption we can find liftP (resp.B) to an almost finitelygenerated projectiveA′-module (resp. to an almost finitely presented etaleA′-algebra)P ′ (resp.


B′). Thence we apply claim 5.5.8 to further liftP ′ to a moduleP (resp. to an algebraB) asstated.

Let now (A, I) be a general tight henselian pair; we can find a henselian pair(R, J) suchthatRa = A, Ja = I andJ ⊂ m0. Denote by(Rh,m0R

h) the henselization of the pair(R,m0R), and letRh := Rh/JRh. Let us writeJ = colim

λ∈ΛJλ, whereJλ runs over the filtered

family of all finitely generated subideals ofJ ; setRλ := R/Jλ andRhλ := Rh ⊗R Rλ for every

λ ∈ Λ. Furthermore, letX := SpecR, Xλ := SpecRλ, Xh := SpecRh, Xhλ := Xh ×X Xλ,

U := X \ V (m0), φh : Uh := U ×X Xh → U , φhλ : Uh

λ := U ×X Xhλ → Uλ := U ×X Xλ

the natural morphisms of schemes. Now, letP be an almost finitely generatedA/I-module andB an almost finitely presented etaleA/I-algebra;P induces a quasi-coherentOa

X-moduleP∼,and by restriction we obtain a quasi-coherentOU -moduleP∼|U . Furthermore, sinceP is almost

finitely presented, we see thatP∼|U is finitely presented; by [29, Ch.IV, Th.8.5.2(ii)] it followsthat for someλ0 ∈ Λ there exists a quasi-coherent finitely presented moduleP onUλ0 whoserestriction to the closed subsetU agrees withP∼|U . By restricting further, we can even achievethatP be locally free of finite rank ([29, Ch.IV, Prop.8.5.5]). Similarly, we can find a locallyfreeOλ0-algebraB such thatB|U ≃ B∼|U .

Claim5.5.11. For every almost finitely generated projective(Rh)a-moduleQ there exists analmost finitely generated projective(Rh)a-moduleQ such thatQ⊗(Rh)a (Rh)a ≃ Q.

Proof of the claim:By theorem 3.2.28(i.b) we know thatQ lifts to an almost finitely generatedprojective moduleQ1 over (Rh/J2Rh)a. ThenQ2 := Q1 ⊗A (A/m2

0A) is an almost finitelygenerated projective(Rh/m2

0Rh)a-module, therefore by claim 5.5.10 we can liftQ2 to an almost

finitely generated projectiveRh-moduleQ (notice that(Rh,m20) is still a henselian pair). It

remains only to show thatQ is a lifting of Q; however, by construction we haveQ1 ⊗(Rh)a

(Rh/m20R

h)a ≃ Q ⊗(Rh)a (Rh/m20R

h)a, so it follows from (ii) thatQ1 ≃ Q/I2Q, whence theclaim.

Claim5.5.12. The natural functor(Rh)a-Etafp → (Rh)a-Etafp is an equivalence of categories.

Proof of the claim:By claim 5.5.10, the natural functors(Rh)a-Etafp → (Rh/m0Rh)a-Etafp

and(Rh)a-Etafp → (Rh/m0Rh)a-Etafp are equivalences of categories. The claim is a formal

consequence.

We apply claim 5.5.11 withQ := P ⊗A (Rh)a; let Q∼ be the quasi-coherentOaXh-module

associated toQ. By construction, the restrictionQ∼|Uh is a quasi-coherentOUh-module of finite

presentation, and we have an isomorphismβ : Q∼|Uh

∼→ φh∗(P∼|U). It then follows by [29,

Ch.IV, Cor.8.5.2.5] that there exists someµ ∈ Λ with Xµ ⊂ Xλ0 , such that the isomorphismβextends to an isomorphismβµ : Q∼

|Uhµ

∼→ φh∗µ (P|Uµ). Similarly, by claim 5.5.12, we can find an

almost finitely presented etale(Rh)a-algebraC with an isomorphismγµ : C∼|Uh

µ≃ φh∗(B∼

|Uhµ).

According to (5.5.4), the global section functors:

OaXλ

-Modqcoh → Raλ-Mod Oa

Xhλ-Modqcoh → (Rh

λ)a-Mod

are equivalences. Clearly the localization functors:

OUλ-Modqcoh → Oa

Uλ-Modqcoh OUh

λ-Modqcoh → Oa

Uhλ-Modqcoh


are equivalences as well, and similarly for the corresponding categories of algebras. By propo-sition 5.5.5(i) it follows that the natural diagrams:

Raλ-Mod //

(Rhλ)a-Mod

OUλ-Modqcoh

φh∗λ // OUh

λ-Modqcoh

Raλ-Alg //

(Rhλ)a-Alg

OUλ

-Algqcoh

φh∗λ // OUh

λ-Algqcoh

are2-cartesian for everyλ ∈ Λ. Hence, the datum(P|Uµ, Q⊗Ra Raµ, βµ) determines uniquely

a Raµ-modulePµ that lifts P , and the datum(B|Uµ, C ⊗Ra Ra

µ, γµ) determines aRaµ-algebra

Bµ that liftsB. Furthermore, since the natural morphismUµ ∐ Xhµ → Xµ is faithfully flat, it

follows from remark 3.2.26(ii) thatPµ andBµ are almost finitely generated projective overRaµ.

For the same reasons,Bµ is unramified, hence etale overRaµ. Finally, we apply claim 5.5.10 to

the henselian pair(Ra, Jaµ) to lift Pµ andBµ all the way toA, thereby concluding the proof ofthe theorem.

Lemma 5.5.13.Suppose thatm has homological dimension≤ 1, and let (A, I) be a tighthenselian pair,A := A/I,Q an almost finitely generated projectiveA-module,M anA-moduleandφ : Q→ M/IM anA-linear epimorphism. Then there exists an almost finitely generatedprojectiveA-moduleQ and a morphismφ : Q→M such thatφ⊗A 1A = φ.


Claim 5.5.14. The lemma holds ifI2 = 0. Furthermore, in this caseφ is an epimorphism.

Proof of the claim:First of all, notice thatM/IM is almost finitely generated, hence the sameholds forM , in view of lemma 3.2.25(ii). If nowQ is an almost finitely generated projectiveA-module andφ : Q→M is a morphism that liftsφ, we haveCoker(φ)⊗A A/I ≃ Coker φ = 0,whenceCoker φ = 0 by lemm 5.1.6. In other words, the second assertion follows from the first.Define theA-moduleN as the fibre product in the cartesian diagram ofA-modules:

Nα //

β

Q

φ

Mπ // M/IM

(whereπ is the natural projection). Notice thatIN = Kerα; indeed, clearlyα(IN) = 0, andon the other handβ(IN) = IM = Ker π ≃ Kerα. We derive an isomorphismψ : Q

∼→N/IN , and clearly it suffices to find a morphismQ → N that lifts ψ. Under our currentassumptions, theorem 5.5.6(i) provides an almost finitely generated projectiveA-moduleQ1

such thatQ1 ⊗A A ≃ Q, which in turns determines an extension ofA-modulesE := (0 →I ⊗A Q → Q1 → Q → 0). Furthermore,ψ induces an epimorphismχ : I ⊗A Q → IN ,whence an extensionχ ∗ E := (0→ IN → Q2 → Q→ 0). Another such extension is definedby F := (0 → IN → N

α→ Q → 0). However, any extensionX of Q by IN induces amorphismu(X) : I ⊗A Q → IN , defined as in (3.2.19). Directly on the definition one cancheck thatu(X) depends only on the class ofX in Ext1

A(Q, IN), and moreover, ifY is anyother such extension, thenu(X + Y ) = u(X) + u(Y ) (whereX + Y denotes the Baer sumof the two extensions). We can therefore compute:u(χ ∗ E − F ) = χ u(E) − u(F ); butthe definition ofE is such thatu(E) = 1I⊗AQ

and by inspecting the construction ofF weget u(F ) = χ. So finally u(χ ∗ E − F ) = 0; this means thatχ ∗ E − F is an extensionof A-modules, that is, its class is contained in the subgroupExt1

A(Q, IN) ⊂ Ext1

A(Q, IN).Notice now thatExt2

A(Q,Kerχ) = 0, due to lemma 2.4.14(i),(ii); sinceχ is an epimorphism,


it then follows that the induced mapExt1A(Q,χ) is surjective. Hence there exists an extension

X := (0 → I ⊗A Q → Q3 → Q → 0) of A-modules, such thatχ ∗ X = χ ∗ E − F , i.e.F = χ ∗ (E − X). SayE ′ := E − X = (0 → I ⊗A Q → Q → Q → 0); by construction,u(E′) = u(E), soQ is a flatA-module (see (3.2.19)) that liftsQ. Finally,Q is almost finitelygenerated projective by lemma 3.2.25(i),(ii). The push-out E ′ → χ ∗ E ′ delivers the promisedmorphismQ→ N .

Next, since the pair(A, In+1) is still tight henselian for everyn ∈ N, an easy induction showsthat the lemma holds whenI is a nilpotent ideal. For the general case, pickn > 0 and a finitelygenerated subidealm0 ⊂ m with In ⊂ m0A; by the foregoing we can find an almost finitelygenerated projectiveA/In+1-moduleQn+1 and an epimorphismφn+1 : Qn+1 → M/In+1M .By theorem 5.5.6(i), we can liftQn+1 to an almost finitely generated projectiveA-moduleQ;the obstruction to lifting the induced morphismQ → M/In+1M to a morphismQ → M isa classω ∈ Ext1

A(Q, In+1M); by the argument of claim 5.1.19 we see that the image ofω inExt1

A(Q, IM) vanishes, whence the claim.

Corollary 5.5.15. LetA be aV a-algebra,I ⊂ rad(A) a tight ideal and setA := A/I; let P analmost finitely generatedA-module, such thatP := P ⊗A A is an almost projectiveA-module.Then the following conditions are equivalent:

(i) P is an almost projectiveA-module.(ii) P is a flatA-module.(iii) P is an almost finitely presentedA-module andTorA1 (P,A) = 0.(iv) The natural morphismP ∗ → (P )∗ is an epimorphism.

Proof. Clearly (i) implies all the other assertions, so it suffices to show that each of the assertions(ii)-(iv) implies (i). Let us first remark that, in view of lemmata 2.3.13, 2.4.13 and theorem2.1.12, we can assume that the homological dimension ofm is≤ 1. Furthermore, let(Ah, Ih)be the henselization of the pair(A, I); according to (5.1.13), the morphismA→ Ah is faithfullyflat. In view of remark 3.2.26(ii) and lemma 2.4.31(i) we deduce that each of the statements (i)-(iii) on P andP is equivalent to the corresponding statement (i)h-(iii) h made on theAh-moduleP ⊗A Ah and the(A ⊗A Ah)-moduleP ⊗A Ah. Moreover, one checks easily that (iv)⇒(iv)h.Thus, up to replacing(A, I) by (Ah, Ih) we can assume that(A, I) is a tight henselian pair(notice as well thatIh = IAh). By lemma 5.5.13 we can find an almost finitely generatedprojectiveA-moduleQ and a morphismφ : Q → P such thatφ ⊗A 1A/I is an isomorphism.By lemma 5.1.6 we deduce easily thatφ is an epimorphism, Suppose now that (ii) holds; thento deduce (i) it remains only to prove the following :

Claim5.5.16. Kerφ = 0.

Proof of the claim:In view of proposition 2.4.28(v), it suffices to show thatE := EQ/A(x) = 0for everyx ∈ Kerφ∗ (see definition 2.4.23(iv)). However, sinceQ isA-flat, we can compute:

0 = TorA1 (A/E , P ) ≃ Ker(Ker(φ)⊗A (A/E )→ Q/EQ) ≃ ((Kerφ) ∩ EQ)/(E ·Kerφ)

that is,(Kerφ) ∩ EQ = E ·Kerφ. By proposition 2.4.28(v) we havex ∈ (EQ)∗; on the otherhand we also know thatKerφ ⊂ IQ, whencex ∈ (E IQ)∗. We apply once again proposition2.4.28(v) to deriveE = E I, so finallyE = 0 in view of lemma 5.1.6.

Next, assume (iii); we compute:0 = TorA1 (P,A) ≃ Ker(Ker(φ)⊗A A→ Ker(φ⊗A 1A)) =Ker(φ)⊗AA. SinceP is almost finitely presented,Kerφ is almost finitely generated by lemma2.3.17, whenceKerφ = 0 by lemma 5.1.6, so (i) holds. Finally, letφ∗ : P ∗ → Q∗ be thetransposed of the morphismφ; by lemma 2.4.31(i), the natural morphismψ : Q∗ → (Q/IQ)∗ isan epimorphism. The compositionψφ∗ factors through the transposed morphism(φ⊗A1A)∗ :(P )∗ → (Q/IQ)∗, so it is an epimorphism when (iv) holds; then lemma 5.1.6 implies easily


thatφ∗ is an epimorphism; since it is obviously a monomorphism, we deduce thatP ∗∼→ Q∗.

so (φ∗)∗ : Q ≃ (Q∗)∗ → (P ∗)∗ is an isomorphism; since the latter factors through the naturalmorphismP → (P ∗)∗, we see thatφ is a monomorphism and (i) follows.

5.6. Smooth locus of an affine almost scheme.Thoughout this section we fix aV a-algebraA and setS := SpecA. LetX be an affineS-scheme. We often identifyX with the functor itrepresents:

X : A-Alg→ Set T 7→ X(T o) := HomA-Algo(T o, X).

The usual argument from faithfully flat descent shows thatX is a sheaf for the fpqc topologyof A-Algo. In this section we aim to study, for every suchX, thesmooth locus ofX overS,which will be a certain natural subsheaf ofX. The starting point is the following:

Definition 5.6.1. Let S andX be as in (5.6). Given an affineS-schemeT andσ ∈ X(T ), wesay thatσ lies in the smooth locus ofX overS if the following two conditions hold:

(a) H1(Lσ∗La

X/S) = 0 and(b) H0(Lσ

∗LaX/S) is an almost finitely generated projectiveOT -module.

We denote byXsm(T ) ⊂ X(T ) the subset of all theT -sections ofX that lie in the smooth locusof X overS.

5.6.2. Using remark 3.2.26(iii) one sees thatXsm is a subsheaf ofX. Just as for usual schemes,in order to get a handle on the smooth locusXsm, one often needs to assume that the almostschemeX satisfies some finiteness conditions. For our purposes, the following will do:

Definition 5.6.3. We say that the affine almostS-schemeX is almost finitely presentedif thereexists an almost finitely generated projectiveOS-module, and an almost finitely generated idealJ of SP := Sym•OS

(P ), such thatX ≃ SpecSP/J .

Lemma 5.6.4. LetX = SpecSF/J , whereSF := Sym•OS(F ) for some flatOS-moduleF , and

J is any ideal. Then there is a natural isomorphism inD(OX-Mod):

τ[−1LaX/S ≃ (0→ J/J2 → OX ⊗OS

P → 0).

Proof. Let us remark the following:

Claim 5.6.5. With the notation of the lemma, there is a natural isomorphism LaSpecSF /S

≃SF ⊗OS

F [0] in D(SF -Mod).

Proof of the claim:By a theorem of Lazard ([51, Ch.I, Th.1.2]), every flatOS∗-module is thefiltered colimit of a family of free modules of finite rank; in particular this holds forF!; sinceboth functorsSym• andL commute with filtered colimits (proposition 2.5.33), we canthenreduce to the case whereSF ≃ OS[T1, ..., Tn] for somen ∈ N. In this case, we have naturalisomorphismsLa

SpecSF /S≃ La

OS∗[T1,...,Tn]/OS∗in light of proposition 8.1.7(ii). The claim follows.

Using claim 5.6.5, the assertion can be shown as in the proof of [46, Ch.III, Cor.1.2.9.1].

5.6.6. LetX be an almost finitely presentedS-scheme andt ∈ m any element. ThenOS[t−1]

is a (usual)V [t−1]-algebra, and we letSt := Spec OS[t−1],Xt := X ×S St. BothSt andXt are

represented by (usual) affine schemes overSpecV [t−1], and obviouslyXt is finitely presentedoverSt. Using lemma 5.6.4 it is also easy to see that the subfunctorXsm,t := Xsm ∩Xt of thefunctorX is represented by the smooth locus ofXt overSt, which is an open subscheme of thelatter scheme.


5.6.7. In the situation of (5.6.6), suppose moreover thatI ⊂ OS is a given ideal, and thattis regular inOS. LetR := OS∗; thent is a non-zero-divisor inR, and we have a well defined(t, I∗)-adic topology onXt(R[t−1]) = Xt(St). Furthermore, it is clear that the restriction map:

X(S)→ Xt(St) σ 7→ σ∗[t−1](5.6.8)

is injective. Consequently we can endowX(S) with the (t, I)-adic topology, defined as thetopology induced by the(t, I∗)-adic topology ofXt(St).

Lemma 5.6.9. In the situation of(5.6.7), the map(5.6.8)is an open imbedding for the respec-tive (t, I)-adic and(t, I∗)-adic topologies.

Proof. Let us writeX = SpecSP/J , whereSP is the symmetric algebra of an almost finitelygenerated projectiveOS-moduleP , and setP := SpecSP . ThenX is a closed subscheme ofP andXt is a closed subscheme ofPt, which is a vector bundle of finite rank overSt. The(t, I∗)-adic topology ofXt(St) is induced by the(t, I∗)-adic topology ofPt(St), and conse-quently the(t, I)-adic topology ofX(S) is induced by the(t, I)-adic topology ofP(S). Sincethe commutative diagram of sets

X(S) //

Xt(St)

P(S) // Pt(St)

(5.6.10)

is cartesian, we reduce to showing that the restriction mapP(S) → Pt(St) is open. To thisaim, we setP [t−1]∗ := HomOS

(P,OS[t−1]) and we consider the diagram

HomOS(P,OS)

∼ //

P(S)

P [t−1]∗

∼ // Pt(St)

whose horizontal arrows are given by the rule:φ 7→ Sym•OSφ. SinceP [t−1]∗ is a finitely

generatedOS[t−1]-module, it is endowed with a well defined(t, I∗)-preadic topology. We define

a linear(t, I)-adic topologyon HomOS(P,OS), by declaring that the system of submodules

(HomOS(P, tnI) | n ∈ N) forms a cofinal family of open neighborhoods of zero.

Claim5.6.11. With these(t, I)-adic and(t, I∗)-adic topologies, (5.6.10) is a diagram of con-tinuous maps, and the horizontal arrows are homeomorphisms.

Proof of the claim:In caseP [t−1] ≃ R[t−1]n for somen ∈ N, we havePt(St) ≃ R[t−1]n, andthen the bottom arrow of (5.6.10) is a homeomorphism, essentially by definition. The generalcase can be reduced to the case of a free module, by writingR[t−1]n = P [t−1] ⊕ Q for someprojectiveR[t−1]-moduleQ, and remarking that the(t, I∗)-adic topologies are compatible withcartesian products. To prove that the top arrow is a homeomorphism, it suffices therefore toshow that the(t, I)-adic topology ofHomOS

(P,OS) is induced by the(t, I∗)-adic topology ofP [t−1]∗. To this aim, pick a finitely generatedR-moduleP0 ⊂ P∗ with tP∗ ⊂ P0. ClearlyHomR(P0,OS[t

−1]) = P [t−1]∗, and again by reducing to the case whereP [t−1] is free, oneverifies that the(t, I∗)-adic topology onP [t−1]∗ admits the system(HomR(P0, t

nI∗) | n ∈ N)as a cofinal family of open neighborhoods of zero. For everyn ∈ N setUn := φ : P →OS | φ(P0) ⊂ tnI∗; we have

HomOS(P, tnI) ⊂ Un ⊂ HomOS

(P, tn−1I) for everyn ∈ N(5.6.12)

which implies the claim.


In view of claim 5.6.11, we are reduced to showing that the mapHomOS(P,OS) → P [t−1]∗

is open. But this is again a direct consequence of (5.6.12).

Proposition 5.6.13.LetX, T be affineS-schemes andσ ∈ X(T ) a T -section,I ⊂ rad(OT )an ideal, and setT0 := Spec OT/I; suppose that the restrictionσ0 ∈ X(T0) of σ lies in thesmooth locus ofX. Suppose moreover that either:

(a) I is nilpotent, or(b) I is tight andX is almost finitely presented overS.

Thenσ ∈ Xsm(T ).

Proof. Suppose that (a) holds; for any quasi-coherentOT -moduleF , let us denote byFil•IF

the I-adic filtration onF . We can writeτ[−1Lσ∗La

X/S ≃ (0 → Nφ→ P → 0) for two OT -

modulesN andP , and we can assume thatP is almost projective overOT , so that the naturalmorphism

τ[−1(OT0

L

⊗OTLσ∗La

X/S)→ (0→ N/INgr0Iφ−→ P/IP → 0)

is an isomorphism inD(OT0-Mod). Hence, the assumption onσ means thatgr0Iφ is a monomor-

phism with almost finitely generated projective cokernel overOT0 . We consider, for every inte-geri ∈ N the commutative diagram:

griINgri

Iφ // griIP

gr0IN ⊗OT

griIOT

αi

OO

gr0Iφ⊗OT1gri

IOT // gr0

IP ⊗OTgriIOT

βi

OO

SinceP is almost projective (especially, flat)βi is an isomorphism. Moreover, sincegr0Iφis a monomorphism with almost projective cokernel, the longexact Tor sequence shows thatgr0Iφ ⊗OT

1griIOT

is a monomorphism. It follows thatαi is a monomorphism for everyi ∈ N,and since it is obviously an epimorphism, we deduce thatαi is an isomorphism andgr•Iφ isa monomorphism, therefore the same holds forφ. Let C := Coker(N → P ); we deduceeasily thatTorOT

1 (OT0, C) = 0, and then it follows from the local flatness criterion (see [54,Ch.8,Th.22.3]) thatC is a flatOT -module. Finally lemma 3.2.25(i),(ii) says thatC is almostfinitely generated projective, whence the claim, in case (a).

Next, suppose that assumption (b) holds; by lemma 5.6.4 there is an isomorphism :

τ[−1LaX/S ≃ (0→ N

φ→ Q→ 0)

whereN is almost finitely generated andQ is almost finitely generated projective overOX .

SinceQ is almost projective, we haveτ[−1(OT0

L

⊗OTLσ∗La

X/S) ≃ (0→ N/INφ0→ Q/IQ→ 0),

and by assumptionKerφ0 = 0 andCokerφ0 is an almost finitely generated projectiveOT0-module. Using the long exact Ext sequence we deduce thatN/IN is almost projective. Thus,the bottom arrow of the natural commutative diagram

Q∗φ∗ //

α

N∗

β

(Q/IQ)∗φ∗0 // (N/IN)∗

is an epimorphism. Invoking twice corollary 5.5.15 we find first thatα is an epimorphism(whence so isβ), and then thatN is almost projective. It then follows thatφ∗ ⊗OS

1OS0is an

epimorphism as well, hence the same holds forφ∗, by applying Nakayama’s lemma 5.1.6 to


Coker φ∗. The long exact Ext sequence then shows thatKerφ∗ is almost projective, and the lat-ter is almost finitely generated as well, sinceQ is. Dualizing, we see thatφ is a monomorphismandCoker φ ≃ (Kerφ∗)∗ is almost finitely generated projective, which is the claim.

Lemma 5.6.14.LetX be an affineS-scheme, and suppose we are given a cartesian diagramof OS-algebras as(3.4.8). There follows a cartesian diagram of sets:

Xsm(SpecA0) //

Xsm(SpecA2)

Xsm(SpecA1) // Xsm(SpecA3).

Proof. We have to check that every sectionσ ∈ X(SpecA0) whose restrictions toSpecA1

andSpecA2 lie in the smooth locus ofX overS, lies itself in the smooth locus. Hence, let

τ[−1Lσ∗LX/S ≃ (0 → N

φ→ P → 0), for someA0-modulesN andP , chosen so thatP is

almost projective. HenceAiL

⊗A0 τ[−1Lσ∗LX/S ≃ (0→ Ni

φi→ Pi → 0) (whereNi := Ai⊗A0 Nand likewise forPi, i = 1, 2). By assumption,Kerφi = 0 andCoker φi is almost finitelygenerated projective overAi (i = 1, 2). It follows thatNi is almost projective fori = 1, 2.Using proposition 3.4.21 we deduce thatCoker φ is almost finitely generated projective overA0 andN is almost projective. In particular,N is flat and consequentlyN ⊂ N1 ⊕ N2, soKerφ = 0, and the assertion follows.

Lemma 5.6.15.LetX andY be two affineS-schemes, and suppose thatTorOSi (OX,OY ) = 0

for everyi > 0. Then we have a natural isomorphism of sheaves:

(X ×S Y )sm ≃ Xsm × Ysm.

Proof. Let πX : X ×S Y → X be the natural projection, and define likewiseπY . By theorem2.5.35, our assumptions imply that the natural morphismπ∗XLa

X/S ⊕ π∗Y LaY/S → La

X×SY/Sis a

quasi-isomorphism. Let nowT be an affineS-scheme and(σ, τ) ∈ X ×S Y (T ) = X(T ) ×Y (T ); we derive a natural isomorphism:

L(σ, τ)∗LX×SY/S ≃ Lσ∗LX/S ⊕ Lτ ∗LY/S

from which the claim follows straightforwardly.

Theorem 5.6.16.Assume thathom.dimV m ≤ 1. LetX be an almost finitely presented affineS-scheme,I ⊂ OS an ideal such that the pair(OS, I) is henselian,m0 ⊂ m a finitely generatedsubideal, and setSn := Spec OS/m

n0I for everyn ∈ N. Then we have:

(i) Im(Xsm(S1)→ Xsm(S0)) = Im(Xsm(S)→ Xsm(S0)).(ii) SetX∧sm(S) := lim

n∈NXsm(Sn), and endowX∧sm(S) with the corresponding pro-discrete

topology. Then the natural mapXsm(S)→ X∧sm(S) has dense image.

Proof. We begin with an easy reduction:

Claim5.6.17. In order to prove (i), we can assume thatI is a tight ideal.

Proof of the claim:Indeed, letm0 ⊂ m be any subideal, and choosem1 ⊂ m such thatm0 ⊂ m21.

SetI ′ := m1I, S ′0 := Spec OS/I′, S ′1 := Spec OS/m1I

′. Notice thatI ′ is tight, and supposethat the assertion is known for this ideal. By a simple chase on the commutative diagram:

Xsm(S) // Xsm(S0) Xsm(S1)

oo

Xsm(S) // Xsm(S ′0)

OO

Xsm(S ′1)oo


the assertion can then be deduced forI as well.Let σ0 ∈ Xsm(S0) that admits an extensionσ ∈ Xsm(S1), pick finitely many generators

ε1, ..., εk for m0, and define a mapφ : I⊕k → m0I by the rule:(a1, ..., ak) 7→∑k

i εi · ai.Claim 5.6.18. Suppose thatI2 = 0 (so thatI is anOS0-module); thenExt1OS0

(Lσ∗0LaX/S , φ) is

onto andExt1OS0(Lσ∗0La

X/S, I)a = 0.

Proof of the claim:For the first assertion we use the spectral sequence

Epq2 := ExtpOS0

(Hq(Lσ∗0La

X/S), I⊕k)⇒ Extp+qOS0

(Lσ∗0LaX/S , I

⊕k)

and a similar one form0I. Sinceσ0 lies in the smooth locus ofX, theE012 terms vanish, hence

we are reduced to verifying that the mapExt1OS0

(H0(Lσ∗0La

X/S), φ) is surjective. However, the

cokernel of this map is a submodule ofExt2OS0

(H0(Lσ∗0La

X/S),Kerφ); again the assumptionthatσ0 lies in the smooth locus ofX, andhom.dimV m ≤ 1, imply that the latter Ext groupvanishes by lemma 2.4.14(ii). The same spectral sequence argument also proves the secondassertion.

Claim 5.6.19. Assertion (i) holds ifI2 = 0.

Proof of the claim:We need to show that there exists a morphismσ : S → X extendingσ0;thenσ ∈ Xsm(S) in view of proposition 5.6.13. In other words, we have to findσ♯ that fits intoa morphism of extensions ofOS-algebras:

0 // 0 //

OX

σ♯

OX//

σ♯0

0

0 // I // OS// OS0

// 0.

By proposition 3.2.16, the obstruction to the existence ofσ♯ is a class

ω ∈ Ext1OX(La

X/S , I) ≃ Ext1OS0(Lσ∗0La

X/S , I).

Likewise, the obstruction to extendingσ is a classω ∈ Ext1OS0(Lσ∗0La

X/S ,m0I), andω is the

image ofω under the mapExt1OS0(Lσ∗0La

X/S , j) (where we have denoted byj : m0I → I the

inclusion). From claim 5.6.18 one deduces easily first: thatExt1OS0(Lσ∗0La

X/S , j φ) vanishes,

and therefore, second: thatExt1OS0(Lσ∗0L

aX/S , j) must already vanish. Thusω = 0, and the

assertion holds.

Claim 5.6.20. Choose a finitely generated subidealm1 ⊂ m such thatm0 ⊂ m21. The sectionσ0

can be lifted to an element oflimn∈N

Xsm(Spec OS/m1In).

Proof of the claim:For everyn > 0, let Tn := Spec OS/m21In andjn : Spec OS/m1I

n → Tnthe natural morphism; we construct by induction onn ∈ N a sequence of sectionσn ∈ X(Tn),such that the family(σiji | i > 0) defines an element oflim

n∈NXsm(Spec OS/m1I

n). To this aim,

we takeσ1 equal to the restriction ofσ; suppose then thatn > 1 and thatσn−1 is already given.Notice that the imageJ ⊂ OSn of m1I

n−1 satisfiesJ2 = 0. By the claim 5.6.19, it follows thatσn−1 jn−1 extends to a section inXsm(Tn), and this we callσn.

Let us now show how to deduce assertion (i) from (ii). By claim5.6.17 we can suppose thatIm ⊂ m0OS for somem ≥ 0 and a finitely generated subidealm0 ⊂ m; let m1 ⊂ m be as inclaim 5.6.20; clearly for everyn ∈ N there existsn ≥ 0 such thatm1I

m ⊂ mn0I, hence claim

5.6.20 shows thatσ0 can be lifted to an element ofX∧sm(S) and then (ii) yields (i) trivially.Hence, it remains only to show (ii).


Claim5.6.21. In order to prove (ii) we can assume thatm0 is a principal ideal.

Proof of the claim:We argue by induction on the number of generators ofm0. Thus, letε1, ..., εkbe a finite system of generators form0, and suppose that the assertion is known for all idealsgenerated by less thank elements. Letm(n)

0 be the ideal generated byεn1 , ..., εnk , and setS(n) :=

Spec OS/m(n)0 I; for everyn ∈ N there existsN ∈ N such thatmN

0 ⊂ m(n)0 ⊂ mn

0 , whence anisomorphism of pro-discrete spaces:X∧sm(S) ≃ lim

n∈NXsm(S(n)). Thus, we can suppose that we

are given a compatible system of sectionsσn ∈ Xsm(S(n)), and wish to show that, for everyn ∈ N, there existsσ ∈ Xsm(S) whose restriction toS(n) agrees withσn. Fix N > 0, setT := Spec OS/ε

N1 I and letm1 ⊂ m be the ideal generated byε2, ..., εk. Let alsoT(n) :=

T ×S S(n) for everyn ∈ N, and denote byσn|T ∈ Xsm(T(n)) the restriction ofσn. Clearly

T(n) ≃ Spec OT/m(n)1 OT for everyn ≥ N , and the pair(OT , IOT ) is henselian (cp. remark

5.1.9(v)). Hence, by inductive assumption, for everyn ≥ N we can findσT ∈ X(T ) whoserestriction toT(n) agrees withσn|T . We can then apply claim 5.6.20, in order to find a compatiblesystem of sections(σ′n ∈ X(Spec OS/ε

n1I) | n ≥ N), whose restriction toSpec OS/ε

N−11 I

agrees with the restriction ofσT . Finally, if we assume that (ii) is known wheneverm0 isprincipal, we can find a sectionσ ∈ Xsm(S) whose restriction toSpec OS/ε

N−11 OS agrees with

σ′N , whence the claim.

For a givenε ∈ m, setK(ε) :=⋃n∈N AnnOS

(εn).

Claim5.6.22. K(ε)∗ =⋃n∈N AnnOS∗

(εn).

Proof of the claim:Clearly we have only to show the inclusionK(ε)∗ ⊂⋃n∈N AnnOS∗

(εn). By

applying the left exact functorM 7→ M∗ to the left exact sequence0 → AnnOS(εn) → OS

εn

→OS we deduce thatAnnOS

(εn)∗ = AnnOS∗(εn). However,ε · K(ε)∗ ⊂

⋃n∈N AnnOS

(εn)∗, sothe claim follows easily.

Claim5.6.23. (K(ε) ∩ εnI)∗ ⊂ nil(OS∗) for everyn > 0.

Proof of the claim:We have(K(ε) ∩ εnI)∗ = K(ε)∗ ∩ (εnI)∗. Let x ∈ (εnI)∗; according tolemma 5.1.14, we havexk ∈ εnOS∗ for every sufficiently largek ∈ N. On the other hand, itis easy to check thatAnnOS∗

(εm) ∩ εnOS∗ ⊂ nil(OS∗) for everyn,m > 0. In view of claim5.6.22, the assertion follows.

Claim5.6.24. If n > 0, every almost finitely generated subideal ofK(ε) ∩ εnI is nilpotent.

Proof of the claim:Let I be such an ideal; we can find a finitely generated idealI0 ⊂ (K(ε)∩εnI)∗ such thatεI∗ ⊂ I0 ⊂ I∗. SinceI ⊂ εnI, lemma 5.1.14 says that there existsN ∈ Nsuch that(I∗)N ⊂ εnOS∗, hence(I∗)N+1 ⊂ I0. From claim 5.6.23 we deduce thatI0 is anilpotent ideal, so the same holds forI .

Claim5.6.25. In order to prove (ii) we can assume thatm0 = εV , whereε ∈ m is anOS-regularelement.

Proof of the claim:Let us writeX = SpecSP/J , whereSP := Sym•OSP for some almost

finitely projectiveOS-moduleP , andJ ⊂ SP is an almost finitely generated ideal. By claim5.6.21 we can assume thatm0 = εV for someε ∈ m. SetS := Spec OS/K(ε) andSn :=S×S Sn for everyn ∈ N. Letσ∧ ∈ X∧sm(S); by definitionσ∧ is a compatible family of sectionsσn : Sn → X lying in the smooth locus ofX overS. In turns,σn can be viewed as the datumof anOS-linear morphismτn : P → OS/ε

nI, such that the induced morphism ofOS-algebrasSym•OS

τ : Sym•OSP → OS/ε

nI satisfies the condition:Sym•OSτ(J) = 0. For everyn ∈ N we


have a cartesian diagram ofOS-algebras:

OS/(K(ε) ∩ εnI) //

OS

πn

OSn

pn // OSn.

Notice thatε is regular inOS and suppose that assertion (ii) holds for all almost finitelypre-sentedS-schemes, especially forX := X ×S S. Let σn : Sn → X be the restriction ofσn,for everyn ∈ N. It then follows that, for everyn ∈ N, σn extends to a sectionσ ∈ Xsm(S).The datum ofσ is equivalent to the datum of anOS-linear morphismτ : P → OS such thatSym•OS

τ (J) = 0 and such thatπn τ = pn τn. The pair(τn, τ ) determines a morphismωn : P → OS/(K(ε) ∩ εnI), and by construction we have:Sym•OS

ωn(J) = 0, i.e. ωn inducesa sectionσ′ : Spec OS/(K(ε) ∩ εnI)→ X, and thenσ′ must lie in the smooth locus ofX overS, in view of lemma 5.6.14. The obstruction to the existence ofa lifting ω : P → OS of ωn, isa classαn ∈ Ext1

OS(P,K(ε) ∩ εnI). A simple verification shows that

K(ε) ∩ εnI = ε · (K(ε) ∩ εn−1I) for all n > 0.(5.6.26)

From (5.6.26), an argument as in the proof of claim 5.1.19 allows to conclude that the imageof αn in Ext1

OS(P,K(ε) ∩ εn−1I) vanishes; however, this image is none other thanαn−1, so

actuallyαn = 0, and the sought lifting can be found for everyn > 0. By construction wehaveJ := Sym•OS

ω(J) ⊂ K(ε) ∩ εnI. Now, ω induces a sectionσ′′ : Spec OS/J → X,which by construction extendsσ′; moreover, the pair(OS/J, I/J) is henselian (cp. remark5.1.9(v)), henceεnI/J is a tight radical ideal. Thus, using proposition 5.6.13(ii) we derivethatσ′′ lies as well in the smooth locus ofX overS. SinceJ is almost finitely generated,Jis nilpotent in view of claim 5.6.24. Moreover, by (5.6.26) we can writeJ = εJ for someideal J ⊂ K(ε) ∩ εn−1I; J is nilpotent ifn ≥ 2, since in that caseJ 2 ⊂ J . Hence wecan apply claim 5.6.20, to deduce that the restriction ofσ′′ to Spec OS/J extends to a sectionσ ∈ Xsm(S); by constructionσ extendsσn−1. Sincen can be taken to be arbitrarily large, theclaim follows.

So finally we suppose thatm0 = tV , with t anOS-regular element. SetO∧S := limn∈N

OSn and

S∧ := Spec O∧S . We have a natural bijection

X(S∧) ≃ X∧(S) := limn∈N

X(Sn).(5.6.27)

Let I∧∗ ⊂ O∧S∗ ≃ limn∈N

OSn∗ be the topological closure ofI∗; sincetnI∧∗ is the topological closure

of tnI∗, one verifies easily that (5.6.27) identifies the pro-discrete topology ofX∧(S) with the(t, I∧∗ )-adic topology ofX(S∧) (see (5.6.7) and the proof of lemma 5.6.9).

Claim 5.6.28. The homeomorphism (5.6.27) induces a bijection:Xsm(S∧) ≃ X∧sm(S).

Proof of the claim: Let (σn | n ∈ N) be an element ofXsm(S∧), andσ∧ ∈ X(S∧) thecorresponding section. We have to show thatσ∧ lies in the smooth locus ofX. Thus, letI∧ := Ker(O∧S → OS/I); we have a natural isomorphism:O∧S /tI

∧ ≃ OS/tI, and by assump-tion, the restriction ofσ∧ to Spec O∧S /tI

∧ lies in the smooth locus ofX. The assertion thenfollows from proposition 5.6.13.

Under the standing assumptions, we have a cartesian diagramof almost algebras:

OS//

OS[t−1]

O∧S // O∧S [t−1]


whence a cartesian diagram of sets:

X(S) //

Xt(St)

α

X(S∧)

β // Xt(S∧t ).

Now, let (σn | n ∈ N) ∈ X∧sm(S); by claim 5.6.28, the corresponding sectionσ∧ : S∧ → Xlies in the smooth locus ofX. Let σ∧t ∈ Xsm,t(S

∧t ) be the restriction ofσ∧. By corollary

5.1.16(ii), the pair(OS∗, tI∗) is henselian; then by (5.6.6) and proposition 5.4.21, the restrictionαsm : Xsm,t(St) → Xsm,t(S

∧t ) has dense image for the(t, I∧∗ )-adic topology. Hence, we can

approximateσ∧t arbitrarily (t, I∧∗ )-adically close by a section of the formτ∧t := α(τt), whereτt ∈ Xsm,t(St). Furthermore,β is an open imbedding, by lemma 5.6.9. Hence, ifτ∧t is closeenough toσ∧, we can findτ∧ ∈ X(S∧) such thatβ(τ∧) = τ∧t . In view of proposition 5.6.13we can also achieve thatτ∧ lies in the smooth locus ofX. The pair(τt, τ∧) determines a uniquesectionτ ∈ X(S), and by constructionτ can be obtained as(t, I)-adically close to(σn | n ∈ N)as desired. Especially, we can achieve thatτ lies in the smooth locus ofX, which concludes theproof of the theorem.

5.7. Quasi-projective almost schemes.LetR be aV -algebra; we denote by(R-Alg)ofpqc thelarge fpqc site ofaffineR-schemes, and similarly for the site(Ra-Alg)ofpqc. The localizationfunctorR-Alg→ Ra-Alg defines a morphism of sites:

j : (Ra-Alg)ofpqc → (R-Alg)ofpqc.

If F is any sheaf on(R-Alg)ofpqc, thenF a := j∗F can be described as the sheaf associ-ated to the presheafB 7→ F (B∗). Especially, we can regard anyR-schemeX as a sheaf on(R-Alg)ofpqc, and hence we obtain thealmost schemeXa associated toX. If X is affine, thisnotation agrees with that of (3.3.3). For a generalX, pick a Zariski hypercoveringZ• → X,where eachZi is a disjoint union of affineR-schemes; thenX ≃ colim

∆oZ• as fpqc-sheaves, and

thusXa ≃ colim∆o

Za• in the topos of sheaves on(Ra-Alg)ofpqc. Furthermore, we have

j∗SpecB = SpecB!! ⊗Ra!!R for everyRa-algebraB.

If G is anR-group scheme, thenGa is clearly anRa-group scheme, and ifY → X is anyG-torsor over anR-schemeX (for the fpqc topology ofX), thenY a → Xa is aGa-torsor.

5.7.1. Let nowA be anRa-algebra,J ⊂ A∗ a finitely generated ideal. We define thequasi-affineRa-schemeSpecA \ V (J) as the almost schemej∗(SpecA∗ \ V (J)). Let f1, ..., fk be afinite set of generators ofJ ; by (5.7) this can be realized as the subsheaf

⋃ki=1 SpecA[f−1

i ] ⊂SpecA, where the union takes place in the category of sheaves on(Ra-Alg)ofpqc.

Lemma 5.7.2. LetB be aV a-algebra andg1, ..., gk ∈ B∗. Then∑k

i=1 giB = B if and only ifthe natural morphismB → ∏k

i=1B[g−1i ] is faithfully flat.

Proof. Supppose that∑k

i=1 giB = B; we have to show that∏k

i=1C[g−1i ] 6= 0 for every non-

zero quotientC of B. ReplacingB by C, we are reduced to showing that, ifB[g−1i ] = 0 for

everyi ≤ k, thenB = 0. However, the conditionB[g−1i ] = 0 implies that for everyε ∈ m

there existsni ∈ N such thatε · gnii = 0; setn := max(ni | i ≤ k). We have

∑ki=1 g

ni B = B,

henceε ·B = 0. Sinceε is arbitrary, the claim follows. Conversely, letB′ :=∏k

i=1B[g−1i ] and

I :=∑k

i=1 giB; clearlyIB′ = B′, thereforeI = B providedB′ is faithfully flat overB.


Lemma 5.7.3. With the notation of(5.7.1), the almost schemeSpecA \V (J) is the subfunctorof SpecA, whoseT -section are the morphismsσ : T → SpecA such thatσ♯(J)OT = OT , forevery affineRa-schemeT .

Proof. Let σ be aT -section ofSpecA \ V (J); by definition, this is the same as saying that thenatural morphism∐ki=1T ×SpecA SpecA[f−1

i ]→ T is a fpqc-covering ofT . Hence the claim isjust a restatement of lemma 5.7.2.

5.7.4. Alternatively, one can regard a quasi-affineRa-scheme as a difference of two affineRa-schemes. This viewpoint is elaborated in the following:

Definition 5.7.5. Let φ : F → G be a morphisms of sheaves on(Ra-Alg)ofpqc.(i) We say thatφ is a closed imbeddingif, for every affineRa-schemeT and every section

T → G , the induced morphismT ×G F → T is a closed imbedding of affineRa-schemes.(ii) The differenceG \F is the subsheaf ofG whoseT -sections are all theT -sectionsT → G

of G such thatT ×G F = ∅, for every affineRa-schemeT . Here∅ denotes the initialobject of the topos of sheaves on(Ra-Alg)ofpqc.

Using lemma 5.7.3 one checks easily that the sheaf-theoretic differenceSpecA \ SpecA/Jis the same as the almost scheme considered in (5.7.1).

Lemma 5.7.6. Letφ : F → G be a morphism of sheaves on(R-Alg)ofpqc. Then:

j∗(G \F ) ⊂ j∗G \ j∗F .

Proof. G \F is also the largest subobject of ofG that has empty intersection withIm(φ); inparticular this is defined for every topos. Then the statement holds for any morphism of topoi(in this casej), and it follows easily from the facts that the pull-back functor (herej∗) is leftexact and sends empty objects to empty objects.

Example 5.7.7.LetE be anRa-module.(i) We define the presheafPRa(E) on(Ra-Alg)ofpqc as follows. For every affineRa-schemeT ,

theT -sections ofP(E) are the strictly invertible quotients ofE⊗RaOT , i.e. the equivalenceclasses of epimorphismsE ⊗Ra OT → L whereL is a strictly invertibleOT -module (seedefinition (4.4.32)).

(ii) More generally, for every integerr ∈ N we can define the presheafGrassrRa(E); its T -sections are the rankr almostOT -projective quotientsE ⊗Ra OT → P .

By faithfully flat descent it is easy to verify that both thesepresheaves are in fact fpqc-sheaves.

5.7.8. LetE be anR-module,B anRa-algebra; a sectionσ ∈ GrassrR(E)(SpecB∗) is aquotient mapσ : E ⊗R B∗ → P , onto a projectiveB∗-moduleP of rankr. To σ we associatethe quotientσa : Ea⊗RB → P a; after passing to the associated sheaves we obtain a morphism:

j∗GrassrR(E)→ GrassrRa(Ea) σ 7→ σa.(5.7.9)

SinceP ≃ P a∗ , it is clear that the ruleσ 7→ σa defines an injective map of presheaves, hence

(5.7.9) is a monomorphism.

Lemma 5.7.10.Suppose thatE is a finitely generatedR-module. Then(5.7.9) is an isomor-phism.

Proof. We have to show that (5.7.9) is an epimorphism of fpqc-sheaves; hence, letT be anyaffineRa-scheme,σ : Ea ⊗Ra OT → P a quotient morphism, withP almost projective overOT of rank r. The contention is that there exists a faithfully flat morphismf : U → T suchthat f ∗σ is in the image ofGrassrR(E)(U∗) (whereU∗ := Spec OU∗). By theorem 4.4.24 wecan then reduce to the case whenP = Or

T . We deduce an epimorphismΛrOTσ : Λr

OT(Ea ⊗Ra


OT ) → ΛrOT

(OrT ) ≃ OT . Let e1, ..., ek be a finite set of generators ofE, and for every subset

J := j1, ..., jr ⊂ 1, ..., k of cardinalityr, let eJ := ej1 ∧ ... ∧ ejr ∈ ΛrOT

(Ea ⊗Ra OT ); itfollows that the set(tJ := Λr

OTσ(eJ) | J ⊂ 1, ..., k) generatesOT . SetUJ := Spec OT [t−1

J ]

for everyJ ⊂ 1, ..., k as above; by constructionσ induces surjective mapsE ⊗R OT∗[t−1J ]→

OrT∗[t

−1J ]; after tensoring byOT [t−1

J ]∗ we obtain surjectionsE ⊗R OT [t−1J ]∗ → OT [t−1

J ]r∗. Inother words, the restrictionσ|UJ

lies in the image ofGrassrR(E)(UJ∗) for every suchJ ; on theother hand, lemma 5.7.2 says that the natural morphism∐JUJ → T is a fpqc covering, whencethe claim.

Lemma 5.7.10 says in particular thatGrassrRa(Ea) is an almost scheme associated to ascheme wheneverE is finitely generated. Next we wish to define the projectiveRa-schemeassociated a gradedRa-algebra.

Definition 5.7.11. Let A := ⊕i∈NAi be a gradedRa-algebra which is generated byA1 overA0 := Ra, in the sense that the natural morphism of gradedRa-algebrasSym•RaA1 → A is anepimorphism. We letProj(A) be the sheaf on(Ra-Alg)ofpqc defined as follows. For every affineRa-schemeT , theT -points ofProj(A) are the strictly invertible quotientsA1 ⊗Ra OT → L ofA1 ⊗Ra OT , inducing morphisms of gradedRa-algebrasA→ Sym•OT

L.

Lemma 5.7.12.With the notation of(5.7.11), the natural map of sheavesProj(A)→ PRa(A1)is a closed imbedding.

Proof. Let T be any affineRa-scheme andσ : T → PRa(A1) a morphism of fpqc-sheaves.We have to show thatT ′ := T ×PRa (A1) Proj(A) is a closed subscheme ofT . By definition,σ : A1 ⊗Ra OT → L is a strictly invertible quotient; we deduce viaSym•OT

σ a natural structureof Sym•RaA1-module onSym•OT

L. LetU be an affineRa-scheme andU → T ′ a morphism offpqc-sheaves; setJn := Ker (Symn

RaA1 → A) for everyn ∈ N. By definition:

Jn · SymnOTL ⊂ Ker (Symn

OTL→ OU ⊗OT

SymnOTL). for everyn ∈ N

SinceL is flat, this is the same as:

Jn · SymnOTL ⊂ Ker (OT → OU) · Symn

OTL. for everyn ∈ N(5.7.13)

For everyn ∈ N denote byevn : SymnOTL ⊗OT

SymnOTL∗ → OT the evaluation morphism of

theOT -moduleSymnOTL. Then (5.7.13) is equivalent to:

∑

n∈N

evn(Jn · SymnOTL⊗OT

SymnOTL∗) ⊂ Ker (OT → OU).

Conversely, let

I :=∑

n∈N

evn(Jn · SymnOTL⊗OT

SymnOTL∗).(5.7.14)

Then the restriction ofσ to Spec OT/I is a section ofProj(A). All in all, this shows thatT ′

represents the closed subschemeSpec OT/I of T , whence the claim.

Lemma 5.7.15.Let S := ⊕i∈NSi be a gradedR-algebra, withS0 = R; suppose thatS isgenerated byS1 and thatS1 is a finitely generatedR-module. Then the natural map

j∗Proj(S)→ Proj(Sa)

is an isomorphism.


Proof. LetB be anRa-algebra, andσ : Sa1 ⊗Ra B → L a strictly invertibleB-module quotient.By lemma 5.7.10 we know that there is a faithfully flatB-algebraC such that the induced mapσ′ : S1 ⊗R C∗ → (L⊗B C)∗ is a rank one projective quotient. Suppose now thatσ is a sectionof Proj(Sa) and letJ := Ker(Sym•R(S1) → S); by definition we haveJa ⊂ Ker Sym•Bσ, andsinceSym•C∗

(L⊗B C)∗ does not containm-torsion, it follows thatJ ⊂ Ker Sym•C∗σ′.

5.7.16. Next we consider quasi-projective almost schemes.Hence, letA be as in definition5.7.11 andJ := ⊕i∈NJi ⊂ A a graded ideal. We setX := Proj(A) \ Proj(A/J ).

Lemma 5.7.17.Keep the assumptions of(5.7.16), and suppose moreover thatA = Sa andJ = Ja for a gradedR-algebraS and a finitely generated graded idealJ ⊂ S. Supposemoreover thatS1 is a finitely generatedR-module and thatS1 generatesS. Then

X ≃ j∗(Proj(S) \ Proj(S/J)).

Proof. SetY := Proj(S), Y0 := Proj(S/J), X ′ := Y \ Y0; in view of lemmata 5.7.6 and5.7.15, we have only to show that

j∗Y \ j∗Y0 ⊂ j∗X ′.

Hence, letB be anRa-algebra andσ ∈ j∗Y (SpecB); by lemma 5.7.15 we can find a faithfullyflat morphismT → SpecB and a projectiveOT∗-moduleL∗ of rank one such that the restrictionof σ toT is induced by a surjective mapτ : S1⊗ROT∗ → L∗. Suppose now thatσ is a section ofj∗Y \ j∗Y0; it follows thatτ0 := τ×Y Y0 = ∅. From lemma 5.7.12 and its proof we deduce easilythatτ0 is represented theRa-schemeSpec OT/I, whereI is defined as in (5.7.14); consequentlyI = OT . SinceSym•OT

L∗ is generated by(L∗)∗, which is a finitely generatedOT∗-module, andsinceJ is finitely generated by assumption, we see thatI is generated by the imagesx1, ..., xkof finitely many elements inJ ·Sym•OT∗

L∗⊗OTSym•OT∗

(L∗)∗. It then follows from lemma 5.7.2

that there exists a covering morphismU → T such that∑k

i=1 xiOU∗ = OU∗, which means thatthe restriction ofσ toU is a section ofj∗X ′.

5.7.18. In the situation of (5.7.16), letY := SpecA\Spec (A/JA+), whereA+ := ⊕i>0Ai;thenY represents the functor that assigns to everyRa-schemeT the set of all maps ofRa-algebrasφ : A→ OT such that

φ(JA+) · OT = OT .(5.7.19)

Moreover, we have a map ofRa-schemesπ : Y → X, representing the natural transformationof functors that assigns to anyT -sectionφ : A → OT of Y the induced map ofOT -modulesφ1 : A1⊗ROT → OT . Condition (5.7.19) ensures thatφ1 is surjective andφ1 /∈ Proj(A/J )(T )wheneverT is non-empty, hence this rule yields a well definedT -section ofX. It is a standardfact thatπ is aGm-torsor. The action ofGm(T ) onY (T ) can be described explicitly as follows.To a given pair(φ, u) ∈ Y (T )×O×T one assigns the uniqueR-algebra mapφu : A→ OT suchthatφu(x) = uφ(x) for everyx ∈ A1.

Lemma 5.7.20.Keep the notation of(5.7.18). The subsheaf

Ysm := (SpecA)sm \ Spec (A/JA+) ⊂ Y

is a subtorsor ofY for the naturalGm-action onY .

Proof. ClearlyGm = (Gm)sm; hence, by lemma 5.6.15, we have a natural identification:

(Gm ×Ra Y )sm ≃ Gm × Ysm.

However, the natural transformation of functorsGm ×Ra Y → Y ×X Y defined by the rule:

(u, φ) 7→ (φ, φu) for every local sectionu of Gm andφ of Y


is an isomorphism, therefore(Gm ×Ra Y )sm ≃ (Y ×Xa Y )sm. Let T be anRa-scheme,φ ∈Y (T ) andu ∈ Gm(T ); it follows that (φ, φu) ∈ (Y ×X Y )sm(T ) if and only if φ ∈ Ysm(T ).Furthermore, it is clear that(φ, φu) is in the smooth locus ofY ×X Y if and only if (φu, φ) is(use theX-automorphism ofY ×X Y that swaps the two factors). However, the pair(φu, φ) canbe written in the form(ψ, ψv) for ψ = φu andv = u−1, so thatψ is in the smooth locus if andonly if (ψ, ψv) is. Ergo, φ is in the smooth locus if and only ifφu is, as required.

Definition 5.7.21. Let X andY be as in (5.7.18). We define thesmooth locus ofX as thequotient (in the category of sheaves on(Ra-Alg)ofpqc):

Xsm := Ysm/Gm.

The notationXsm remains somewhat ambiguous, until one shows that the smoothlocus ofXdoes not depend on the choice of presentation ofX as quotient of a quasi-affine almost schemeY . While we do not completely elucidate this issue, we want at least to make the followingremark:

Lemma 5.7.22. In the situation of(5.7.18), suppose thatL is a strictly invertibleRa-module,and setB := ⊕r∈NAr ⊗Ra L ⊗r, JB := ⊕r∈NJr ⊗Ra L ⊗r, soB is a gradedRa-algebra andJB ⊂ B a graded ideal. Then there are natural isomorphisms

X ≃ XB := Proj(B) \ Proj(B/JB) Xsm ≃ XB,sm.

Proof. We define as follows a natural isomorphism of functorsX∼→ XB. To everyRa-scheme

T and everyT -sectionA1⊗Ra OT → L, we assign the strictly invertible quotientA1⊗Ra L ⊗Ra

OT → L ⊗Ra L. SetYB := SpecB \Spec (B/JBB+); it follows thatYB/Gm ≃ XB. Hence,the identityXsm ≃ XB,sm can be checked locally on the fpqc topology. However,L is locallyfree of rank one on(Ra-Alg)ofpqc, and the claim is obvious in caseL is free.

Theorem 5.7.23.Supposem has homological dimension≤ 1, and that(Ra, I) is a henselianpair. LetP be an almost finitely generated projectiveRa-module andI ,J ⊂ SP := Sym•RaPtwo graded ideals, withJ almost finitely generated. SetX := Proj(SP/J ) \ V (I ). Then,for every finitely generated subidealm0 ⊂ m, we have:

Im(Xsm(SpecRa/m0I)→ Xsm(SpecRa/I)) = Im(Xsm(SpecRa)→ Xsm(SpecRa/I)).

Proof. First of all, arguing as in the proof of claim 5.6.17 we reduceto the case whereI is atight ideal. Next, setJ := ⊕i∈NJi, I := ⊕i∈NIi and letσ : (P/J1) ⊗Ra Ra/m0I → Lbe anRa/m0I-section ofX. By theorem 5.5.6 we can liftL to an almost finitely generatedprojectiveRa-moduleL. We haveΛ2

RaL ⊗Ra Ra/m0I ≃ Λ2RaL = 0, whenceΛ2

RaL = 0 byNakayama’s lemma 5.1.6, soL is strictly invertible. We setS ′ := Sym•Ra(P ⊗Ra L∗), I ′ :=⊕r∈NIr ⊗Ra L∗⊗r, J ′ := ⊕r∈NJr ⊗Ra L∗⊗r,B := S ′/J ′, Y ′ := SpecB \ Spec (B/I ′B+)andX ′ := Proj(B) \ Proj(B/I ′). By lemma 5.7.22 we have an isomorphismβ : X ′ ≃ Xpreserving the smooth loci; by inspecting the definition we find thatβ−1(σ) lies in the im-age of the projectionY ′sm(SpecRa/m0I) → X ′sm(SpecRa/m0I). Hence we can replaceP byP ⊗Ra L∗ and assume from start thatσ lies in the image of the mapYsm(SpecRa/m0I) →Xsm(SpecRa/m0I). Let τ ∈ Ysm(SpecRa/m0I) that maps toσ. By theorem 5.6.16(i), the im-ageτ0 of τ in (SpecSP/J )sm lifts to a sectionτ ∈ (SpecSP/J )sm(SpecRa). To conclude,it suffices to show the following:

Claim5.7.24. Suppose thatI is tight. Thenτ ∈ Y (SpecRa).

Proof of the claim:Let A := SP/J , so thatτ : A → Ra is a morphism ofRa-algebras, andsetJ := τ(A+I ); by assumptionτ0 ∈ Y (SpecRa/I), which means thatJ · (Ra/I) = (Ra/I),i.e. J + I = Ra. SetM := Ra/JRa; it then follows thatM = IM , henceM = 0 by lemma5.1.6; henceJRa = Ra, which is the claim.


5.8. Lifting and descent of torsors. We begin this section with some generalities about groupschemes and their linear representations, that are preliminary to our later results about liftingsof torsors in an almost setting (theorems 5.8.19 and 5.8.21).

Furthermore, we apply the results of section 5.4 to derive a descent theorem forG-torsors,whereG is a group scheme satisfying some fairly general conditions.

5.8.1. LetR be a ring,G an affine group scheme defined overR. We setO(G) := Γ(G,OG);soO(G) is anR-algebra and the multiplication map ofG determines a structure of co-algebraonO(G). In (3.3.7) we have defined the notion of a (left or right)G-action on a quasi-coherentOX-moduleM , whereX is a scheme acted on byG. We specialize now to the case whereX = SpecR, andG acts trivially onX. In such situation, aG-action onM is the same as amap of fpqc-sheaves, from the sheaf represented byG to the sheaf of automorphisms ofM ; i.e.the datum, for everyR-schemeT , of a functorial group homomorphism:

G(T )→ AutOT(M ⊗R OT ).(5.8.2)

Explicitly, given a map as in (5.8.2), takeT := G; then the identity mapG → G determinesan O(G)-linear automorphismβ of M ⊗R O(G), and one verifies easily thatβ fulfills theconditions of (3.3.7). Conversely, an automorphismβ as in (3.3.7) extends uniquely to a welldefined map of sheaves (5.8.2). Furthermore, theG-action onM can be prescribed by choosing,instead of aβ as above, anR-linear map:

γ : M →M ⊗R O(G)

satisfying certain identities analogous to (3.3.9) and (3.3.10); of course one can then recover thecorrespondingβ by extension of scalars. One says that the pair(M, γ) is anO(G)-comodule;more precisely, one defines right and leftO(G)-comodules, and the bijection just sketched setsup an equivalence from the category ofR-modules with leftG-actions to the category of rightO(G)-comodules. One says that anO(G)-comodule isfinitely generated(resp. projective) ifits underlyingR-module has the same property.

Lemma 5.8.3. Suppose thatR is a Dedekind domain and thatO(G) is flat overR. Then:

(i) EveryO(G)-comodule is the filtered union of its finitely generatedO(G)-comodules.(ii) Every finitely generatedO(G)-comodule is quotient of a projectiveO(G)-comodule.

Proof. (i) is [63, §1.5, Cor.] and (ii) is [63,§2.2, Prop.3].

5.8.4. LetR be a ring,G be a flat group scheme of finite presentation overR, and suppose thatG admits a closed imbedding as a subgroup schemeG ⊂ GLn. We suppose moreover that thequotientXG := GLn/G (for the right action ofG onGLn) is representable by a quasi-projectiveR-scheme, and thatXG admits an ampleGLn-equivariant line bundle. ThenXG is necessarilyof finite presentation by [30, lemme 17.7.5]. Furthermore,XG is smooth; indeedXG is flat overSpecR, hence smoothness can be checked on the geometric fibres overthe points ofSpecR,which reduces to the case whereR is an algebraically closed field. In such case, smoothnessoverR is the same as regularity and the latter follows sinceGLn is regular and the quotient mapGLn → XG is faithfully flat. The following lemma 5.8.5 provides plenty of examples of thesituation envisaged in this paragraph.

Lemma 5.8.5. Suppose thatR is a Dedekind domain andG is a flat affine group scheme offinite type overR. ThenG fulfills the conditions of(5.8.4).

Proof. The proof is obtained by assembling several references to the existing literature. To startwith, letP be a finitely generated subrepresentation of the left (or right) regular representationof G onO(G) which generatesO(G) as anR-algebra. By [19, 1.4.5]P is a finitely generatedprojectiveR-module and the action ofG on P is faithful, in the sense that it gives a closed


immersion:G → AutR(P ). Taking the direct sum with a trivial representation one gets aclosed immersion ofR-group schemes:G → GLn,R. By a theorem of M.Artin (see [1, 3.1.1])the quotient fppf sheafGLn/G is represented by an algebraic space of finite presentation overSpecR. Then by [1, Th.4.C] this algebraic space is a scheme. LetK be the fraction field ofR; by classical results of Chevalley and Chow,GLn,K/GK is quasi-projective overK and by[59, VIII.2] it follows that GLn/G is quasi-projective overSpecR. The ample line bundleLon GLn/G that one gets always has aGLn-linearization. This follows from the proof of [66,lemma 1.2]. For our purpose it suffices to know that the weakerassertion that some tensormultiple of L is GLn-linearizable. We may assume thatL is trivial on the identity sectionS := SpecR→ GLn,R/G.

Claim5.8.6. Ker(Pic GLn,Re∗→ PicS) = 0 (wheree : S → GLn,R is the identity section).

Proof of the claim:GLn,K is open in an affine space, hence it has trivial Picard group, henceevery divisor onGLn,R is linearly equivalent to a divisor whose irreducible components do notdominateS, i.e. a pull-back of a divisor onS, which gives the claim.

The assertion now follows from [59, VII, Prop.1.5].

Lemma 5.8.7. Under the assumptions of(5.8.4), there is aGLn-equivariant isomorphism ofR-schemes:

XG∼→ Proj((Sym•RP)/I ) \ V (J )(5.8.8)

whereP is a projectiveGLn-comodule, andI ,J ⊂ Sym•RP are two finitely generatedgraded ideals, that are sub-O(G)-comodules ofSym•RP.

Proof. Let L be an ampleGLn-equivariant line bundle onXG. SinceXG is quasi-compact,we can findn ∈ N large enough, so thatL ⊗n is very ample. We can then replaceL by L ⊗n

and therefore assume thatXG admits a locally closed imbedding inPR(Γ(XG,L )). Again bycompactness argument, we can find a finitely generatedR-submoduleW ⊂ Γ(XG,L ) suchthatXG already imbeds intoPR(W ). Let f1, ..., fk be a finite set of generators of theR-moduleW . By lemma 5.8.3(i) we can find a finitely generatedZ-moduleW0 ⊂W which is aO(GLn)-comodule containing the(fi | i ≤ k). Up to replacingW by theR-module generated byW0,we can further assume thatW is anO(GLn)-comodule.

Claim5.8.9. W is the quotient of a projectiveO(G)-comodule of finite typeL.

Proof of the claim:By the foregoing we can assume thatW is generated by a finitely generatedZ-submoduleW0 ⊂W which is anO(G)-comodule. By lemma 5.8.3(ii) we can writeW0 as aquotient of a projectiveO(G)-comoduleL0; henceW is a quotient ofL := L0 ⊗Z R.

It follows thatXG is a locally closed subscheme ofP(L). Let Y be the schematic closureof the image ofXG; we can writeY ≃ Proj((Sym•RL)/I), for some graded idealI, andI isnecessarily anO(G)-comodule. Furthermore, we haveY \XG = V (J), whereJ ⊂ Sym•RL isanother graded ideal, also anO(G)-comodule. Arguing as in the foregoing we can writeJ =⋃α Jα, whereJα runs over the filtered family of the finitely generatedO(G)-sub-comodules of

J . By quasi-compactness one shows easily thatY \XG = V (Jα) for Jα large enough; we setJ := Jα. Likewise, we can find a finitely generatedO(G)-subcomoduleI ⊂ I such that(P(L) \ V (J )) ∩ V (I) = (P(L) \ V (J )) ∩ V (I ).

5.8.10. Under the assumptions of (5.8.4), letT be aGLn-torsor. ThenG acts onT via theimbeddingG ⊂ GLn, and the functorT/G is representable by a smoothR-scheme.

Lemma 5.8.11.The functorT/G : R-Scheme → Set is naturally isomorphic to the functorthat assigns to everyR-schemeS the set of all pairs(H,ω), whereH is aG-torsor onSfpqc

andω : H ×G GLn∼→ T is an isomorphism ofGLn-torsors.


Proof. Indeed, letσ : S → T/G be any morphism ofR-schemes; setHσ := S ×T/G T , whichis aG-torsor onSfpqc. The natural morphismHσ → T induces a well defined isomorphismof GLn-torsorsωσ : Hσ ×G GLn

∼→ T . Conversely, given(H,ω) overS, we deduce aG-equivariant morphismH → T ; after taking quotients by the left action ofG, we deduce amorphismS ≃ H/G → T/G. It is easy to verify that these two rules define mutually inversenatural transformations of functors, whence the claim.

5.8.12. The rightGLn-torsorT is endowed by a natural left action by the groupAutGLn(T ).The latter is a group scheme locally isomorphic toGLn in the Zariski topology; especially it issmooth overR. After taking the quotient ofT by the right action ofG we deduce a left actionof AutGLn(T ) onT/G :

AutGLn(T )×R T/G→ T/G.(5.8.13)

Theorem 5.8.14.Resume the assumptions of proposition5.4.21. LetG be a group scheme overSpecR[t−1] which satisfies (relative toR[t−1]) the conditions of(5.8.4). Then the natural map

H1(SpecR[t−1]fpqc, G)→ H1(SpecR∧[t−1]fpqc, G)

is a bijection.


Claim 5.8.15. The theorem holds whenG = GLn.

Proof of the claim:Indeed, aGLn torsor over a schemeX is the same as a locally freeOX-module of rankn. Hence the assertion is just a restatement of corollary 5.4.42.

For a givenGLn-torsorT , letT∧ := T ×R[t−1] R∧[t−1] and denote by

H1T ⊂ H1(SpecR[t−1]fpqc, G)

the subset consisting of all classes ofG-torsorsH such thatH ×G GLn ≃ T ; define likewisethe subsetH1

T∧ ⊂ H1(SpecR∧[t−1]fpqc, G). According to claim 5.8.15 it suffices to show:

Claim 5.8.16. The restrictionH1T → H1

T∧ is a bijection.

Proof of the claim:The morphism (5.8.13) defines a groupoidT/G of quasi-projectiveR[t−1]-schemes;T/G is locally isomorphic toXG in the Zariski topology, hence it is smooth (see5.8.4) and we can therefore apply theorem 5.4.38. The assertion follows directly after one hasremarked that, for everyR[t−1]-schemeS, the setπ0(T/G(S)) is in natural bijection with theset of isomorphism classes ofG-torsorsH on Sfpqc such thatH ×G GLn ≃ T ×SpecR[t−1] S.

5.8.17. Finally we come to the lifting problems forGa-torsors. LetA be aV a-algebra, setR := A∗ and letG a smooth affine group scheme of finite type overSpecR. We suppose thatG is a closed subgroup scheme ofGLn,R, for somen ∈ N. ClearlyGa is a group scheme overS := SpecA. Unless otherwise stated,Ga will be acting on the right on any such torsor.

Lemma 5.8.18.Keep the notation of(5.8.17). The category ofGLn-torsors overSfpqc is natu-rally equivalent to the category whose objects are the almost projectiveOS-modules of constantrankn, and whose morphisms are the linear isomorphisms.

Proof. Let O∗ be the structure sheaf ofSfpqc, i.e. the sheaf of rings defined by the rule:T 7→OT∗. To a givenGLn-torsorP we assign theO∗-moduleFP := P ×GLn On

∗ (for the naturalleft action ofGLn on O∗). Locally in Sfpqc the sheafFP is a freeO∗-module of rankn, andthe assignmentP 7→ FP is an equivalence fromGLn-torsors to the category of all such locallyfree O∗-modulesF of rank n, whose inverse is the rule:F 7→ IsoO∗

(On∗ , F ). On the other


hand, theorem 4.4.24 says that any almost projectiveOS-moduleM of constant rankn definesa locally freeO∗-module inSfpqc, by the ruleT 7→ (OT ⊗OS

M)∗ and by faithfully flat descentit is also clear that the resulting functor is an equivalenceas well.

Theorem 5.8.19.Assume thathom.dimV m ≤ 1. LetS andG be as in(5.8.17), andI ⊂ OS anideal such that the pair(OS, I) is henselian. Let alsom0 ⊂ m be a finitely generated subideal,and setSn := Spec OS/m

n0I for n = 0, 1. LetP andQ be twoGa-torsors overS, and suppose

thatβ : P ×S S1∼→ Q×S S1 is an isomorphism ofGa×S S1-torsors. Thenβ ×S 1S0 lifts to an

isomorphismβ : P∼→ Q ofGa-torsors.

Proof. Let Iso(P,Q) denote the functor that assigns to everyS-schemeT the set of isomor-phisms ofGa×S T -torsorsP ×S T ∼→ Q×S T . It is easy to see thatIso(P,Q) is a sheaf for thefpqc topology ofS. Since, locally inSfpqc, this sheaf is represented by the almost schemeGa, itfollows by faitfully flat descent thatIso(P,Q) is represented by an affineS-scheme, which wedenote by the same name. To the (right)Ga-torsorP we associate a leftGa-torsorP ′, whoseunderlyingS-scheme is the same asP , and whoseGa-action is defined by the rule:g ·x := xg−1

for everyS-schemeT , everyg ∈ Ga(T ) and everyx ∈ P (T ). Furthermore, we letH denotethe trivial left and rightGa-torsor whose underlyingS-scheme isGa, and whose left and rightGa-actions are induced by the multiplication mapGa ×S Ga → Ga.

Claim5.8.20. There is a natural isomorphism of sheaves onSfpqc:

ω : Iso(P,Q) ≃ Q×Ga

H ×Ga

P ′.

Proof of the claim:Recall that the meaning of the right-hand side is as follows.For everyS-schemeT , one consider the presheaf whoseT -sections of are the equivalence classes of triples(x, h, y) ∈ Q(T )×H(T )×P ′(T ), modulo the equivalence relation such that(x, g1 ·h ·g2, y) ∼(x · g1, h, g2 · y) for every such(x, h, y) and everyg1, g2 ∈ Ga(T ). ThenQ ×Ga

H ×GaP ′ is

the sheaf associated to this presheaf. The sought isomorphism is defined as follows. LetT beanS-scheme, andβ a T -section ofIso(P,Q). We pick a covering morphismU → T in Sfpqc

such thatP (U) 6= ∅; let y ∈ P (U); we setω(β, y) := (β(y), 1, y). If now z ∈ P (U) is anyother section, we havez = yg for someg ∈ Ga(U), therefore:ω(β, y) = (β(y), g · g−1, y) ∼(β(y) · g, 1, g−1y) = (β(yg), 1, yg) = ω(β, z), i.e. the equivalence class ofω(β, y) does notdepend on the choice ofy, hence we have a well definedU-sectionω(β) of Q ×Ga

H ×GaP ′.

Furthermore, letpi : U ×T U → U , i = 1, 2 be the two projections; the sectionsp∗1ω(β, y) andp∗2ω(β, y) differ by an element ofGa(U ×T U) so, by the same argument, they lie in the sameequivalence class, which means that actuallyω(β) comes fromT , as required. We leave to thereader the verification thatω thus defined is an isomorphism.

Composing the closed imbeddingGa ⊂ GLn,S with the standard group homomorphismGLn,S ⊂ SLn+1,S, we can viewGa as a closed subgroup scheme ofSLn,S, whence a closedGa-equivariant imbedding ofS-schemesGa ⊂ Mn,S := Spec OS[xij | i, j ≤ n]. From claim5.8.20 we derive a closed imbedding ofS-schemes:

Iso(P,Q) ⊂ X := Q×Ga

Mn,S ×Ga

P ′

(whereGa acts on the left and right onMn,S in the obvious way).X is locally isomorphicto Mn,S in Sfpqc, hence it is of the formSpec (Sym•OS

M), for an almost finitely generatedprojectiveOS-moduleM ; especially,X is almost finitely presented overS, whence the sameholds forIso(P,Q). Now, the givenβ gives a section ofIso(P,Q) overS1, so the theorem is animmediate consequence of theorem 5.6.16(i).

Theorem 5.8.21.Keep the notation and assumptions of theorem5.8.19, and suppose further-more thatI is tight, and thatG fulfills the conditions of(5.8.4). LetP0 be anyGa-torsor overS0. Then there exists aGa-torsorP overS with an isomorphism ofGa-torsors:P0 ≃ P ×S S0.


Proof. By assumption there exists a finitely generated subidealm0 ⊂ m and an integern > 0such thatIn ⊂ m0OS; then by theorem 3.3.38(ii) we can liftP0 to aG-torsor overS1 :=OS/m0I. We form theGLn-torsorQ := P ×Ga

GLn. By lemma 5.8.18,Q is the same asan almost projectiveOS1-module of constant rankn; by theorem 5.5.6(i) we can then find analmost projectiveOS-moduleQ that lifts Q, and then a standard application of Nakayama’slemma 5.1.6 shows thatQ has constant rank equal ton (cp. the proof of theorem 5.7.23). Thus,Q is aGLn-torsor with an isomorphism ofGLn-torsorsω : P ×Ga

GLn∼→ Q ×S S1. By (the

almost version of) lemma 5.8.11, the datum of(P, ω) is the same as the datum of a morphismσ : S1 → Q/Ga of sheaves onSfpqc. The contention is thatσ ×S S0 lifts to a morphismσ : S → Q/Ga. However, we have a naturalGLn-equivariant isomorphism

Q/Ga ≃ Q×GLn (GLn/Ga) ≃ Q×GLn (GLn/G)a.

By lemmata 5.7.17 and 5.8.7 it follows that there is aGLn-equivariant isomorphism:

Q/Ga ≃ Proj(Sym•OSP ′)/I ′) \ V (J ′)

with P ′ := Q×GLn P, I ′ := I ×GLn P andJ ′ := J ×GLn P, whereP, I andJ areas in (5.8.8). By lemma 3.2.25(ii),(iii) and remark 3.2.26(i),(ii) we see thatP ′ is almost finitelygenerated projective andI ′, J ′ are almost finitely generated. Finally,Q/Ga is smooth sinceXG is (see 5.8.4), so the existence of the sectionσ follows from theorem 5.7.23.


6. VALUATION THEORY

This chapter is an extended detour into valuation theory. The first two sections contain noth-ing new, and are only meant to gather in a single place some useful material that is known toexperts, but for which satisfactory references are hard to find. The main theme of sections 6.3through 6.5 is the study of the cotangent complex of an extension of valuation rings. To give asample of our results, suppose thatk is a perfect field, and letW be a valuation ring containingk; then we show thatΩW/k is a torsion-freeW -module. Notice that this assertion would bean easy consequence of the existence of resolution of singularities fork-schemes; our methodsenable us to prove it unconditionally, as well as several other statements and variants for log-arithmic differentials. Furthermore, consider a finite separable extensionK ⊂ L of henselianvalued fields of rank one; it is not difficult to see that the corresponding extension of valuationringsK+ ⊂ L+ is almost finite, hence one can define the different idealDL+/K+ as in chapter 4.In case the valuation ofK is discrete, it is well known that the length of the module of relativedifferentialsΩL+/K+ equals the length ofL+/DL+/K+; theorem 6.3.20 generalizes this identityto the case of arbitrary rank one valuations; notice that in this case the usual notion of lengthwon’t do, since the modules under considerations are only almost finitely generated.

Section 6.6 ties up with earlier work of Coates and Greenberg[21], in which the notion ofdeeply ramified extensionof a local field was introduced, and applied to the study ofp-divisiblegroups attached to abelian varieties defined over suchp-adic fields. Essentially, section 2 of [21]rediscovers the results of Fresnel and Matignon [37], although via a different route, closer to theoriginal treatment of Tate in [64]. In particular, an algebraic extensionE ofK is deeply ramifiedif and only if DE+/K+ = (0) according to the terminology of [37]. We adopt Coates andGreenberg’s terminology for our section 6.6, and we give some complements which were notobserved in [21]; notably, proposition 6.6.2, which we regard as the ultimate generalization ofthe one-dimensional case of the almost purity theorem. Our proof generalizes Faltings’ method,which relied on the above mentioned relationship between differentials and the different ideal;of course, in the present setting we need to appeal to our theorem 6.3.20, rather than estimatinglengths the way Faltings did. Finally, we extend the definition of deeply ramified extension toinclude valued fields of arbitrary rank.

6.1. Ordered groups and valuations. In this section we gather some generalities on valua-tions and related ordered groups, which will be used in latersections.

6.1.1. As usual, avalued field(K, | · |K) consists of a fieldK endowed with a surjective grouphomomorphism| · |K : K× → ΓK onto an ordered abelian group(ΓK ,≤), such that

|x+ y|K ≤ max(|x|K , |y|K)(6.1.2)

wheneverx+y 6= 0. We denote by1 the neutral element ofΓK , and the composition law ofΓKwill be denoted by:(x, y) 7→ x · y. It is customary to extend the map| · |K to the whole ofK,by adding a new element0 to the setΓK , and setting|0| := 0. One can then extend the orderingof ΓK to ΓK ∪ 0 by declaring that0 is the smallest element of the resulting ordered set. Inthis way, (6.1.2) holds for everyx, y ∈ K. The map| · |K is called thevaluationof K andΓKis itsvalue group.

6.1.3. Anextension of valued fields(K, |·|K) ⊂ (E, |·|E) consists of a field extensionK ⊂ E,and a valuation| · |E : E → ΓE ∪ 0 together with an imbeddingj : ΓK ⊂ ΓE, such that therestriction toK of | · |E equalsj | · |K .

Example 6.1.4.Let | · | : K → Γ ∪ 0 be a valuation on the fieldK.(i) Given a field extensionK ⊂ E, it is known that there always exist valuations onE which

extend| · | (cp. [16, Ch.VI,§1, n.3, Cor.3]).


(ii) If the field extensionK ⊂ E is algebraic and purely inseparable, then the extension of| · | is unique. (cp. [16, Ch.VI,§8, n.7, Cor.2]).

(iii) We can construct extensions of| · | on the polynomial ringK[X], in the following way.Let Γ′ be an ordered group with an imbedding of ordered groupsΓ ⊂ Γ′. For everyx0 ∈ K,and everyρ ∈ Γ′, we define theGauss valuation| · |(x0,ρ) : K[X] → Γ′ ∪ 0 centered atx0

and with radiusρ (cp. [16, Ch.VI,§10, n.1, Lemma 1]) by the rule:

a0 + a1(X − x0) + ... + an(X − x0)n 7→ max|ai| · ρi | i = 0, 1, ..., n.

(iv) The construction of (iii) can be iterated : for instance, suppose that we are given asequence ofk elementsρ := (ρ1, ρ2, ..., ρk) of the ordered abelian groupΓ′ of (iv). Thenwe can define a Gauss valuation| · |(0,ρ) on the fraction field ofK[X1, X2, ..., Xk], with valuesin Γ′, by the rule:

∑α∈Nk aαX

α 7→ max|aα| · ρα | α ∈ N.(v) Suppose again it is given an ordered groupΓ′ with an imbedding of ordered groupsΓ ⊂

Γ′. LetT ⊂ Γ′/Γ be a finite torsion subgroup, sayT ≃ Z/n1Z⊕ ...⊕Z/nkZ. For everyi ≤ k,pick an elementγi ∈ Γ′ whose class inΓ′/Γ generates the direct summandsZ/niZ of T . Letxi ∈ K such that|xi| = ni · γi. For everyi = 1, ..., k pick an elementyi in a fixed algebraicclosureEa of E, such thatyni

i = xi; then the fieldE := K(y1, ..., yk) has degree overK equalto the order ofT , and it admits a unique valuation| · |E extending| · |. Of course,|yi|E = γi foreveryi ≤ k.

6.1.5. We want to explain a construction which is a simultaneous generalization of examples6.1.4(iv),(v). Suppose it is given the datumG := (G, j,N,≤) consisting of:

(a) an abelian groupG with an imbeddingj : K× → G such thatG/j(V ×) is torsion-free;(b) a subgroupN of G/j(V ×) such that the natural map:

Γ∼→ K×/V × → ΓG := G/(N + j(V ×))

is injective;(c) an ordering≤ onΓG such that the injective map:Γ→ ΓG is order-preserving.

Let us denote byK[G] (resp.K[K×]) the groupK-algebra of the abelian groupG (resp.K×).Any element ofK[G] can be written uniquely as a formal linear combination

∑g∈G ag · [g],

whereag ∈ K for everyg ∈ G, andag = 0 for all but a finite number ofg ∈ G. We augmentK[K×] overK via theK-algebra homomorphism

K[K×]→ K : [a] 7→ a for everya ∈ K×.(6.1.6)

Then we letK[G] := K[G] ⊗K[K×] K, where theK[K×]-algebra structure onK is definedby the augmentation (6.1.6). It is easy to verify thatK[G] is the maximal quotient algebra ofK[G] that identifies the classes of[g · a] anda · [g], for everyg ∈ G anda ∈ K×. Pick, forevery classγ ∈ G/K×, a representativegγ ∈ G. It follows that every element ofK[G] canbe written uniquely as a formalK-linear combination

∑γ∈G/K× aγ · [gγ]. We define a map

| · |G : K[G]→ ΓG∪ 0 by the rule:∑

γ∈G/K×

aγ · [gγ] 7→ maxγ∈G/K×

|aγ| · |gγ|(6.1.7)

where|gγ| ∈ ΓG denotes the class ofgγ . One verifies easily that| · |G does not depend on thechoice of representativesgγ. Indeed, if(hγ | γ ∈ G/K×) is another choice then, for everyγ ∈ G/K× we havegγ = j(xγ) · hγ for somexγ ∈ K×; therefore[gγ] = xγ · [hγ ] and|gγ| = |xγ | · |hγ |.Lemma 6.1.8.K[G] is an integral domain, and| · |G extends to a valuation:

| · |G : K(G) := Frac(K[G])→ ΓG ∪ 0.


Proof. Let (Gα | α ∈ I) be the filtered system of the subgroupsGα of G such thatK× ⊂ Gα

andGα/K× is finitely generated. EachGα defines a datumGα := (Gα, j, N∩(Gα/j(V

×)),≤),and clearlyK[G] = colim

α∈IK[Gα]. We can therefore reduce to the case whereG/K× is finitely

generated. WriteG/K× = T ⊕F , whereT is a torsion group andF is torsion-free. There existunique subgroupsT , F ⊃ K× inGwith T /K× = T andF /K× = F . LetGT := (T , j, 0,≤)

andGF := (F , j, N,≤) be the corresponding data. The functorH 7→ K[H ] preserves colimits,since it is left adjoint to the forgetful functor fromK-algebras to abelian groups; it followseasily thatK[G] ≃ K[GT ]⊗K K[GF ]. By inspecting (6.1.7), one can easily show thatK[GT ]is of the type of example 6.1.4(v) andK[GF ] is of the type of example 6.1.4(iv). Especially,K[G] is a domain, and| · |G is induced by a Gauss valuation of a free algebra over the finite fieldextensionK[GT ] of K.

The next result shows that an arbitrary valuation is always “close” to some Gauss valuation.

Lemma 6.1.9. Let (E, | · |E) be a valued field extension of(K, | · |). Letx ∈ E \ K, and let(ai | i ∈ I) be a net of elements ofK (indexed by the directed set(I,≤)) with the followingproperty. For everyb ∈ K there existsi0 ∈ I such that|x − ai|E ≤ |x − b|E for everyi ≥ i0.Letf(X) ∈ K[X] be a polynomial that splits inK[X] as a product of linear polynomials. Thenthere existsi0 ∈ I such that|f(x)|E = |f(X)|(ai,|x−ai|E) for everyi ≥ i0.

Proof. To prove the claim, it suffices to consider the case whenf(X) = X− b for someb ∈ K.However, from the definition of the sequence(ai | i ∈ I) we havemax(|x − ai|E, |ai − b|) ≥|x−b|E ≥ |x−ai|E for every sufficiently largei ∈ I. Therefore,|x−b|E = max(|x−ai|E, |b−ai|) = |X − b|(ai,|x−ai|E).

6.1.10. To see how to apply lemma 6.1.9, let us consider the case whereK is algebraicallyclosed andE = K(X), the field of fractions of the freeK-algebra in one generator, whichwe suppose endowed with some valuation| · |E with values inΓE. We apply lemma 6.1.9 tothe elementx := X ∈ E. Suppose first that there exists an elementa ∈ K that minimizesthe functionK → ΓE : b 7→ |X − b|E. In this case the trivial neta fulfills the conditionof the lemma. Since every polynomial ofK[X] splits overK, we see that| · |E is the Gaussvaluation centered ata and with radius|X − a|E. Suppose, on the other hand, that the functionb 7→ |X − b|E does not admit a minimum. It will still be possible to choose anet of elementsai | i ∈ I fulfilling the conditions of lemma 6.1.9 (indexed, for instance, by a subset of thepartially ordered setΓ). Then| · |E is determined by the identity:

|f(X)|E = limi∈I|f(X)|(ai,|X−ai|E) for everyf(X) ∈ E.

6.1.11. Given a valuation| · | on a fieldK, the subsetK+ := x ∈ | |x| ≤ 1 is avaluationring of K, i.e., a subring ofK such that, for everyx ∈ K \ 0, eitherx ∈ K+ or x−1 ∈ K+.The subset(K+)× of units ofK+ consists precisely of the elementsx ∈ K such that|x| = 1.Conversely, letV be a valuation ring ofK with maximal idealm; V induces a valuation| · | onK whose value group isΓK := K×/V × (then| · | is just the natural projection). The orderingon ΓK is defined as follows. For given classesx, y ∈ ΓK , we declare thatx < y if and only ifx/y ∈ m.

Remark 6.1.12. (i) It follows easily from (6.1.11) that every finitely generated ideal of a valu-ation ring is principal. Indeed, ifa1, ..., an is a set of generators for an idealI, pick i0 ≤ n suchthat|ai0 | = maxi≤n |ai|; thenI = (ai0).

(ii) It is also easy to show that any finitely generated torsion-freeK+-module is free and anytorsion-freeK+-module is flat (cp. [16, Ch.VI,§3, n.6, Lemma 1]).


(iii) Let E be a field extension of the valued field(K, | · |K). Then the integral closureW ofK+ in E is the intersection of all the valuation rings ofE containingK+ (cp. [16, Ch.VI,§1,n.3, Cor.3]). In particular,K+ is integrally closed.

(iv) Furthermore, ifE is an algebraic extension ofK, thenW is aPrufer domain, that is, forevery prime idealp ⊂ W , the localizationWp is a valuation ring. Moreover, the assignmentm 7→Wm establishes a bijection between the set of maximal ideals ofW and the set of valuationringsV of E whose associated valuation| · |V extends| · |K (cp. [16, Ch.VI,§8, n.6, Prop.6]).

(v) Let R andS be local rings contained in a fieldK, mR andmS their respective maximalideals. One says thatR dominatesS if S ⊂ R andmS = S ∩mR. It is clear that the relation ofdominance establishes a partial order structure on the set of local subrings ofK. Then a localsubring ofK is a valuation ring ofK if and only if it is maximal for the dominance relation (cp.[16, Ch.VI,§1, n.2, Th.1]).

(vi) Let K+ be a valuation ring ofK with maximal idealm, andK+h a henselization ofK+. One knows thatK+h is an ind-etale localK+-algebra (cp. [60, Ch.VIII, Th.1]), henceit is integral and integrally closed (cp. [60, Ch.VII,§2, Prop.2]). Denote byKh the field offractions ofK+h andW the integral closure ofK+ in Kh. It follows thatW ⊂ K+h. Let mh

be the maximal ideal ofK+h; sincemh ∩K+ = m, we deduce thatq := mh ∩W is a maximalideal ofW ; then by (iv),Wq is a valuation ring ofKh dominated byK+h; by (v) it followsthatK+h = Wq, in particular this shows that the henselization of a valuation ring is again avaluation ring. The same argument works also for strict henselizations.

The following lemma provides a simple method to construct extensions of valuation rings,which is sometimes useful.

Lemma 6.1.13.Let(K, | · |) be a valued field,κ the residue field ofK+,R aK+-algebra whichis finitely generated free as aK+-module, and suppose thatR ⊗K+ κ is a field. ThenR is avaluation ring, and the morphismK+ → R induces an isomorphism of value groupsΓK

∼→ ΓR.

Proof. Let e1, ..., en be aK+-basis ofR. Let us define a map| · |R : R → ΓR ∪ 0 in thefollowing way. Givenx ∈ R, write x =

∑ni=1 xi · ei; then|x|R := max|xi| | i = 1, ..., n.

If |x| = 1, then the imagex of x in R ⊗K+ κ is not zero, hence it is invertible by hypothesis.By Nakayama’s lemma it follows easily thatx itself is invertible inR. Hence, every elementyof R can be written in the formy = u · b, whereu ∈ R× andb ∈ K+ is an element such that|b| = |y|R. It follows easily thatR is an integral domain. Moreover, it is also clear that, givenanyx ∈ Frac(R) \ 0, eitherx ∈ R or x−1 ∈ R, soR is indeed a valuation ring and| · |R is itsvaluation.

Lemma 6.1.14.Every finitely presented torsionK+-moduleM is isomorphic to a direct sumof the form

(K+/a1K+)⊕ ...⊕ (K+/anK

+)

wherea1, ..., an ∈ K+. More precisely, ifFφ→ M is any surjection from a freeK+-module

F of rankn, then there is a basise1, ..., en of F and elementsa1, ..., an ∈ K+ \ 0 such thatKerφ = (a1K

+)⊕ ...⊕ (anK+).

Proof. We proceed by induction on the rankn of F . Forn = 1 the claim follows easily fromremark 6.1.12(i). Supposen > 1; first of all,S := Ker(φ) is finitely generated by [16, Ch.I,§2,n.8, lemme 9]. ThenS is a freeK+-module, in light of remark 6.1.12(ii); its rank is necessarilyequal ton, sinceS ⊗K+ K = F ⊗K+ K.

The image of the evaluation mapS ⊗K+ F ∗ → K+ given byf ⊗ α 7→ α(f) is a finitelygenerated idealI 6= 0 of K+, hence it is principal, by remark 6.1.12(i). Let

∑i=1 fi ⊗ αi be an

element whose image generatesI; this means that∑

i=1 αi(fi) is a generator ofI, hence one ofthe terms in the sum, sayα1(f1), is already a generator. The mapα1 : S → I is surjective onto


a free rank oneK+-module, therefore it splits, which shows thatS = (f1K+)⊕ (S ∩Ker α1).

In particular,S ′ := S ∩Ker α1 is a finitely generated torsion-free, hence freeK+-module. Lete1, ..., en be a basis ofF ; thenf1 =

∑ni=1 ai · ei for someai ∈ K+. Consider the projection

πi : F → K+ such thatπi(ej) = δij for j = 1, ..., n; clearlyπi(f1) = ai ∈ I. This shows thatf1 = α1(f1) · g for someg ∈ F . It follows thatα1(g) = 1, whenceF = (gK+)⊕Kerα1. SetF ′ := Kerα1; we have shown thatM ≃ (K+/α1(f1))⊕ (F ′/S ′). ButF ′ is a freeK+-moduleof rankn− 1, hence we conclude by induction.

6.1.15. In later sections we will be concerned with almost ring theory in the special case wherethe basic setup(V,m) (see 2.1.1) consists of a valuation ringV . In preparation for this, we fixthe following terminology, which will stand throughout therest of this work. IfV is a valuationring, then thestandard setupattached toV is the pair(V,m) wherem := V in case the valuegroup ofV is isomorphic toZ (i.e. V is a discrete valuation ring), and otherwisem is themaximal ideal ofV .

6.1.16. LetK → ΓK ∪ 0 be a valuation on the fieldK, andK+ its valuation ring. Weconsider the categoryK+a-Mod relative to the standard setup(K+,m). The topological groupDiv(K+a) of fractional idealsof K+a is the subspace ofIK+a(Ka) which consists of all thesubmodulesI 6= Ka of the almostK+-moduleKa, such that the natural morphismI ⊗K+a

Ka → Ka is an isomorphism. The group structure is induced by the multiplication of fractionalideals.

Remark 6.1.17. One verifies easily thatDiv(K+a) is isomorphic to the groupD(K+) definedin [16, Ch.VII, §1, n.1].

The structure of the ideals ofK+ can be largely read off from the value groupΓ. In order toexplain this, we are led to introduce some notions for general ordered abelian groups.

6.1.18. We endow an ordered groupΓ with the uniform structure defined in the followingway. For everyγ ∈ Γ such thatγ > 1, the subset ofΓ× Γ given byE(γ) := (α, β) | γ−1 <α−1 · β < γ is an entourage for the uniform structure, and the subsets ofthis kind form afundamental system of entourages. LetΓ∧ be the completion ofΓ for this uniform structure.

Lemma 6.1.19.With the notation of(6.1.16), there exists a natural isomorphism of topologicalgroups:Div(K+a)

∼→ Γ∧K .

Proof. We only indicate how to construct the morphism, and leave thedetails to the reader. Inlight of remark 6.1.12(i), for every idealI ⊂ K+a we can find a netJi | i ∈ S of principalideals converging toI (for some filtered ordered set(S,≤)). Let γi ∈ ΓK be the value of agenerator ofJi. One verifies that the netγi | i ∈ S converges inΓ∧K to some elementγ. Thenwe assign:I 7→ γ. One verifies that this rule is well-defined and that it extends uniquely to thewhole ofDiv(K+a).

Definition 6.1.20. Let Γ be any ordered abelian group with neutral element1.(i) We denote byΓ+ ⊂ Γ the subset of all theγ ∈ Γ such thatγ ≤ 1.

(ii) A subgroup∆ of Γ is said to beconvexif it satisfies the following property. Ifx ∈ ∆+ and1 > y > x, theny ∈ ∆. The setSpec Γ of all the convex subgroups ofΓ will be called thespectrumof Γ. We define theconvex rankof Γ as the supremumc.rk(Γ) over the lengthsr of the chains0 ( ∆1 ( ... ( ∆r := Γ, such that all the∆i are convex subgroups. Ingeneralc.rk(Γ) ∈ N∪∞, but we will mainly encounter situations for which the convexrank is a positive integer. It is easy to see that the convex rank is always less than or equalto the usual rank, defined asrk(Γ) := dimQ(Γ ⊗Z Q). To keep the two apart, we callrational rankthe latter.


Example 6.1.21.(i) If Γ is an ordered abelian group, there exists a unique ordered group struc-ture onΓ ⊗Z Q such that the natural mapΓ → Γ ⊗Z Q is order-preserving. Indeed, ifΓ is thevalue group of a valuation| · | on a fieldK, and| · |Ka is any extension of| · |K to the algebraicclosureKa of K, then it is easy to see (e.g.using example 6.1.4(v)) thatΓKa ≃ Γ⊗Z Q.

(ii) Furthermore, letKs ⊂ Ka be the separable closure ofK; we claim that| · |Ka mapsKs

surjectively ontoΓKa. Indeed, ifa ∈ Ka is inseparable overKs, then the minimal polynomialm(X) ∈ Ks[X] of a is of the formXpm − b for someb ∈ Ks. For c ∈ K×, let mc(X) ∈Ks[X] be the polynomialm(X) + cX; if a′ is a root ofmc(X), thena′ ∈ Ks; moreover,|(a− a′)pm|Ka = |c · a′|Ka, hence for|c|Ka sufficiently small we have|a|Ka = |a′|Ka.

(iii) For any valued field(K, | · |), and everyγ ∈ ΓK , let Uγ := x ∈ K | |x| ≤ γ.One defines thevaluation topologyon K as the unique group topology such that the family(Uγ | γ ∈ Γ) is a fundamental system of open neighborhoods of0. The argument in (ii) showsmore precisely thatKs is dense inKa for the valuation topology of(Ka, | · |Ka).

(iv) If ∆ ⊂ Γ is any subgroup, thenc.rk(Γ) ≤ c.rk(∆) + rk(Γ/∆) (cp. [16, Ch.VI,§10, n.2,Prop.3]).

(v) A subgroup∆ ⊂ Γ is convex if and only if there is an ordered group structure onΓ/∆such that the natural mapΓ→ Γ/∆ is order-preserving. Then the ordered group structure withthis property is unique.

(vi) If c.rk(Γ) = 1, we can find an order-preserving imbedding

ρ : (Γ, ·,≤) → (R,+,≤).

Indeed, pick an elementg ∈ Γ with g > 1. For everyh ∈ Γ, and every positive integern,there exists a largest integerk(n) such thatgk(n) < hn. Then(k(n)/n | n ∈ N) is a Cauchysequence and we letρ(h) := lim

n→∞k(n)/n. One verifies easily thatρ is an order-preserving

group homomorphism, and since the convex rank ofΓ equals one, it follows thatρ is injective.

6.1.22. There is an inclusion-reversing bijection betweenthe set of convex subgroups of thevalue groupΓ of a valuation| · | and the set of prime ideals of its valuation ringK+. Thisbijection assigns to a convex subgroup∆ ⊂ Γ, the prime idealp∆ := x ∈ K+ | γ >|x| for everyγ ∈ ∆. Conversely, to a prime idealp, there corresponds the convex subgroup∆p := γ ∈ Γ | γ > |x| for all x ∈ p. Then, the value group of the valuation ringK+

p is(naturally isomorphic to)Γ/∆p. Furthermore,K+/p is a valuation ring of its field of fractions,and its value group is∆p.

6.1.23. Therank of a valuationis defined as the convex rank of its value group. It is clearfrom (6.1.22) that this is the same as the Krull dimension of the associated valuation ring.

6.1.24. For any field extensionF1 ⊂ F2, denote bytr.d(F2 : F1) the transcendence degreeof F2 over F1. Let E be a field extension of the valued fieldK, and | · |E : E× → ΓEan extension of the valuation| · |K : K× → ΓK of K to E. Let κ (resp. κ(E)) be theresidue field of the valuation ring of(K, | · |) (resp. of(E, | · |)). Then we have the inequality:rk(ΓE/ΓK) + tr.d(κ(E) : κ) ≤ tr.d(E : K) (cp. [16, Ch.VI,§10, n.3, Cor.1]).

6.1.25. The image ofK+ \ 0 in Γ is the monoidΓ+. The submonoids ofΓ+ are in bijectivecorrespondence with the multiplicative subsets ofK+\0which contain(K+)×. The bijectionis exhibited by the following ”short exact sequence” of monoids:

1→ (K+)× → K+ \ 0 π→ Γ+ → 1.

Then, to a monoidM ⊂ Γ+ one assigns the multiplicative subsetπ−1(M).


6.1.26. Let us say that a submonoidN of a monoidM is convexif the following holds. Ifγ, δ ∈ M andγ · δ ∈ N , thenγ ∈ N andδ ∈ N . For every submonoidN there is a smallestconvex submonoidN con such thatN ⊂ N con. One deduces a natural bijection between convexsubmonoids ofΓ+ and prime ideals ofK+, by assigning on one hand, to a convex monoidM ,the idealp(M) := K+ \ π−1(M), and on the other hand, to a prime idealp, the convex monoidM(p) := π(K+ \ p).

6.1.27. The subsets of the formM \ N , whereN is a convex submonoid of the monoidM ,are the first examples of ideals in a monoid. More generally, one says that a subsetI ⊂M is anidealofM , if I ·M ⊂ I. Then we say thatI is aprime idealif I is an ideal such that, for everyx, y ∈M with x · y ∈ I, we have eitherx ∈ I or y ∈ I. Equivalently, an idealI is a prime idealif and only ifM \ I is a submonoid; in this caseM \ I is necessarily a convex submonoid. Fora monoidM , let us denote bySpecM the set of all the prime ideals ofM . Taking into account(6.1.22), we derive bijections

Spec Γ∼→ SpecK+ ∼→ Spec Γ+ : ∆ 7→ p∆ 7→ π(p∆) = Γ+ \∆+.

Furthermore, the bijectionSpecK+ ∼→ Spec Γ+ extends to an inclusion-preserving bijectionbetween the ideals ofK+ and the ideals ofΓ+.

In the sequel, it will be sometimes convenient to study a monoid via the system of its finitelygenerated submonoids. In preparation for this, we want to delve a little further into the theoryof general commutative monoids.

Definition 6.1.28. LetM be a commutative monoid.(i) We say thatM is integral if we havea = b, whenevera, b, c ∈ M anda · c = b · c.

(ii) We say thatM is free if it isomorphic toN(I) for some index setI. In this case, a minimalset of generators forM will be called abasis.

6.1.29. LetMnd be the category of commutative monoids. The natural forgetful functorZ-Mod → Mnd admits a left adjoint functorM 7→ Mgp. Given a monoidM , the abeliangroupMgp can be realized as the set of equivalence classes of pairs(a, b) ∈ M ×M , where(a, b) ∼ (a′, b′) if there existsc ∈ M such thata · b′ · c = a′ · b · c; the addition is definedtermwise, and the unit of the adjunction is the mapφ : M → Mgp : a 7→ (a, 1) for everya ∈M . It is easy to see thatφ is injective if and only ifM is integral.

6.1.30. The categoryMnd admits arbitrary limits and colimits. In particular, it admits directsums. The functorM 7→Mgp commutes with limits and colimits.

Theorem 6.1.31.Let∆ be an ordered abelian group,N ⊂ ∆+ a finitely generated submonoid.Then there exists a free finitely generated submonoidN ′ ⊂ ∆+ such thatN ⊂ N ′.

Proof. SinceN is a submonoid of a group, it is integral, soN ⊂ Ngp. The group homomor-phismNgp ⊂ ∆ induced by the imbeddingN ⊂ ∆ is injective as well. The verification isstraightforward, using the description ofNgp in (6.1.29). ThenNgp inherits a structure of or-dered group from∆, and we can replace∆ by Ngp, thereby reducing to the case where∆ isfinitely generated andN spans∆. Thus, in our situation, the convex rankr of ∆ is finite; wewill argue by induction onr. Suppose then thatr = 1. In this case we will argue by inductionon the rankn of ∆. If n = 1, then one has only to observe thatZ+ is a free monoid. Supposenext thatn = 2; in this case, letg1, g2 ∈ ∆ be a basis. We can suppose thatg1 < g2 < 1;indeed, ifg1 > 1, we can replace it byg−1

1 ; then, sincer = 1, we can find an integerk such thatg′2 := g2·gk1 < 1 andg′2 > g1; clearlyg1, g

′2 is still a basis of∆. We define inductively a sequence

of elementsgi ∈ ∆+, for everyi > 2, in the following way. Suppose thati > 2 and that the ele-mentsg3 < g4 < ... < gi−1 have already been assigned; letki := supn ∈ N | gi−1 · g−ni−2 ≤ 1;notice that, since the convex rank of∆ equals1, we havek <∞. We setgi := gi−1 · g−ki

i−2 .


Claim 6.1.32. We havegNi · gN

i+1 ⊂ gNi+1 · gN

i+2 for everyi ≥ 1, and∆+ =⋃i≥1(g

Ni · gN

i+1).

Proof of the claim:The first assertion is obvious. We prove the second assertion. Let g ∈ ∆+;for everyi ≥ 1 we can writeg = gai

i · gbii+1 for uniqueai, bi ∈ Z. Notice thatai andbi cannotboth be negative. Suppose that eitherai+1 or bi+1 is not inN; we show that in this case

|ai+1|+ |bi+1| < |ai|+ |bi|.(6.1.33)

Indeed, we must have eitherai < 0 and bi > 0, or ai > 0 and bi < 0. However,ai+1 =ai · ki+1 + bi andbi+1 = ai; thus, ifai < 0, thenbi+1 < 0, and consequentlyai+1 > 0, whence

|ai+1| < |bi|(6.1.34)

and if ai > 0, thenai+1 < 0, so again (6.1.34) holds. From (6.1.33) it now follows thateventuallyai andbi become both positive.

SinceN is finitely generated, claim 6.1.32 shows thatN ⊂ gNi · gN

i+1 for i > 0 sufficientlylarge, so the claim follows in this case.

Next, suppose that the convex rankr = 1 andn := rk(∆) > 2. Write ∆ = H ⊕ G for twosubgroups such thatrk(H) = n− 1 andG = gZ for someg ∈ ∆.

Claim 6.1.35. For everyδ ∈ ∆+ \ 1 we can finda, b ∈ H such thatδ < a · g−1 < 1 andδ < b−1 · g < 1.

Proof of the claim:Let ρ : ∆ → R be an order-preserving imbedding as in example 6.1.21(vi);sincerk(H) > 1, it is easy to see thatρ(H) is dense inρ(∆). The claim is an immediateconsequence.

Let g1, ..., gk be a set of generators forN . For everyi ≤ k we can writegi = hi · gni foruniquehi ∈ H andni ∈ Z. Suppose thatni ≥ 0; it follows easily from claim 6.1.35 that thereexistsa, b ∈ H such thata < g < b andgi < (b−1 · g)ni < 1 (resp.gi < (a−1 · g)ni < 1) foreveryi such thatni ≥ 0 (resp.ni < 0). Then forni ≥ 0 seth′i := hi · bni and forni < 0 seth′i := hi · ani . Notice thath′i < 1. Seth0 := a · b−1 and letM be the submonoid ofH+ spannedby h0, h

′1, ..., h

′k,; we can imbedM in a larger submonoidM ′ ⊂ H+ such that(M ′)gp = H.

Then, by inductive assumption, we can imbedM ′ in a free submonoidL ⊂ H+. Let l1, ..., ln−1

be a basis forL.We can writeh0 =

∏ti=1 lki

for some integersk1, ..., kt ∈ 1, ..., n − 1. Notice thath0 ≤b−1 · g and letr < t be the largest integer such that

∏ri=1 lki

> g · b−1; setl′ := g · b−1 ·∏ri=1 l

−1ki

andl′′ := g−1 · b ·∏r+1i=1 lki

.

Claim 6.1.36. The submonoidL′ generated byl1, ..., ln−1, l′, l′′ \ lkr+1 containsN .

Proof of the claim:Indeed, sincel′ · l′′ = lkr+1, it follows thatL ⊂ L′; moreover,g · b−1 =

l′ ·∏ri=1 lki

andg−1 · a = l′′ ·∏ti=r+2 lki

so g · b−1, g−1 · a ∈ L′. Now the claim follows byremarking thatgi = h′i · (g−1 · a)−ni if ni < 0, andgi = h′i · (g · b−1)ni if ni ≥ 0.

Now, it is clear thatL ⊂ ∆+; since moreoverL spans∆ and is generated byn elements, itfollows thatL is a free monoid, so the proof is concluded in casec.rk(∆) = 1.

Finally, supposer > 1 and pick a convex subgroup0 6= ∆0 ( ∆; then the ordering on∆induces a unique ordering on∆/∆0 such that the projection mapπ : ∆ → ∆/∆0 is order-preserving. LetN0 := π(N). By induction,N0 can be imbedded into a finitely generatedfree submonoidF0 of (∆/∆0)

+. By lifting a minimal set of generators ofF0 to elementsf1, ..., fn ∈ ∆+, we obtain a free finitely generated monoidF ⊂ ∆+ with π(F ) = F0. Now,choose a finite setS of generators forN ; we can partitionS = S1 ∪ S2, whereS1 = S ∩ ∆0

andS2 = S \∆0. By construction, for everyx ∈ S2 there exist integerski,x ≥ 0 (i = 1, ..., n)such thatyx := x · ∏n

i=1 f−ki,x

i ∈ ∆0. Let g := maxyx | x ∈ S2; if g < 1, let ei := fi,otherwise letei := fi · g for everyi ≤ n. Since∆0 is convex, we have in any case:ei < 1


for i ≤ n. Moreover, the elementszx := x ·∏ni=1 e

−ki,x

i are contained in∆+0 . By induction,

the submonoid of∆+0 generated byS1 ∪ zx | x ∈ S2 is contained in a free finitely generated

monoidF ′ ⊂ ∆+0 . Using the convexity of∆0 one verifies easily thatN ′ := F · F ′ is a free

monoid. ClearlyN ⊂ N ′, so the assertion follows.

Remark 6.1.37. Another proof of theorem 6.1.31 can be found in [32, Th.2.2].Moreover, thistheorem can also be deduced from the resolution of singularities of toric varieties ([49, Ch.I,Th.11]).

6.2. Basic ramification theory. This section is a review of some basic ramification theory inthe setting of general valuation rings and their algebraic extensions.

6.2.1. Throughout this section we fix a valued field(K, | · |). Its valuation ring will be denotedK+ and the residue field ofK+ will be denoted byκ. If (E, | · |E) is any valued field extensionof K, we will denote byE+ the valuation ring ofE, by κ(E) its residue field and byΓE itsvalue group. Furthermore, we letKa be an algebraic closure ofK, andKs the separable closureof K contained inKa.

6.2.2. LetE ⊂ Ka be a finite extension ofK. LetW be the integral closure ofK+ in E; byremark 6.1.12(iv), to every maximal idealp of W we can associate a valuation| · |p : E× → Γp

extending| · |, and (up to isomorphisms of value groups) every extension of| · | toE is obtainedin this way. Setκ(p) := W/p; it is known that

∑p∈Max(W )[Γp : Γ] · [κ(p) : κ] ≤ [E : K] (cp.

[16, Ch.VI §8, n.3 Th.1]).

6.2.3. Suppose now thatE is a Galois extension ofK. ThenGal(E/K) acts transitively onMax(W ). For a givenp ∈ Max(W ), thedecomposition subgroupDp ⊂ Gal(E/K) of p isthe stabilizer ofp. Thenκ(p) is a normal extension ofκ and the natural morphismDp →Aut(κ(p)/κ) is surjective; its kernelIp is the inertia subgroupat p (cp. [16, Ch.V,§2, n.2,Th.2] for the case of a finite Galois extension; the general case is obtained by passage to thelimit over the family of finite Galois extensions ofK contained inE).

6.2.4. If nowE is a finite Galois extension ofK, then it follows easily from (6.2.2) and(6.2.3) that the integers[Γp : Γ] and[κ(p) : κ] are independent ofp, and therefore, ifW admitsn maximal ideals, we have :n · [Γp : Γ] · [κ(p) : κ] ≤ [E : K].

Lemma 6.2.5. Let K+sh be a strict henselization ofK+; thenK+sh is a valuation ring andΓK+sh = Γ.

Proof. It was shown in remark 6.1.12(vi) thatK+sh is a valuation ring. To show the secondassertion, letR be more generally any integrally closed local domain; the (strict) henselizationofR can be constructed as follows (cp. [60, Ch.X,§2, Th.2]). LetF := Frac(R),F s a separableclosure ofF , p any maximal ideal of the integral closureW of R in F s, D andI respectivelythe decomposition and inertia subgroups ofp; let WD (resp.W I) be the subring of elementsof W fixed byD (resp. byI) and setpD := WD ∩ p, pI := W I ∩ p. Then the localizationRh := (WD)pD (resp. Rsh := (W I)pI ) is a henselization (resp. strict henselization) ofR.Now, let us makeR := K+, soF := K andF s := Ks; let E ⊂ Ks be any finite Galoisextension ofK; WE := W ∩ E is the integral closure ofK+ in E; setDE := D ∩Gal(E/K),IE := I ∩ Gal(E/K), E ′ := EDE , E ′′ := EIE . Let p′ := p ∩ E ′; it then follows from [62,Ch.VI,§12, Th.23] that[Γp′ : Γ] = 1. Clearly the value groupΓKh of K+h is the filtered unionof all suchΓp′, so we deduceΓKh = Γ. Therefore, in order to prove the lemma, we can assumethatK = Kh. In this caseGal(E/K) coincides with the decomposition subgroup ofp′ andIE is a normal subgroup ofGal(K/E) such that[Gal(K/E) : IE] equals the cardinalitynof Aut(κ(E)/κ). By the definition ofIE it follows that the natural map :Aut(κ(E)/κ) →


Aut(κ(E ′′)/κ) is an isomorphism. We derive :[E ′′ : K] = n ≤ [κ(E ′′) : κ]; then from (6.2.4)we obtainΓE′′ = Γ and the claim follows.

6.2.6. We suppose now thatK+ is ahenselianvaluation ring, with value groupΓ. Then, onany algebraic extensionE ⊂ Ka ofK, there is a unique valuation| · |E extending| · |, and thus aunique inertia subgroup, which we denote simply byI. By remark 6.1.12(iv),E+ is the integralclosure ofK+ in E.

Remark 6.2.7. (i) In the situation of (6.2.6), the inequality of (6.2.2) simplifies to :

[κ(E) : κ] · [ΓE : ΓK ] ≤ [E : K].

(ii) Sometimes this inequality is actually an equality; this is for instance the case when thevaluation ofK is discrete and the extensionK ⊂ E is finite and separable (cp. [16, Ch.VI,§8,n.5, Cor.1]).

(iii) However, even when the valuation ofK is discrete, it may happen that the inequality (i)is strict, ifE is inseparable overK. As an example, letκ be a perfect field of positive character-istic, and choose a power seriesf(T ) ∈ κ[[T ]] which is transcendental over the subfieldκ[T ].EndowF := Frac(κ(T 1/p, f(T ))) with theT -adic valuation, and letK be the henselization ofF . Then the residue field ofK is κ and the valuation ofK is discrete. LetE := K[f(T )1/p].Then[E : K] = p, ΓE = ΓK andκ(E) = κ.

6.2.8. For a fieldF , we denote byµ(F ) the torsion subgroup ofF×. LetE be a finite Galoisextension ofK (with K+ still henselian). One defines a pairing

I × (ΓE/ΓK)→ µ(κ(E))(6.2.9)

in the following way. For(σ, γ) ∈ I × ΓE, let x ∈ E× such that|x| = γ; then let(σ, γ) 7→σ(x)/x ( mod mE). One verifies easily that this definition is independent of the choice ofx;moreover, ifx ∈ K×, thenσ acts trivially onx, so the definition is seen to depend only on theclass ofγ in ΓE/ΓK .

6.2.10. Suppose furthermore thatκ is separably closed. Then the inertia subgroup coincideswith the Galois groupGal(E/K) and moreoverµ(κ(E)) = µ(κ). The pairing (6.2.9) inducesa group homomorphism

Gal(E/K)→ HomZ(ΓE/ΓK ,µ(κ)).(6.2.11)

Let p := char(κ). For a groupG, let us denote byG(p) the maximal abelian quotient ofG thatdoes not containp-torsion.

Proposition 6.2.12.Under the assumptions of(6.2.10), the map(6.2.11)is surjective and itskernel is ap-group.

Proof. Let n := [E : K]. Notice thatµ(κ) does not containp-torsion, hence every homomor-phismΓE/ΓK → µ(κ) factors through(ΓE/ΓK)(p). Let m be the order of(ΓE/ΓK)(p). Letus recall the definition of the Kummer pairing: one takes the Galois cohomology of the exactsequence ofGal(Ks/K)-modules

1→ µm → (Ks)×(−)m

−→ (Ks)× → 1

and applies Hilbert90, to derive an isomorphismK×/(K×)m ≃ H1(Gal(Ks/K),µm). Now,since(m, p) = 1 andκ is separably closed, the equationXm = 1 admitsm distinct solutionsin κ. SinceK+ is henselian, these solutions lift to roots of1 in K, i.e., µm ⊂ K×, whenceH1(Gal(Ks/K),µm) ≃ Homcont(Gal(Ks/K),µm). By working out the identifications, onechecks easily that the resulting group isomorphism

K×/(K×)m ≃ Homcont(Gal(Ks/K),µm)


can be described as follows. To a givena ∈ K×, we assign the group homomorphism

Gal(Ks/K)→ µm : σ 7→ σ(a1/m)/a1/m.

Notice as well that, sinceκ is separably closed, more generally every equation of the formXm = u admitsm distinct solutions inκ, providedu 6= 0; again by the henselian property wededuce that every unit ofK+ is anm-th power inK×; thereforeK×/(K×)m ≃ ΓK/mΓK .

Dualizing, we obtain an isomorphism

HomZ(Γ/mΓ,µm) ≃ HomZ(Homcont(Gal(Ks/K),µm),µm).

However,

Homcont(Gal(Ks/K),µm) = colimH⊂Gal(Ks/K)

HomZ(Gal(Ks/K)/H,µm)

whereH runs over the cofiltered system of open normal subgroups ofGal(Ks/K) such thatGal(Ks/K)/H is abelian with exponent dividingm. It follows that

HomZ(Homcont(Gal(Ks/K)/H,µm),µm) ≃ limH⊂Gal(Ks/K)

Gal(Ks/K)/H

where the right-hand side is a quotient ofGal(Ks/K)(p). Hence, we have obtained a surjectivegroup homomorphism

Gal(Ks/K)→ HomZ(m−1ΓK/ΓK ,µm)∼→ HomZ(m−1ΓK/ΓK ,µ(κ)).(6.2.13)

(SinceΓK is torsion-free, we can identify naturallyΓK/mΓK to the subgroupm−1ΓK/ΓK ⊂(ΓK ⊗Z Q)/ΓK). Let j : ΓE/ΓK → m−1ΓK/ΓK be the inclusion map. One verifies directlyfrom the definitions, that the maps (6.2.11) and (6.2.13) fit into a commutative diagram

Gal(Ks/K) //

Gal(E/K)

HomZ(m−1ΓK/ΓK ,µ(κ))

ρ // HomZ(ΓE/ΓK ,µ(κ))

where the top map is the natural surjection, andρ := HomZ(j,µ(κ)). Finally, an easy applica-tion of Zorn’s lemma shows thatρ is surjective, and therefore, so is (6.2.11).

It remains to show that the kernelH of (6.2.11) is ap-group. Suppose thatσ ∈ H andneverthelessp does not divide the orderl of σ; then we claim that theK-linear mapφ : E → E

given byx 7→ ∑l−1i=0 σ

i(x) is an isometry. Indeed,φ(x) = l · x +∑l−1

i=1(σi(x) − x); it suffices

then to remark that|l · x| = |x| and|σi(x) − x| < |x|, sinceσi ∈ H for i = 0, ..., l − 1. Next,for everyx ∈ E we can write0 = σl(x) − x = φ(x − σ(x)); henceσ(x) = x, that is,σ is theneutral element ofGal(E/K), as asserted.

Corollary 6.2.14. Keep the assumptions of(6.2.10), and suppose moreover that(p, [E : K]) =1. ThenΓE/ΓK ≃ HomZ(Gal(E/K),µ(K)). Moreover, ifΓE/ΓK ≃ Z/q1Z ⊕ ... ⊕ Z/qkZ,then there exista1, ..., ak ∈ K withE = K[a

1/q11 , ..., a

1/qkk ].

Proof. To start out, since(p, [E : K]) = 1, proposition 6.2.12 tells us that the map (6.2.11)is an isomorphism. In particular,Gal(E/K) is abelian, and[ΓE : ΓK ] ≥ [E : K], whence[ΓE : ΓK ] = [E : K] by remark 6.2.7(ii). ThereforeGal(E/K) ≃ Z/q1Z ⊕ ... ⊕ Z/qkZ andE is a compositum of cyclic extensionsE1, ..., Ek of orderq1, ..., qk. It follows as well thatΓE/ΓK ≃ HomZ(Gal(E/K),µ(κ)), so the first assertion holds; furthermore the latter holdsalso for every extension ofK contained inE. We deduce :

Claim6.2.15. The Galois correspondence establishes an inclusion preserving bijection betweenthe subgroups ofΓE containingΓK , and the subfields ofE containingK.


To prove the second assertion, we are thus reduced to the casewhereE is a cyclic extensionof prime power order, sayGal(E/K) ≃ Z/qZ, with (q, p) = 1. Let γ ∈ ΓE be an elementwhose class inΓE/ΓK is a generator; we can finda ∈ K such that|a| = γq. LetE ′ := K[a1/m]andF := E · E ′. SinceΓF is torsion-free, one sees easily that its subgroupsΓE andΓE′

coincide. However,F satisfies again the assumptions of the corollary, thereforeclaim 6.2.15applies toF , and yieldsE = E ′.

Definition 6.2.16. Let (K, | · |) be a valued field. We denote byK+sh be the strict henselisa-tion of K+ and setKsh := Frac(K+sh). Themaximal tame extensionKt of K in its separa-ble closureKs is the union of all the finite Galois extensionsE of Ksh insideKs, such that([E : Ksh], p) = 1. Notice that, by corollary 6.2.14, every such extension is abelian and thecompositum of two such extensions is again of the same type, so the family of all such finiteextension is filtered, and therefore their union is their colimit, so the definition makes sense.

6.2.17. SinceΓKsh = ΓK , one verifies easily from the foregoing that there is a natural isomor-phism of topological groupsGal(Kt/Ksh) ≃ HomZ(ΓK ⊗Z Z(p)/ΓK ,µ

(p)), whereµ denotesthe group of roots of1 in Ksh and where we endow the target with the profinite topology.

6.2.18. LetE ⊂ Ks be any separable extension ofK. Then it is easy to check thatEt =E ·Kt. Indeed, one knows thatEsh = E ·Ksh; then letF be a finite separable extension ofEsuch that([F : E], p) = 1. By taking roots of elements ofK we can find an extensionF ′ of Ksuch that([F ′ : K], 1) = 1 and(ΓF/ΓK)(p) = (ΓF ′/ΓK)(p) and thenE · F ′ ·Ksh ⊃ F .

6.3. Algebraic extensions.In this section we return to almost rings: we suppose it is given avalued field(K, | · |), and then we will study exclusively the almost ring theory relative to thestandard setup attached toK+ (see (6.1.15)). For an extensionE ofK, we will use the notationof (6.2.1). Furthermore, we will denoteWE the integral closure ofK+ in E.

Lemma 6.3.1. LetR be a ring and0 → M1 → M2 → M3 → 0 a short exact sequence offinitely generated torsionR-modules, and suppose that theTor-dimension ofM3 is≤ 1. ThenF0(M2) = F0(M1) · F0(M3).

Proof. We can find epimorphismsφi : Rni → Mi for i ≤ 3, with n2 = n1 + n3, fitting into acommutative diagram with exact rows:

0 // Rn1 //

φ1

Rn2 //

φ2

Rn3 //

φ3

0

0 // M1// M2

// M3// 0.

LetNi := Kerφi (i ≤ 3). By snake lemma we have a short exact sequence:0→ N1 → N2π→

N3 → 0. Since the Tor-dimension of theM3 is≤ 1, it follows thatN3 is a flatR-module.

Claim 6.3.2. Λn3+1R N3 = 0.

Proof of the claim:SinceN3 is flat, the antisymmetrizer operatorak : ΛkRN3 → N⊗k3 is injective

for everyk ≥ 0 (cp. the proof of proposition 4.3.26). On the other hand,Λn3+1R Rn3+1 = 0,

thus it suffices to show that the natural mapj⊗k : N⊗k3 → (Rn3)⊗k is injective for everyk ≥ 0. This is clear fork = 0. Suppose that injectivity is known forj⊗k; we havej⊗k+1 =(1R⊗k ⊗R j) (j⊗k ⊗R 1N3). SinceN3 is flat, we conclude by induction onk.

Next recall that, for everyk ≥ 0 there are exact sequences

N1 ⊗R ΛkN2 → Λk+1N2π∧k+1

−→ Λk+1N3.(6.3.3)


(To show that such sequences are exact, one uses the universality of Λk+1R N3 for (k + 1)-linear

alternating maps toR-modules). From (6.3.3) and claim 6.3.2, a simple argument by inductionon k shows that the natural mapψ : Λn1

R N1 ⊗ Λn3R N2 → Λn2

R N2 is surjective. Finally, by

definition, we haveF0(Ni) = Im(ΛniRNi

j∧nii−→ Λni

RRni∼→ R). To conclude, it suffices therefore

to remark that the diagram:

Λn1R N1 ⊗ Λn3

R N2

1Λ

n1R

N1⊗π∧k+1

ψ // Λn2R N2

j∧n22

Λn1R N1 ⊗ Λn3

R N3// Λn2

R Rn2

(6.3.4)

commutes. We leave to the reader the task of verifying that the commutativity of (6.3.4) boilsdown to a well-known identity for determinants of matrices.

Remark 6.3.5. (i) Lemma 6.3.1 applies especially to a short exact sequenceof finitely pre-sented torsionK+-modules, since by lemma 6.1.14, any such module has homological dimen-sion≤ 1.

(ii) By the usual density arguments (cp. the proof of proposition 2.3.24), it then follows thatlemma 6.3.1 holds trueverbatim, even when we replaceR byK+ and theR-modulesM1,M2,M3 by uniformly almost finitely generated torsionK+a-modules.

Proposition 6.3.6. Suppose thatK+ is a valuation ring of rank one. LetE be a finite separableextension ofK. ThenW a

E andΩW aE/K

+a are uniformly almost finitely generatedK+a-moduleswhich admit the uniform bounds[E : K] and respectively[E : K]2. Moreover,W a

E is an almostprojectiveK+a-module.

Proof. In view of the presentation (2.5.27), the assertion forΩW aE/K

+a is an immediate conse-quence of the assertion forW a

E. The trace pairingtE/K : E × E → K is perfect sinceE isseparable overK. Let e1, ..., en be a basis of theK-vector spaceE ande∗1, ..., e

∗n the dual basis

under the trace morphism, so thattE/K(ei ⊗ e∗j) = δij for everyi, j ≤ n. We can assume thatei ∈WE and we can finda ∈ K+ \0 such thata · e∗i ∈WE for everyi ≤ n. Letw ∈WE; wecan writew =

∑ni=1 ai · ei for someai ∈ K. We havetE/K(w ⊗ a · e∗j) ∈ K+ for everyj ≤ n;

on the other hand,tE/K(w ⊗ a · e∗j ) = a · aj . Thus, if we letφ : Kn → E be the isomorphism(x1, ..., xn) 7→

∑ni=1 xi · ei, we see that

(K+)n ⊂ φ−1(WE) ⊂ a−1 · (K+)n.(6.3.7)

We can writeWE as the colimit of the familyW of all its finitely generatedK+-submodulescontaininge1, ..., en; if W0 ∈ W , thenW0 is a freeK+-module by remark 6.1.12(ii); then it isclear from (6.3.7) that the rank ofW0 must be equal ton. The proof follows straightforwardlyfrom the following:

Claim6.3.8. Let ε ∈ m; there existsW0 ∈ W such thatε ·WE ⊂W0.

Proof of the claim: Indeed, suppose that this is not the case. Then we can find an infinitesequence of finitely generated submodules⊕ni=1ei · K+ ⊂ W0 ⊂ W1 ⊂ W2 ⊂ ... ⊂ WE

such thatε ·Wi+1 * Wi for everyi ≥ 0. From (6.3.7) and lemma 6.3.1 it follows easily thatF0((K

+)n/a · (K+)n) ⊂ F0(Wk+1/W0) =∏k

i=0 F0(Wi+1/Wi) for everyk ≥ 0. However,F0(Wi+1/Wi) ⊂ AnnK+(Wi+1/Wi) ⊂ ε ·K+ for everyi ≥ 0. We deduce that|a|n < |ε|k foreveryk ≥ 0, which is absurd, since the valuation ofK has rank one.


6.3.9. Suppose that the valuation ring ofK has rank one. LetK ⊂ E ⊂ F be a tower of finiteseparable extensions. Letp ⊂WE be any prime ideal; thenWE,p is a valuation ring (see remark6.1.12(iii)), andWF,p is the integral closure ofWE,p in F . It then follows from proposition6.3.6 and remark 6.1.12(ii) thatW a

F,p is an almost finitely generated projectiveW aE,p-module;

we deduce thatW aF is an almost finitely generated projectiveW a

E-module, therefore we candefine the different ideal ofW a

F overW aE. To ease notation, we will denote it byDWF /WE

. If| · |F is a valuation ofF extending| · |, thenF+ = WF,p for some prime idealp ⊂WF ; moreover,if | · |E is the restriction of| · |F toE, thenE+ = WE,q, whereq = p∩WE. For this reason, weare led to defineDF+/E+ := (DWF /WE

)p.

Lemma 6.3.10.LetK ⊂ E ⊂ F be a tower of finite separable extensions ofK. Then:

(i) TheW aE-module(W a

E)∗ is invertible.(ii) DWE/K+ ·DWF /WE

= DWF /K+.

Proof. In view of proposition 4.1.27, (ii) follows from (i). We show(i): from proposition6.3.6 we can find, for everyε ∈ m, a finitely generatedK+-submoduleM ⊂ WE such thatε ·WE ⊂M . By remark 6.1.12(ii) it follows thatM is a freeK+-module, so the same holds forM∗ := HomK+(M,K+). The scalar multiplicationM∗ → M∗ : φ 7→ ε · φ factors through amapM∗ → W ∗

E, and if we letN be theWE-module generated by image of the latter map, thenε ·W ∗

E ⊂ N . Furthermore, for every prime idealp ⊂ WE, the localizationNp is a torsion-freeWE,p-module; sinceWE,p is a valuation ring, it follows thatNp is free of finite rank, again byremark 6.1.12(ii). Hence,N is a projectiveWE-module. In particular, this shows that(W a

E)∗ isalmost finitely generated projective as aW a

E-module. To show that(W aE)∗ is also invertible, it

will suffice to show that the rank ofN equals one. However, the rank ofN can be computed asdimE N ⊗WE

E. We haveN ⊗WEE = W ∗

E ⊗K+ K = HomK(E,K), so the assertion followsby comparing the dimensions of the two sides.

Proposition 6.3.11.Suppose thatK+ has rank one. LetK ⊂ E be a finite field extension suchthat l := [E : K] is a prime. Letp := char(κ). Suppose that either:

(a) l 6= p andK = Ksh, or(b) l = p andK = Kt, or(c) the valuation ofK is discrete and henselian, andE is separable overK, or(d) the valuation ofK is discrete and henselian,ΓE = ΓK andκ(E) = κ.

Then :

(i) In case(a), (b) or (d) holds, there existsx ∈ E \K such thatE+ is the filtered union of afamily of finiteK+-subalgebras of the formE+

i := K+[aix+ bi], (i ∈ N) whereai, bi ∈ Kare elements with|aix+ bi| ≤ 1.

(ii) In case(c) holds, there exists an elementx ∈ E+ such thatE+ = K+[x].(iii) Furthermore, ifE is a separable extension ofK, thenHj(LE+/K+) = 0 for everyj > 0.(iv) If E is an inseparable extension ofK, thenHj(LE+/K+) = 0 for everyj > 1, and

moreoverH1(LE+/K+) is a torsion-freeE+-module.

Proof. Let us first show how assertions (iii) and (iv) follow from (i)and (ii). Indeed, since thecotangent complex commutes with colimits of algebras, by (i) and (ii) we reduce to dealingwith an algebra of the formK+[w] for w ∈ E+. Such an algebra is a complete intersectionK+-algebra, quotient of the free algebraK+[X] by the idealI ⊂ K+[X] generated by the minimalpolynomialm(X) of w. In view of [46, Ch.III, Cor.3.2.7], one has a natural isomorphism inD(K+[w]-Mod)

LK+[w]/K+ ≃ (0→ I/I2 δ→ ΩK+[X]/K+ ⊗K+ K+[w]→ 0).


If we identify ΩK+[X]/K+ ⊗K+ K+[w] to the rank one freeK+[w]-module generated bydX,thenδ can be given explicitly by the rule:f(X) 7→ f ′(w)dX, for everyf(X) ∈ (m(X)).However,E is separable overK if and only ifm′(w) 6= 0. It follows thatδ is injective if andonly if E is separable overK, which proves (iii). IfE is inseparable overK, thenδ vanishesidentically by the same token. This shows (iv).

We prove (ii). Since the valuation is discrete, we must have either e := [ΓE : Γ] = l orf := [κ(E) : κ] = l (see remark 6.2.7(ii)). Ife = l, then pick any uniformizera ∈ E; everyelement ofE can be written as a sum

∑l−1i=0 xi ·ai with xi ∈ K for everyi < l. Then it is easy to

see that such a sum is inE+ if and onlyxi ∈ K+ for everyi < l. In other words,E+ = K+[a].In casef = l, we can writeκ(E) = κ[u] for some unitu ∈ (E+)×; moreover,mE = mE+; thenK+[u] + mE+ = E+; since in this caseE+ is a finiteK+-module, we deduceE+ = K+[u] byNakayama’s lemma.

We prove (i). Suppose that (a) holds; then by corollary 6.2.14 it follows thatΓE/ΓK ≃ Z/lZandE = K[a1/l] for somea ∈ K. Hence:

|ai/l| /∈ Γ for everyi = 1, ..., l − 1.(6.3.12)

We can suppose that the valuation ofK is not discrete, otherwise we fall back on case (c); then,for everyε ∈ m, there existsbε ∈ K such that|ε| < |blε · a| < 1. Let x0, ..., xl−1 ∈ K and setw :=

∑l−1i=0 xi · ai/l. Clearly every element ofE can be written in this form. From (6.3.12) we

derive that the values|xi · ai/l| such thatxi 6= 0 are all distinct. Hence,|w| = max0≤i<l

|xi · ai/l|.Suppose now thatw ∈ E+; it follows that |xi · ai/l| ≤ 1 for i = 0, ..., l − 1, and in fact|xi · ai/l| < 1 for i 6= 0. Let ε ∈ m such that|εl−1| > |xi · ai/l| for everyi 6= 0. A simplecalculation shows that|xi · b−iε | < 1 for everyi 6= 0, in other words,w ∈ K+[bε · a1/l], whichproves the claim in this case.

In order to deal with cases (b) and (d) we need some preparation. Let x ∈ E \ K be anyelement, and set:

ρ(x) := infa∈K|x− a| ∈ Γ∧E .

We consider case (b). Notice that the hypothesisK = Kt implies that the valuation ofKis not discrete. For anyy ∈ E we can writey = f(x) for somef(X) := b0 + b1X + ... +bdX

d ∈ K[X] with d := deg f(X) < p. The degree of the minimal Galois extensionF of Kcontaining all the roots off(X) dividesd!, henceF ⊂ Kt = K. In other words, we can writey = ak ·

∏di=1(x− αi) for someα1, ..., αd ∈ K.

We distinguish two cases: first, suppose that there existsa ∈ K with |x − a| = ρ(x).Replacingx by x − a we may achieve that|x| ≤ |x − a| for everya ∈ K. Then the constantsequence(an := 0 | n ∈ N) fulfills the condition of lemma 6.1.9. Thus, ify = f(x) as above isin E+, we must have|f(X)|(0,ρ(x)) ≤ 1; in other terms:

|bi| · ρ(x)i ≤ 1 for everyi ≤ d.(6.3.13)

Now, if ρ(x) ∈ ΓK , we can findc ∈ K such thatx0 := x · c still generatesE and |x0| = 1,whence|bi/ci| ≤ 1 for everyi ≤ 1; however,y = b0 + (b1/c) · x0 + (b2/c

2)x20 + ...+ (bd/c

d)xd,thusy ∈ K+[x0], so in this case,E+ itself is one of theE+

i .In caseρ(x) /∈ ΓK , since anywayΓK is of rank one and not discrete, we can find a sequence

of elementsc1, c2, ... ∈ K such that, lettingxi := x · ci, we have

|xj − a| ≥ |xj| for everya ∈ K, j ∈ N; |xj | < 1 and |xj | → 1.

Claim6.3.14. If x /∈ ΓK , then|xl| /∈ ΓK for every0 < l < p.

Proof of the claim:Indeed, suppose that|xl| ∈ ΓK for some0 < l < p; sinceΓK is l-divisible,we can multiplyx by somea ∈ K to have|xl| = 1, therefore|x| = 1, a contradiction.


From (6.3.13) and claim 6.3.14 we deduce that actually|bi| · ρ(x)i < 1 wheneveri > 0. Itfollows that, forj sufficiently large, we will have1 > |xij | > |bi| ·ρ(x)i for everyi > 0. Writingy = b0 +(b1/cj)xj +(b2/c

2j)x

2j + ...+(bd/c

dj )x

dj we deducey ∈ K+[xj ], therefore the sequence

of K+-subalgebraK+[ci · x] will do in this case.Finally we have to consider the case where the infimumρ(x) is not attained for anyx ∈ E.

In this case, since the valuation is not discrete and of rank1, we can find, for everyx ∈ E, asequence of elementsa0, a1, a2, ... ∈ K such that

γj := |x− aj | → ρ(x).(6.3.15)

In particular, forj sufficiently large we will have|x| > |x − aj |, therefore|x| = |aj|. Thisshows:

ΓE = ΓK .(6.3.16)

Now, pickx ∈ E \K and any sequence of elementsai ∈ K such that (6.3.15) holds; it is clearthat(ai | i ∈ N) fulfills the condition of lemma 6.1.9. Consequently

|y| = |f(X)|(aj ,γj) for every sufficiently largej.(6.3.17)

Let f(X) = b0,j + b1,j(X − aj) + ...+ bd,j(X − aj)d. (6.3.17) says that|bi,j| · γij ≤ 1 wheneverj is sufficiently large. However, from (6.3.16) we know thatγj ∈ ΓK . Pick cj ∈ K such that|cj | = γ−1

j and setxj := cj(x − aj). It follows that|bi,j/cij| ≤ 1 andy = b0,j + (b1,j/c1,j)xj +

... + (bd,j/cdj )x

dj . Hencey ∈ K+[xj ]. It is then easy to verify that the family of all suchK+-

subalgebras is filtered by inclusion, and thus conclude the proof of case (b).At last, we turn to case (d). Notice that, by remark 6.2.7(ii), this case can occur only ifE is

inseparable overK, and thenl = p. Let x ∈ E \ K; let a ∈ m be a uniformizer; for givenn ∈ N, suppose thatbn ∈ K has been found such that|x − bn| ≤ |an|. Sinceκ(E) = κ, wecan find an elementc ∈ K+ such thatc ≡ (x − bn)/a

n (mod m). Setbn+1 := bn + c · an;then|x − bn+1| ≤ |an+1|. This shows thatρ(x) = 0, and the resulting sequence(bn | n ∈ N)converges tox in them-adic topology. Lety ∈ E; we can writey = f(x) for a polynomialf(X) ∈ K[X] of degreed < p. LetF be the minimal field extension ofK that contains all theroots off(X). Notice that[F : K] dividesd!, henceF is separable overK, and[E ·F : F ] = p.Let f(X) = c ·∏d

i=0(X−αi) be the factorization off(X) in F [X]. By lemma 6.1.9 we deducethat, for every sufficiently largen ∈ N we have:|y| = |f(X)|(bn,|x−bn|), where| · |(bn,|x−bn|)is the Gauss valuation onF (X). One then argues as in the proof of case (b), to show thaty ∈ E+

n := K+[cn(x− bn)], with cn ∈ K such that|cn(x− bn)| = 1. Again, it is easy to verifythatE+

i ⊂ E+i+1 for everyi ∈ N, so the proof is complete.

Corollary 6.3.18. LetE be a finite field extension ofK of prime degreel.

(i) If E satisfies condition(a)of proposition6.3.11, and the valuation ofK is not discrete (butstill of rank one), thenΩE+/K+ = 0, LE+/K+ ≃ 0 andDE+/K+ = E+a.

(ii) If E satisfies condition(c) of proposition6.3.11, then we have :F0(ΩE+/K+) = DE+/K+

andHi(LE+/K+) = 0 for i > 0.

Proof. (i): Since condition (a) holds, proposition 6.3.11 and its proof show that there existsa ∈K such thatE+ is the increasing union of allK+-subalgebras of the formE+

b := K+[b · a1/l],whereb ∈ K+ ranges over all elements such that|bl · a| < 1. Consequently,ΩE+/K+ =colim

bΩE+

b /K+ , andLE+/K+ = colim

bLE+

b /K+. Then, again from proposition 6.3.11 it follows

thatHj(LE+/K+) = 0 for every j > 0. Hence, in order to show the first two assertions, itsuffices to show that the filtered system of theΩE+

b /K+ is essentially zero. However, theE+

b -

moduleΩE+b /K

+ is generated byωb := d(b · a1/l), and clearlyl · (bl · a)(l−1)/l · ωb = 0. Since

(l, p) = 1, it follows that(bl · a)(l−1)/l · ωb = 0. On the other hand, for|b| < |c| we can write:


ωb = b·c−1·ωc. Therefore, the image ofωb in ΩE+c /K+ vanishes, whenever|b·c−1| < |cl·a|(l−1)/l,

i.e., whenever|b · a1/l| < |cl · a| < 1. Since the valuation ofK is of rank one and not discrete,such ac can always be found. To show the last stated equality, let us recall the following generalfact (for whose proof we refer to [60, Ch.VII,§1]).

Claim6.3.19. Suppose thatE = K[w] for somew ∈ E, and letf(X) ∈ K[X] be its minimalpolynomial; the elements1, w, w2, ..., wl−1 form a basis of theK-vector spaceE. Let e∗1, ..., e

∗n

be the corresponding dual basis under the trace pairing; then the basesS := e∗1, ..., e∗n andS ′ := wl−1/f ′(w), wl−2/f ′(w), ..., 1/f ′(w) span the sameE+-submodule ofE.

Let us takew = b · a1/l for someb ∈ V such that|bl · a| < 1. It follows from claim 6.3.19that(DE+/K+)−1 ⊂ f ′(w)−1 ·E+a, whencef ′(w) ∈ DE+/K+∗. However,f ′(w) = l ·wl−1, andfrom the definition ofw we see that|f ′(w)| can be made arbitrarily close to1, by choosing|b|closer and closer to|a|1/l.

(ii): the claim about the cotangent complex is just a restatement of proposition 6.3.11(iii),(iv).By proposition 6.3.11(ii) we can writeVE = K+[w] for somew ∈ E+. Let f(x) ∈ K+[X]be the minimal polynomial ofw. Claim 6.3.19 implies thatDE+/K+ = (f ′(w)); a standardcalculation yieldsΩE+/K+ ≃ E+/(f ′(w)), so the assertion holds.

Theorem 6.3.20.Let (E, | · |E) be a finite separable valued field extension of(K, | · |) andsuppose thatK+ has rank one. ThenF0(ΩE+a/K+a) = DE+/K+ andHi(LE+/K+) = 0 fori > 0.

Proof. We begin with a few reductions:

Claim6.3.21. We can assume thatE is a Galois extension ofK.

Proof of the claim: Indeed, let(L, | · |L) be a Galois valued field extension ofK extending(E, | · |E). We obtain by transitivity ([46, II.2.1.2]) a distinguished triangle

σ−1LL+/E+ → LE+/K+ ⊗E+ L+ → LL+/K+ → LL+/E+ .(6.3.22)

Suppose that the theorem is already known for the Galois extensionsK ⊂ L andE ⊂ L. Then(6.3.22) implies thatHi(LE+/K+) = 0 for i > 0 and moreover provides a short exact sequence

0→ ΩE/K+ ⊗E+ L+ → ΩL+/K+ → ΩL+/E+ → 0.

However, on one hand, by lemma 6.3.10(ii) the different is multiplicative in towers of exten-sions, and the other hand, the Fitting idealF0 is multiplicative for short exact sequences, byvirtue of remark (6.3.5)(ii), so the claim follows.

Claim6.3.23. We can assume thatK+ is strictly henselian.

Proof of the claim:Indeed, letK+sh be the strict henselisation ofK+ andKsh := Frac(K+sh).It is known thatK+sh is an ind-etale extension ofK+, thereforeE+⊗K+K+sh is a reduced nor-mal semilocal integral and flatK+sh-algebra, whence a product of reduced normal local integraland flatK+sh-algebrasW1, ...,Wk. Each suchWi is necessarily the integral closure ofK+sh inEi := Frac(Wi). It follows thatLE+/K+ ⊗K+ K+sh ≃ LE+⊗K+K+sh/K+sh ≃ ⊕ki=1LWi⊗K+K+sh.Furthermore:DE+/K+ ⊗K+a (K+sh)a ≃ ⊕ki=1DE+

i /K+sh and similarly for the modules of dif-

ferentials. We remark as well that the formation of Fitting ideals commutes with arbitrary basechanges. In conclusion, it is clear that the assertions of the theorem hold for the extensionK ⊂ E if and only if they hold for each extensionKsh ⊂ Ei.

Claim6.3.24. SupposeK = Ksh. We can assume thatGal(E/K) is ap-group.


Proof of the claim:Indeed, letP be the kernel of (6.2.11). By proposition 6.2.12,P is p-group;let L be the fixed field ofP . ThenL ⊂ Kt and, by virtue of corollary 6.2.14, we see thatLadmits a chain of subextensionsK := L0 ⊂ L1 ⊂ ... ⊂ Lk := L such that eachLi ⊂ Li+1

satisfies either condition (a) or (c) of proposition 6.3.11.Then, by corollary 6.3.18 it followsthat the assertions of the theorem are already known for the extensionsLi ⊂ Li+1. From here,using transitivity of the cotangent complex and multiplicativity of the different in towers ofextensions, and of the Fitting ideals for short exact sequences, one shows that the assertionshold also for the extensionK ⊂ L (cp. the proof of claim 6.3.21). Now, if the assertions areknown to hold as well for the extensionL ⊂ E, again the same argument proves them forK ⊂ E.

Claim 6.3.25. The theorem holds if the valuation ofK is discrete.

Proof of the claim:By claim 6.3.24, we can suppose thatGal(E/K) is a p-group. Hence,we can find a sequence of subextensionsE0 := K ⊂ E1 ⊂ E2 ⊂ ... ⊂ En := E with[Ei+1 : Ei] = p, for everyi = 0, ..., n− 1. Arguing like in the proof of claim 6.3.24 we see thatit suffices to prove the claim for each of the extensionsEi ⊂ Ei+1. In this case we are left todealing with an extensionK ⊂ E of degreep, which is taken care of by corollary 6.3.18(ii).

Claim 6.3.26. SupposeK = Ksh, thatGal(E : K) is a p-group and that the valuation ofKis not discrete. LetL be a finite Galois extension ofK such that([L : K], p) = 1. Then thenatural mapE+ ⊗K+ L+ → (E · L)+ is an isomorphism.

Proof of the claim:By corollary 6.2.14 we know thatL admits a tower of subextensions of theform K := L0 ⊂ L1 ⊂ ... ⊂ Lk := L, such that, for eachi ≤ k we haveLi+1 = Li[a

1/l] forsomea ∈ Li and some primel 6= p. By induction oni, we can then reduce to the case whereL = K[a1/l] for somea ∈ K and a primel 6= p. Under the above assumptions, we must haveE∩L = K, hencea /∈ E. ThenE ·L = E[a1/l] and by proposition 6.3.11 and its proof,(E ·L)+

is the filtered union of all its subalgebras of the formE+[b · a1/l], whereb ∈ E ranges over allthe elements such that|bm · a| < 1. However, since the valuation ofK is not discrete and hasrank one,ΓK is dense inΓE, and consequently the subfamily consisting of theE+[b · a1/l] withb ∈ K is cofinal. Finally, forb ∈ K we haveE+[b · a1/l] ≃ E+ ⊗K+ K+[b · a1/l]. By takingcolimits, it follows that(E · L)+ ≃ E+ ⊗K+ L+.

Claim 6.3.27. We can assume thatK is equal toKt.

Proof of the claim:By claim 6.3.23 we can and do assume thatK = Ksh, in which caseKt

is the filtered union of all the finite Galois extensionL of K such that([L : K], p) = 1. ThenKt+ =

⋃L L

+ and(E ·Kt)+ =⋃L(E ·L)+, whereL ranges over all such extensions. By claim

6.3.24 we can also assume thatGal(E/K) is ap-group, in which case, by claim 6.3.26, we haveE+⊗K+L+ ∼→ (E ·L)+ for everyL as above. Taking colimit, we getE+⊗K+Kt+ ∼→ (E ·Kt)+.SinceKt+ is faithfully flat overK+, this shows that, in order to prove the theorem, we canreplaceK byKt; however, by (6.2.18) we have(Kt)t = Kt, whence the claim.

After this preparation, we are ready to finish the proof of thetheorem. We are reduced toconsidering a Galois extensionE of K = Kt such thatGal(E/K) is a p-group; moreover,we can assume that the valuation ofK is not discrete. Then, arguing as in the proof of claim6.3.25, we can further reduce to dealing with an extensionK ⊂ E of degreep; furthermore, theconditionK = Kt still holds, by virtue of (6.2.18). In this situation, condition (b) of proposition6.3.11 is fulfilled, henceHj(LE+/K+) = 0 for j > 0, by proposition 6.3.11(iii). It remains toshow the identityF0(ΩEa+/K+a) = DE+/K+. By proposition 6.3.11(i), there existsx ∈ E suchthatE+ is the filtered union of a family of finiteK+-subalgebrasE+

i := K+[aix + bi] (i ∈ N)of E+. Let f(X) ∈ K+[X] be the minimal polynomial ofx. By construction ofE+

i , it is clear


that they form a Cauchy net inIK+a(E+a) converging toE+a. It then follows from lemma2.5.26, that the netΩE+

i /K+ ⊗E+

iE+ | i ∈ N converges toΩE+/K+ in M (E+). In particular,

F0(ΩE+/K+) = limi→∞

F0(ΩE+i /K

+ ⊗E+iE+). The minimal polynomial ofaixi + bi is fi(X) :=

f(a−1i X − bi), therefore:ΩE+

i /K+ = E+

i /(f′i(aixi + bi)) = E+

i /(a−1i f ′(x)). Consequently,

F0(ΩE+/K+) = limi→∞

(a−1i f ′(x)). On the other hand, claim 6.3.19 yields:DE+

i /K+ = (a−1

i f ′(x))

for everyi ∈ N. Then the claim follows from lemma 4.1.30.

The final theorem of this section completes and extends theorem 6.3.20 to include valuationsof arbitrary rank.

Theorem 6.3.28.Let (K, | · |) be any valued field and(E, | · |E) any algebraic valued fieldextension of(K, | · |). We have :

(i) Hi(LE+/K+) = 0 for i > 1 andH1(LE+/K+) is a torsion-freeE+-module.(ii) If moreover,E is a separable extension ofK, thenHi(LE+/K+) = 0 for i > 0.

Proof. Let us show first how to deduce (ii) from (i). Indeed, suppose thatE is separable overK. ThenLE/K ≃ 0. However, by (i), the natural mapH1(LE+/K+)→ H1(LE+/K+)⊗K+ K ≃H1(LE/K) is injective, so the assertion follows.

In order to prove (i), we reduce easily to the case of a finite algebraic extension. Let us writeK as the filtered union of its subfieldsLα that are finitely generated over the prime field. Foreach suchLα, letKα := (Lα)

a ∩ K andEα := (Lα)a ∩ E. ThenEα is a finite extension of

Kα andK is the filtered union of theKα. It follows easily that we can replace the extensionK ⊂ E by the extensionKα ⊂ Eα, thereby reducing to the case where the transcendencedegree ofK over its prime field is finite. In this situation, the rankr ofK is finite (cp. (6.1.24)).We argue by induction onr. Suppose first thatr = 1. We can split into a tower of extensionsK ⊂ Ks ∩ E ⊂ E; then, by using transitivity (cp. the proof of claim 6.3.21), we reduceeasily to prove the assertion for the subextensionsK ⊂ Ks andKs ∩ E ⊂ E. However, thefirst case is already covered by theorem 6.3.20, so we can assume thatE is purely inseparableoverK. In this case, we can further splitE into a tower of subextensions of degree equalto p; thus we reduce to the case where[E : K] = p. We apply transitivity to the towerK ⊂ Kt ⊂ E · Kt = Et: by proposition 6.3.11(iv) we know thatHi(LEt+/Kt+) vanishesfor i > 1 and is torsion-free fori = 1; by theorem 6.3.20, we haveHi(LKt+/K+) = 0 fori > 0, thereforeHi(LEt+/K+) vanishes fori > 1 and is torsion-free fori = 1. Next we applytransitivity to the towerK ⊂ E ⊂ Et : by theorem 6.3.20 we haveHi(LEt+/E+) = 0 for i > 0,and the claim follows easily.

Next suppose thatr > 1, and that the theorem is already known for ranks< r. Arguing as inthe proof of claim 6.3.23, we can even reduce to the case whereK+ is henselian, and thenE+

is the integral closure ofK+ in E. Letpr := (0) ⊂ pr−1 ⊂ ... ⊂ p0 be the chain of prime idealsof K+, and for everyi ≤ r let qi be the unique prime ideal ofE+ lying overpi. The valuationring E+

q1has rankr − 1, thus, by inductive assumption, the desired assertions areknown for

the extensionK+p1⊂ E+

q1. It suffices therefore to show thatHi(LE+/K+) ⊂ Hi(LE+

q1/K+

p1) for

everyi ≥ 0. Pick a ∈ p0 \ p1. ThenK+p1

= K+[a−1] andE+q1

= E+[a−1] andLE+q1/K+

p1=

LE+/K+ ⊗K+ K+[a−1]. Hence, we are reduced to show that multiplication bya is injective onthe homology ofLE+/K+ . LetR := K+/aK+ andRE := E+ ⊗K+ R. We have a short exactsequence0 → K+ a→ K+ → R → 0, therefore, after tensoring byLE+/K+, a distinguishedtriangle:

LE+/K+a→ LE+/K+ → LE+/K+

L

⊗K+ R→ σLE+/K+ .

On the other hand, according to remark 6.1.12(ii),E+ is flat overK+, thereforeLE+/K+

L

⊗K+

R ≃ LRE/R (by [46, II.2.2.1]). Consequently, it suffices to show thatHi(LRE/R) = 0 for i ≥ 2.


However,R = (K+/p1)⊗K+ R, andRE = (E+/q1)⊗K+ R; moreover,E+/q1 is the integralclosure of the valuation ringK+/p1 in the finite field extensionFrac(E+/q1) of Frac(K+/p1).Therefore we can replaceK+ by K+/p1 andE+ by K+/q1. This turns us back to the casewherer = 1. Then the vanishing ofHi(LE+/K+) for i ≥ 2 yields the vanishing ofHi(LRE/R)for i > 2. Moreover, sinceH1(LE+/K+) is torsion-free, multiplication bya onH1(LE+/K+) isinjective, thereforeH2(LRE/R) vanishes as well.

6.4. Logarithmic differentials. In this sectionK+ is a valuation ring of arbitrary rank. Wekeep the notation of (6.2.1). We start by reviewing some facts on logarithmic structures, forwhich the general reference is [48].

6.4.1. LetMndX (reps.Z-ModX) be the category of sheaves of commutative monoids (resp.of abelian groups) on a topological spaceX. The forgetful functorZ-ModX →MndX admitsa left adjoint functorM 7→M gp. If M is a sheaf of monoids,M gp is the sheaf associated to thepresheaf defined by :U 7→ M(U)gp for every open subsetU ⊂ X.

The functorΓ : MndX →Mnd that associates to every sheaf of monoids its global sections,admits a left adjointMnd → MndX : M 7→ MX . For a monoidM , MX is the sheafassociated to the constant presheaf with valueM .

6.4.2. Recall that apre-log structureon a schemeX is a morphism of sheaves of commutativemonoids :α : M → OX, where the monoid structure ofOX is induced by multiplication oflocal sections. We denote bypre-logX the category of pre-log structures onX.

To a monoidM and a morphism of monoidsφ : M → Γ(X,OX), one can associate apre-log structureφX : MX → OX by composing the induced morphism of constant sheavesMX → Γ(X,OX)X with the counit of the adjunctionΓ(X,OX)X → OX .

6.4.3. To a morphismφ : Y → X of schemes, one can associate a pair of adjoint functorsφ∗ : pre-logX → pre-logY andφ∗ : pre-logY → pre-logX . Let (M,α : M → OX)(resp. (N, β : N → OY )) be a pre-log structure onX (resp. onY ) andφ : OX → φ∗OY

φ♯ : φ−1OX → OY the natural morphisms (unit and counit of the adjunction(φ−1, φ∗) on

sheaves ofZ-modules); thenφ−1Mφ−1α−→ φ−1OX

φ♯

−→ OY definesφ∗(M,α : M → OX) andφ∗(N, β : N → OY ) is the morphism of sheaves of monoidsγ : φ∗N ×φ∗OY

OX → OX whichmakes commute the cartesian diagram

φ∗N ×φ∗OYOX

γ //

OX

φ

φ∗N

φ∗β // φ∗OY .

6.4.4. A pre-log structureα is said to be alog structureif α−1(O×X) ≃ O×X. We denote bylogXthe category of log structures onX. The forgetful functorlogX → pre-logX : M 7→ Mpre-log

admits a left adjoint

pre-logX → logX : M 7→ M log(6.4.5)

and the resulting the diagram:

α−1(O×X) //

M

O×X// M log

(6.4.6)

is cocartesian in the category of pre-log structures. From this, one can easily verify that the unitof the adjunction :M 7→ (Mpre-log)log is an isomorphism for every log structureM .


6.4.7. The categorylogX admits arbitrary colimits; indeed, since the unit of the adjunction(6.4.5) is an isomorphism, it suffices to construct such colimits in the category of pre-log struc-tures, and then apply the functor(−) 7→ (−)log which preserves colimits, since it is a leftadjoint. In particular,logX admits arbitrary direct sums, and for any family(M i | i ∈ I) ofpre-log structures we have(⊕i∈IM i)

log ≃ ⊕i∈IM logi .

6.4.8. For any morphism of schemesY → X we remark that, if(M,α) is a log structure onY , then the pre-log structureφ∗(M,α) is actually a log structure (this can be checked on thestalks). We deduce a pair of adjoint functors(φ∗, φ∗) for log structures, as in (6.4.3). These areformed by composing the corresponding functors for pre-logstructures with the functor (6.4.5).

6.4.9. We say that a log structureM is regular if M = (MX)log for some free monoidM , andthe associated morphism of monoidsφ : M → Γ(X,OX) mapsM into the set of non-zero-divisors ofΓ(X,OX).

6.4.10. For anOX-moduleF , denote byHomOX(F , ∗) the category of all homomorphisms

of OX-modulesF → A (for anyOX-moduleA ). A morphism fromF → A to F → B isa morphismA → B of OX-modules which induces the identity onF . This category admitsarbitrary colimits.

6.4.11. Given a pre-log structureα : M → OX , one defines the sheaf oflogarithmic differ-entialsΩX/Z(logM) as the quotient of theOX-moduleΩX/Z ⊕ (OX ⊗ZX

M gp) by theOX-submodule generated by the local sections of the form(dα(m),−α(m) ⊗m), for every localsectionm of M . (The meaning of this is, that one adds toΩX/Z the logarithmic differentialsα(m)−1dα(m)). For every local sectionm of M , we denote byd log(m) the image of1⊗m inΩX/Z(logM). The assignmentM 7→ (ΩX/Z → ΩX/Z(logM)) defines a (covariant) functor :

Ω : pre-logX → HomOX(ΩX/Z, ∗).

Lemma 6.4.12.LetX be a scheme.

(i) The functorΩ commutes with all colimits.(ii) The functorΩ factors through the functor(6.4.5).(iii) Let j : U → X be a formallyetale morphism of schemes andM a log structure onX.

Then the natural morphism:j∗ΩX/Z(logM)→ ΩU/Z(log j∗M) is an isomorphism.(iv) If M is a regular log structure, thenΩ(M ) is a monomorphism ofOX-modules.

Proof. (i): It is clear thatΩ commutes with filtered colimits. Thus, to show that it commuteswith all colimits, it suffices to show that it commutes with finite direct sums and with coequal-izers. We consider first direct sums. We have to show that, forany two pre-log structuresM 1

andM 2, the natural morphism

ΩX/Z(logM1) ∐ΩX/Z

ΩX/Z(logM 2)→ ΩX/Z(logM 1 ⊕M 2)

is an isomorphism. Notice that the functor(−) 7→ (−)gp of (6.4.1) commutes with colimits,since it is a left adjoint. It follows that the diagram

ΩX/Z//

ΩX/Z ⊕ (OX ⊗ZXM gp

1 )

ΩX/Z ⊕ (OX ⊗ZX

M gp2 ) // ΩX/Z ⊕ (OX ⊗ZX

(M1 ⊕M2)gp)

is cocartesian. Thus, we are reduced to show that the kernel of the map

ΩX/Z ⊕ (OX ⊗ZXM gp)→ ΩX/Z(logM)


is generated by the images of the kernels of the corresponding maps relative toM 1 andM 2.However, any section ofM1⊕M 2 can be written locally in the formx · y for two local sectionsx of M1 andy of M2. Then we have :

(dα(x · y),−α(x · y)⊗ (x · y)) = (α(x) · dα(y) + α(y) · dα(x),−α(x) · α(y)⊗ (x · y))= α(x) · (dα(y),−α(y)⊗ y) + α(y) · (dα(x),−α(x)⊗ x)

so the claim is clear. Next, suppose thatφ, ψ : M → N are two morphisms of pre-log structures.Let : α : Q→ OX be the coequalizer ofφ andψ. Clearly,Q is the coequalizer ofφ andψ in thecategory of sheaves of monoids. The functorM 7→ M gp preserves colimits, so we are reducedto consider the cokernel ofβ := φgp − ψgp. Moreover, clearly we haveCoker(β) ⊗ZX

OX ≃Coker(β ⊗ZX

OX); the claim follows easily.(ii): Let us apply the functorΩ to the cocartesian diagram (6.4.6). In view of (i), the resulting

diagram ofOX-modules is cocartesian. However, it is easy to check thatΩX/Z(logα−1(O×X)) ≃ΩX/Z(log O×X) ≃ ΩX/Z. The assertion follows directly.

(iii): one uses [30, Ch.IV, Cor. 17.2.4]; the details will beleft to the reader.(iv): By (ii), the functorΩ descends to a functor

logX → HomOX(ΩX/Z, ∗).(6.4.13)

Since the unit of the adjunction (6.4.5) is an isomorphism, it follows easily that (6.4.13) com-mutes with all colimits of log structures. Hence, to verify thatΩ(M) is a monomorphism whenM is regular, we are immediately reduced to the case whenM is the regular log structure asso-ciated to a morphism of monoidsφ : N→ Γ(X,OX). Let f := φ(1). It is easy to check that inthis case, the diagram

OXf //

df

OX

d log f

ΩX/Zβ // ΩX/Z(logM)

is cocartesian. By assumption,f is a non-zero-divisor, thus multiplication byf is a monomor-phism ofOX-modules, so the assertion follows.

6.4.14. This general formalism will be applied here to the following situation. We considerthe submonoidM := K+ \ 0 of K+. The imbeddingM ⊂ K+ induces a log structure onSpecK+, which we call thetotal log structureonK+. More generally, we consider the naturalprojectionπ : M → Γ+ (see (6.1.25)); then for every submonoidN ⊂ Γ+, we have a logstructureN corresponding to the imbeddingπ−1(N) ⊂ K+. To ease notation, we will denoteby ΩK+/Z(logN) the correspondingK+-module of logarithmic differentials.

Proposition 6.4.15. In the situation of(6.4.14), let ∆ ⊂ Γ be any subgroup,N a prime idealof ∆+ (cp. (6.1.27)) and suppose that the convex rank ofΣ := ∆/(∆+ \N)gp equals one. Thenwe have a short exact sequence

0→ ΩK+/Z(log ∆+ \N)j→ ΩK+/Z(log ∆+)

ρ→ Σ⊗Z (K+/π−1(N) ·K+)→ 0.

Proof. Let us first remark that the assumptions and the notation makesense : indeed, sinceNis a prime ideal of∆+, it follows thatM := ∆+ \ N is a convex submonoid of∆+, henceM = (Mgp)+ andMgp is a convex subgroup of∆ (cp. (6.1.26)), thereforeΣ is an orderedgroup (cp. example (6.1.21)(v)), and hence it makes sense tosay that its convex rank equalsone.

Let us show thatj is injective. We can write∆+ as the colimit of the filtered family of itsfinitely generated submonoidsFα. For each suchFα, theorem 6.1.31 gives us a free finitelygenerated submonoidLα ⊂ ∆+ such thatFα ⊂ Lα. Clearly∆+ is the colimit of theLα, andMis the colimit of theMα := M∩Lα. Thus∆+ is the colimit of theLα andM is the colimit of the


Mα. LetSα be a basis ofLα. SinceM is convex in∆, we see thatMα is free with basisSα∩MandLα = Mα⊕Nα, whereNα is the free submonoid spanned bySα \M . For eache ∈ Sα \M ,pick arbitrarily an elementxe ∈ K+ such that|xe| = e. The mape 7→ xe can be extendedto a map of monoidsNα → K+, and then to a pre-log structureνα : (Nα)SpecV → OSpecK+.Clearly we have an isomorphism of pre-log structures:Lα = Mα ⊕ να. Since the formation oflogarithmic differentials commutes with colimits of monoids, we are reduced to showing thatthe analogous map

jα : ΩK+/Z(logMα)→ ΩK+/Z(logLα)

is injective. By lemma 6.4.12(i), we haveKer(jα) ≃ Ker(ΩK+/Z → ΩK+/Z(log να)). Bylemma 6.4.12(iv), the latter map is injective, whence the assertion.

Next we proceed to show how to constructρ. Define a map

ρ : X := ΩK+/Z ⊕ (π−1∆⊗Z K+)→ Σ⊗Z (K+/π−1(N) ·K+)

by the rule :(ω, a⊗ b) 7→ π(a)⊗ b, for anyω ∈ ΩK+/Z, a ∈ π−1∆, b ∈ K+.

Claim6.4.16. Kerρ contains the kernel of the surjectionX → ΩK+/Z(log ∆+).

Proof of the claim: It suffices to show thata ⊗ |a| ∈ Kerρ wheneverπ(a) /∈ (∆+ \ N)gp.However,π(a) /∈ (∆+ \ N)gp ⇔ π(a) /∈ ∆+ \N ⇔ π(a) ∈ N ⇔ a ∈ π−1(N), so the claimfollows.

By claim 6.4.16 we deduce thatρ descends to the mapρ as desired. It is now obvious thatρ is surjective and that its kernel contains the image ofj. To conclude the proof, it sufficesto show that the cokernel ofj is annihilated byπ−1N . We are thus reduced to showing thatπ−1(N) annihilates the classes inCoker(j) of the elementsd log(e), for everye ∈ π−1N . Leta ∈ π−1(N). Since the convex rank ofΣ equals one, andπ(e) ∈ N , there existsk ≥ 0andb ∈ K+ such thate = ak · b and |b| < |a|. In particular,|b| ∈ ∆+, and we can write:d log(e) = d log(ak · b) = b · k · d log(a) + ak · d log(b), and it is clear thata annihilates each ofthe terms of this expression.

Corollary 6.4.17. In the situation of(6.4.14), we have :

(i) The natural map :βK+ : ΩK+/Z → ΩK+/Z(log Γ+) is injective.(ii) Suppose moreover thatK+ has finite rank. Letpr := 0 ⊂ pr−1 ⊂ ... ⊂ p0 := mK be the

chain of all the prime ideals ofK+. Denote by∆r := ΓK ⊃ ∆r−1 ⊃ ... ⊃ ∆0 := 0 thecorresponding ascending chain of convex subgroups ofΓK (see(6.1.22)). ThenCoker βK+

admits a finite filtrationFil•(Coker βK+) indexed by the totally ordered setSpec(K+),such that :

grpi(Coker βK+) ≃ (∆i+1/∆i)⊗Z (K+/pi) for everypi ∈ SpecK+.

Proof. (i): Since the formation of differentials and logarithmic differentials commutes withcolimits of Z-algebras and log structures, we can reduce to the case whereK is a field of finitetype over its prime field. In this case the convex rank ofΓ is finite, so the assertion follows fromproposition 6.4.15 and an easy induction.

(ii): is a straightforward consequence of proposition 6.4.15.

6.5. Transcendental extensions.In this section we extend the results of section 6.3 to the caseof arbitrary extensions of valued fields.


6.5.1. We fix the following notation throughout this section. For a valued field extension(K, | · |) ⊂ (E, | · |E), we let

ρE+/K+ : E+ ⊗K+ ΩK+/Z(log Γ+)→ ΩE+/Z(log Γ+E)

be the natural morphism. One of the main results of this section states thatCoker(ρE+/K+) isinjective with torsion-free cokernel whenK is algebraically closed (theorem 6.5.20) or whenKhas characteristic zero (lemma 6.5.12). Lurking behind theresults of this sections there shouldbe some notion of ”logarithmic cotangent complex”, which however is not currently available.

6.5.2. LetG := (G, j,N,≤) be a datum as in (6.1.5). We wish to study the total log structureof the valued field(K(G), | · |G). We consider the morphism of monoids

G→ K[G] : g 7→ [g].(6.5.3)

Let K[G]+ be the subring of the elementsx ∈ K[G] such that|x|G ≤ 1. Let π : G → ΓG

be the projection; for every submonoidM ⊂ Γ+G, the preimageπ−1M is a submonoid ofG,

and the restriction of (6.5.3) induces a morphism of monoidsπ−1(M) → K[G]+, whence apre-log structureπ−1MX onX := SpecK[G]+ (see (6.4.2)). To ease notation, we setM :=(π−1MX)log and we will writeΩX/Z(logM) for the associated sheaf of log differentials.

Lemma 6.5.4. Resume the notation of(6.1.5). Then the natural diagram

K× ⊗Z K[G]+α //

β

G⊗Z K[G]+

η

ΩK+/Z(log Γ+)⊗K+ K[G]+ // ΩK[G]+/Z(log Γ+G)

is cocartesian.

Proof. Let P be the push out ofα andβ. We already have a mapφ : P → ΩK[G]+/Z(log Γ+G),

and by inspecting the definition ofK[G]+ one verifies easily thatφ is surjective; thus we needonly find a left inverse forφ. Let us remark also thatβ, and consequentlyη, is surjective, henceit suffices to exhibit:

(a) a derivationδ : K[G]+ → P such thatη(δa) = da for everya ∈ K[G]+;(b) aZ-linear mapψ : (π−1Γ+

G)gp → G such thatη ψ(γ) = d log(g) for everyg ∈ π−1Γ+G.

Of course we can take forψ the natural identification(π−1Γ+G)gp ∼→ G. To defineδ, choose

arbitrarily a set of representatives(gγ | γ ∈ G/K×) for the classes ofG/K×. Then everya ∈ K[G] can be written in a unique way as aK-linear combinationa =

∑γ∈G/K× aγ · [gγ];

we defineδ′ : K[G]+ → G⊗Z K[G]+ by the rule:a 7→∑γ∈G/K× gγ ⊗ aγ . It is easy to check

that the image ofδ′(a) in P does not depend on the choices of representatives, and this definesour sought derivationδ.

6.5.5. LetK(G)+ be the valuation ring of the valuation| · |G. It is easy to see thatK(G)+ =K[G]+p , wherep is the ideal of elementsx ∈ K[G] such that|x|G < 1. It then follows fromlemma 6.4.12(i),(iii) that the diagram of lemma 6.5.4 remains cocartesian when we replaceeverywhereK[G]+ byK(G)+.

Proposition 6.5.6. Suppose thatK is algebraically closed, let(E, | · |E) be a purely transcen-dental valued field extension of(K, | · |) with tr.deg(E : K) = 1. Then:

(i) ΩE+/K+ is a torsion-freeE+-module andHi(LE+/K+) = 0 for everyi > 0.(ii) The natural map ofE+-modules:ΩK+/Z(log Γ+)⊗K+ E+ → ΩE+/Z(log Γ+

E) is injectivewith torsion-free cokernel.


(iii) Suppose thatΓE = Γ. Then the natural diagram

ΩK+/Z ⊗K+ E+ //

ΩE+/Z

ΩK+/Z(log Γ+)⊗K+ E+ρE+/K+

// ΩE+/Z(log Γ+E)

is cocartesian.

Proof. Let mE be the maximal ideal ofE+ andX ∈ E such thatE = K(X). Following(6.1.10), we distinguish two cases, according to whether there exists or there does not exist anelementa ∈ K which minimizes the functionK → ΓE : b 7→ |X − b|E. Suppose firstthat such an element does not exist. We pick a net(ai | i ∈ I) satisfying the conditions oflemma 6.1.9, relative to the elementx := X. For a givenb ∈ K, choosei ∈ I such that|X − ai|E < |X − b|E; then we have|ai − b| = |X − b|E and it follows easily thatΓE = ΓKin this case. Then, for everyi ∈ I we can findbi ∈ K such that|X − ai|E = |bi|. Letf(X)/g(X) ∈ E+ be the quotient of two elementsf(X), g(X) ∈ K[X]. By lemma 6.1.9,we have|f(X)/g(X)|(ai,|bi|) ≤ 1 andγ := |g(X)|E = |g(X)|(ai,|bi|) for every sufficiently largei ∈ I. Picka ∈ K such that|a| = γ. Arguing as in the proof of case (b) of proposition 6.3.11(i),we deduce thata−1 · g(X), a−1 · f(X) ∈ Ai := K+[(X − ai)/bi], and if we letpi := Ai ∩mE+ ,thenf(X)/g(X) ∈ E+

i := Ai,pi. It is also easy to see that the family of theK+-algebrasE+

i isfiltered by inclusion. ClearlyΩE+

i /K+ is a freeE+

i -module of rank one, andHi(LE+i /K

+) = 0

for everyi > 0, so (i) follows easily in this case. Notice that, sinceΓ = ΓE, the log structureΓ+E

on SpecE+ (notation of (6.4.14)) is the log structure associated to the morphism of monoids(K+) \ 0 → E+. It follows easily that, for everyi ∈ I, we have a cocartesian diagram

ΩK+/Z ⊗K+ Aiαi //

ΩAi/Z

ΩK+/Z(log Γ+)⊗K+ Ai // ΩAi/Z(log Γ+)

(6.5.7)

where moreover,αi is split injective; the diagram of (iii) is obtained from (6.5.7), by localizingat p and taking colimits over the familyI; since both operations preserve colimits, we get (ii)and (iii) in this case. Finally, suppose that there exists anelementa ∈ K such that|X − a| isminimal; we can replaceX byX−a, and thus assume thata = 0. By (6.1.10) it follows that|·|Eis a Gauss valuation; then this case can be realized as the valuation| · |G associated to the datumG := (K×⊕Z, j, N,≤), wherej is the obvious imbedding, andN is eitherZ or 0, dependingon whether|X|E ∈ Γ or otherwise. In either case, (6.5.5) tells us that the map of(ii) is splitinjective, with cokernel isomorphic toE+, so (ii) holds. Suppose first that|X|E ∈ Γ. Thenwe can findb ∈ K such that|X/b|E = 1, and one verifies easily thatE+ is the localization ofA := K+[X/b] at the prime idealmK ·E+. Clearly (6.5.7) remains cocartesian when we replaceAi by A andαi by the corresponding mapα; the latter is still split injective, so (iii) followseasily. (i) is likewise obvious in this case. In case|X|E /∈ Γ, we distinguish three cases. First,suppose that|X|E < |b| for everyb ∈ K×. ThenK+[X/b] ⊂ E+ for everya ∈ K×, and indeedit is easy to check thatE+ is the filtered union of itsK+-subalgebras of the formK+[X/b]pb

,wherepb is the prime ideal generated bymK andX/b. Again (i) follows. The second case, when|X|E > |b| for everyb ∈ K×, is reduced to the former, by replacingX with X−1. It remainsonly to consider the case where there exista0, b0 ∈ K such that|a0| < |X|E < |b0|; then wecan find a net(ai, bi | i ∈ I) consisting of pairs of elements ofK×, such that|ai| < |X|E < |bi|for everyi ∈ I, and moreover, for everya, b ∈ K× such that|a| < |X| < |b|, there existsi0 ∈ Iwith |a| < |ai| and|bi| < |b| wheneveri ≥ i0. In such a situation, one verifies easily thatE+

is the filtered union of itsK+-subalgebras of the formE+i := K+[ai/X,X/bi]pi

, wherepi is


the prime ideal generated bymK and the elementsai/X, X/bi. Each ringE+i is a complete

intersectionK+-algebra, isomorphic toK+[X, Y ]/(X · Y − ai/bi). It follows thatLE+i /K

+ is

acyclic in degrees> 0, andΩE+i /K

+ ≃ X · E+i ⊕ Y ⊕ E+

i /(XdY + Y dX). We leave to the

reader the verification that thisE+i -module is torsion-free.

Theorem 6.5.8.Let (K, | · |) ⊂ (E, | · |E) be any extension of valued fields. Then

(i) Hi(LE+/K+) = 0 for everyi > 1 andH1(LE+/K+) is a torsion-freeE+-module.(ii) If K is perfect, thenHi(LE+/K+) = 0 for everyi > 0.

Proof. Let us show first how to deduce (ii) from (i). Indeed, we reduceeasily to the case whereE is finitely generated overK. Then, ifK is perfect, we can find a subextensionF ⊂ E whichis purely transcendental overK, and such thatE is separable overF ; by transitivity, we deducethatLE/K ≃ E⊗F LF/K ; moreoverHi(LF/K) = 0 for i > 0; by (i) we know thatH1(LE+/K+)imbeds intoH1(LE+/K+)⊗E+ E ≃ H1(LE/K), so the assertion follows.

To show (i), let| · |Ea be a valuation on the algebraic closureEa of E, which extends| · |E;recall thatEa+ is a faithfully flatK+-module by remark 6.1.12(ii). We apply transitivity to thetowerK+ ⊂ E+ ⊂ Ea+ to see that the theorem holds for the extension(K, | · |) ⊂ (E, | · |E)if and only if it holds for (K, | · |) ⊂ (Ea, | · |Ea) and for (E, | · |E) ⊂ (Ea, | · |Ea). Forthe latter extension the assertion is already known by theorem 6.3.28(i), so we are reduced toprove the theorem for the case(K, | · |) ⊂ (Ea, | · |Ea). Similarly, we apply transitivity to thetowerK+ ⊂ Ka+ ⊂ Ea+ to reduce to the case where bothK andE are algebraically closed.Then we can writeE as the filtered union of the algebraic closuresEa

i of its finitely generatedsubfieldsEi, thereby reducing to prove the theorem for the extensionsK ⊂ Ea

i ; hence we canassume thattr.d(E : K) is finite. Again, by transitivity, we further reduce to the case wherethe transcendence degree ofE overK equals one. In this case, we can pick an elementX ∈ Etranscendent overK, and writeE = K(X)a. Using once more transitivity, we reduce to showthe assertion for the purely transcendental extensionK ⊂ K(X), in which case proposition6.5.6(i) applies, and concludes the proof.

Lemma 6.5.9. LetR→ S be a ring homomorphism.

(i) Suppose thatFp ⊂ R, denote byΦR : R → R the Frobenius endomorphism ofR, anddefine similarlyΦS. LetR(Φ) := Φ∗RR andS(Φ) := Φ∗SS (cp. (3.5.7)). Suppose moreoverthat the natural morphism :

R(Φ)

L

⊗R S → S(Φ)

is an isomorphism inD(R-Mod). ThenLS/R ≃ 0 in D(s.S-Mod).(ii) Suppose thatS is a flatR-algebra and letp be a prime integer,b ∈ R a non-zero-divisor

such thatp · R ⊂ bp · R. Suppose moreover that the Frobenius endomorphisms ofR′ :=R/bp · R andS ′ := S/bp · S are surjective. Then the natural morphism :

LS/R → LS[b−1]/R[b−1]

is an isomorphism inD(s.S-Mod).

Proof. (i): Let P • := P •R(S) be the standard simplicial resolution ofS by freeR-algebras. ThenLS/R ≃ ΩP •/R ⊗P • S. Let ΦP • : P • → P •(Φ) be the termwise Frobenius endomorphism of thesimplicial algebraP •. As usual, we can writeΦP • = (ΦR ⊗R 1P •) ΦP •/R, where the relativeFrobeniusΦP •/R : R(Φ) ⊗R P • → P •(Φ) is a morphism of simplicialR(Φ)-algebras. Concretely,if P k = R[Xi | i ∈ I] is a free algebra on generators(Xi | i ∈ I), then

ΦP k/R(Xi) = Xpi for everyi ∈ I.(6.5.10)


Under the assumption of the lemma,ΦP •/R is a quasi-isomorphism of simplicialR(Φ)-algebras.It then follows from [46, Ch.II, Prop.1.2.6.2] thatΦP •/R induces an isomorphism

R(Φ) ⊗R LS/R∼→ LS(Φ)/R(Φ)

≃ LS/R.(6.5.11)

However, (6.5.10) shows that (6.5.11) is represented by a map of simplicial complexes which istermwise the zero map, so the claim follows.

(ii): Under the stated assumptions, the Frobenius map induces an isomorphism ofR-algebras:R/b · R ∼→ R′(Φ) (resp. ofS-algebras:S/b · S ∼→ S ′(Φ)). Thus the map

S ′ ⊗R′ (R′)(Φ) → S ′(Φ) : x⊗ y 7→ ΦS′(x) · yis an isomorphism. Since moreoverS ′ is a flatR′-algebra, we see that the assumption of (i) issatisfied, whenceLS′/R′ ≃ 0. If we now tensor the short exact sequence0→ R→ R→ R′ →0 by LS/R, we obtain a distinguished triangle

LS/Rbp−→ LS/R → LS/R

L

⊗R R′ → σLS/R.

However,LS/R

L

⊗R R′ ≃ LS′/R′ by [46, II.2.2.1], so we have shown thatbp acts as an isomor-phism onLS/R. In other words,LS/R ≃ LS/R ⊗S S[b−1] ≃ LS[b−1]/R[b−1], as claimed.

Lemma 6.5.12.Let (K, | · |) ⊂ (E, | · |E) be an extension of valued fields and suppose thatQ ⊂ κ(K). Then the mapρE+/K+ of (6.5.1)is injective with torsion-free cokernel.

Proof. To begin with, let(F, | · |F ) be any valued field extension of(E, | · |E). We remark that:

ρF+/E+ (ρE+/K+ ⊗E+ 1F+) = ρF+/K+.(6.5.13)

Claim6.5.14. Suppose moreover thatE is an algebraic extension ofK. ThenρE+/K+ is anisomorphism.

Proof of the claim:Applying (6.5.13), withF := Ea, we reduce easily to prove the claim incaseE is algebraically closed. LetKsh be the field of fractions of the strict henselization ofK+

(which we see as imbedded inE+). Let j : SpecKsh+ → SpecK+ be the morphism inducesby the imbeddingK ⊂ Ksh; In view of lemma 6.2.5, the log structureΓ+

Ksh on SpecKsh+

(notation of (6.4.14)) equalsj∗Γ. Since moreoverKsh+ is local ind-etale overK+, we deducefrom 6.4.12(iii) thatρKsh+/K+ is an isomorphism. Then arguing as in the foregoing, we see thatit suffices to prove the claim for the case whenK = Ksh. Since everything in sight commuteswith filtered unions of field extensions, we can even reduce tothe case whereE is a finite(Galois) extension ofK. Then, by corollary 6.2.14, this case can be realized as the extensionassociated to some datumG := (G, j,N,≤), where moreoverG/K× is a finite torsion group.Since by assumptionQ ⊂ K[G]+, the claim follows by lemma 6.5.4 and (6.5.5).

Now, if K ⊂ E is an arbitrary extension, we can apply (6.5.13) withF := Ea and claim6.5.14 to the extensionE ⊂ Ea to reduce to the case whereE is algebraically closed. Thenwe can apply again claim 6.5.14 to the extensionK ⊂ Ka and (6.5.13) withE := Ka andF := E, to reduce to the case where alsoK is algebraically closed. Then, by the usual argumentwe reduce to the case of an extension of finite transcendence degree, and even to the case oftranscendence degree equal to one. We factor the latter as a tower of extensionsK ⊂ K(X) ⊂E, whereX is transcendental overK, henceE algebraic overK(X). So finally we are reducedto the caseE = K(X), in which case we conclude by proposition 6.5.6(ii).

Theorem 6.5.15.Let(K, |·|) ⊂ (E, |·|E) be an extension of valued fields, withK algebraicallyclosed. ThenΩE+/K+ is a torsion-freeE+-module.


Proof. Pick a valuation| · |Ea of the fieldEa extending| · |E. We have an exact sequence:H1(LEa+/E+) → Ea+ ⊗E+ ΩE+/K+ → ΩEa+/K+, where the leftmost term is torsion-free bytheorem 6.3.28, so it suffices to show thatΩEa+/K+ is torsion-free, and we can therefore assumethatE is algebraically closed. In this case, if nowchar(K) > 0, it follows that the Frobeniusendomorphism ofE+ is surjective; then, for anya ∈ E+ we can writeda = d(a1/p)p =p · a(p−1)/p · da1/p = 0, so actuallyΩE+/K+ = 0. In casechar(K) = 0 andchar(κ(K)) =p, let us pick an elementb ∈ K+ such that|bp| ≥ |p|. SinceK andE are algebraicallyclosed, the Frobenius endomorphisms onK+/bpK+ andE+/bpE+ are surjective, soLE+/K+ ≃LE+[b−1]/K+[b−1] by lemma 6.5.9(ii). Now,K+[b−1] is the valuation ring of a valuation| · |′ onK, which extends to a valuation| · |′E onE+ whose valuation ring isE+[b−1]. Furthermore, theresidue fields of these valuations are fields of characteristic zero. Hence, we have reduced theproof of the theorem to the case whereκ(K) ⊃ Q. We can further reduce to the case wheretr.d(E : K) is finite andK is the algebraic closure of an extension of finite type of its primefield. By lemma 6.5.12 we have a commutative diagram with exact rows:

E+ ⊗K+ ΩK+/Z//

1E+⊗βK+

ΩE+/Z//

βE+

ΩE+/K+ //

γ

0

0 // E+ ⊗K+ ΩK+/Z(log Γ+) // ΩE+/Z(log Γ+E) // Coker(ρE+/K+) // 0

whereβK+ andβE+ are the maps of corollary 6.4.17(i). By virtue of lemma 6.5.12, it suffices toshow thatγ is injective. SinceβE+ is injective by corollary 6.4.17(i), the snake lemma reducesus to prove :

Claim 6.5.16. The induced mapE+ ⊗K+ Coker βK+ → Coker βE+ is injective.

Proof of the claim:Under our current assumptions,K+ andE+ are valuation rings of finiteKrull dimension, by (6.1.24). Letpr := 0 ⊂ pr−1 ⊂ ... ⊂ p0 := mE be the chain of all the primeideals ofE+. Denote by∆r := ΓE ⊃ ∆r−1 ⊃ ... ⊃ ∆0 := 0 the corresponding ascendingchain of convex subgroups ofΓE (see (6.1.22)). LetFil•(Coker βE+) (resp.Fil•(Coker βK+))be the finite filtration indexed by the totally ordered setSpecE+ (resp.SpecK+), provided bycorollary 6.4.17(ii). Since it is preferable to work with a single indexing set, we use the surjec-tion SpecE+ → SpecK+, to replace bySpecE+ the indexing of the filtration onCoker βK+;of course in this way some of the graded subquotients become trivial, but we do not mind. Withthis notation we can write down the identities:

E+ ⊗K+ grpi(Coker βK+) ≃ E+ ⊗K+ ((∆i+1 ∩ Γ)/(∆i ∩ Γ))⊗Z (K+/pi ∩K+)

for everypi ∈ Spec(E+). Furthermore, our mapφ : E+⊗K+Coker βK+ → Coker βE+ respectsthese filtrations. If nowpi ∈ Spec(E+) is the radical of the extension of a prime ideal ofK+,then clearly the mapgrpi(φ) : E+ ⊗K+ grpi(Coker βK+)→ grpi(Coker βE+) is induced by theimbeddings(∆i+1 ∩ Γ)/(∆i ∩ Γ) ⊂ ∆i+1/∆i andK+/pi ∩K+ ⊂ E+/pi, and it is thereforeinjective. On the other hand, ifpi is not the radical of an ideal extended fromK+, we havegrpi(Coker βK+) = 0, sogrpi(φ) is trivially injective in this case as well. Since the mapgr•(φ)is injective, the same holds forφ, which concludes the proof of the claim and of the theorem.

Theorem 6.5.17.Let | · |Ks be a valuation on the separable closureKs of K, extending thevaluation ofK. Then the mapρ := ρKs+/K+ is injective.

Proof. Suppose first thatΓ is divisible. By the usual reductions, we can assume thatK isfinitely generated over its prime field, hence that the convexrank ofΓKs is finite. We consider


the commutative diagram :

ΩK+/Z ⊗K+ Ks+ α //

βK+⊗1Ks+

ΩKs+/Z

βKs+

ΩK+/Z(log Γ+)⊗K+ Ks+ ρ // ΩKs+/Z(log Γ+Ks)

whereβK+ andβKs+ are the maps of corollary 6.4.17(i). By theorem 6.3.28(ii),α is injective,and the same holds forβKs+ , by corollary 6.4.17(i). It follows thatIm(βK+⊗K+1Ks+)∩Ker ρ =0, in other words, the induced map

Ker ρ→ Ks+ ⊗K+ Coker βK+(6.5.18)

is injective. By corollary 6.4.17(ii), there is a filtrationFil•(Coker βK+) onCoker βK+, indexedby SpecK+, such thatgrpi(Coker βK+) ≃ (∆i+1/∆i) ⊗Z (K+/pi), where∆i,∆i+1 are twoconvex subgroups ofΓK . However, since we assume thatΓ is divisible, the same holds for∆i+1/∆i; we deduce thatgrpi(Coker βK+) vanishes wheneverFrac(K+/pi) is a field of positivecharacteristic. What this means is that the filtrationFil•(Coker βK+) is actually indexed bySpecK+ ⊗Z Q ⊂ SpecK+, and the natural map :

CokerβK+ → Q⊗Z Coker βK+(6.5.19)

is an isomorphism. The same holds also forCoker βKs+ . If K+Q := K+ ⊗Z Q = 0, then

Coker βK+ = Coker βKs+ = 0, henceKer ρ = 0, which is what we had to show.In caseK+

Q 6= 0, thenK+Q is a valuation ring ofK with residue field of characteristic

zero. However, by lemma 6.4.12(iii) it follows easily thatQ ⊗Z Coker βK+ ≃ Coker βK+Q

,

and likewiseρKs+/K+ ⊗Z 1Q = ρKs+Q/K+

Q, whereKs+

Q := Ks+ ⊗Z Q is a valuation ring ofKs

whose valuation extends that ofK+Q . Since (6.5.19) is an isomorphism, (6.5.18) factors through

KerρKs+Q/K+

Q; however, the latter vanishes by lemma 6.5.12. Since (6.5.18) is injective, we

deriveKer ρ = 0, so the theorem holds in this case.In caseΓ is not necessarily divisible, let us choose a datumG := (G, j,N,≤) as in (6.1.5),

such thatG := (Ks)× ⊕ F , whereF is a torsion-free abelian group (whose composition lawwe write in multiplicative notation) andN is the graph of a surjective group homomorphismφ : F → (Ks)×. Notice that in this caseΓG ≃ ΓKs ≃ Γ ⊗Z Q, and the restriction toF ofthe projectionG → ΓG is the mapx 7→ |φ(x−1)|Ks. Let nowH := K× ⊕ F and define anew datumH := (H, j,H ∩ N,≤); sinceφ is surjective, clearly we still haveΓH ≃ Γ ⊗Z Q.Notice as well thatKs(G) is separable overK(H). SetρH := ρK(H)+/K+, ρG := ρKs(G)+/Ks+

andρG/H := ρKs(G)+/K(H)+ . We consider the diagram :

ΩK+/Z(log Γ+)⊗K+ Ks(G)+

ρH⊗1Ks(G)+

ρ⊗1Ks(G)+ // ΩKs+/Z(log Γ+Ks)⊗Ks+ Ks(G)+

ρG

ΩK(H)+/Z(log Γ+H)⊗K(H)+ K

s(G)+ρG/H // ΩKs(G)+/Z(log Γ+

G).

SinceF is torsion-free, it follows easily from (6.5.5) and lemma 6.5.4 thatρH andρG are injec-tive with torsion-free cokernels. Hence, in order to prove thatρ is injective, it suffices to showthatρKs(G)+/K(H)+ is. Finally, letE be the separable closure ofK(H) and choose a valuation onE which extends the valuation ofKs(G); we notice thatKerρKs(G)+/K(H)+ ⊂ KerρE+/K(H)+ .Therefore, we can replaceK by K(H) and reduce to the case whereΓ is divisible, which hasalready been dealt with.

Theorem 6.5.20.Let (K, | · |) ⊂ (E, | ⊂ |E) be an extension of valued fields, withK alge-braically closed. ThenρE+/K+ is injective with torsion-free cokernel.


Proof. By the usual arguments, we can suppose thatE is finitely generated overK. Let| · |Es be an extension of the valuation| · |E to the separable closureEs of E. We haveKerρE+/K+ ⊂ KerρEs+/K+, and evenCokerρE+/K+ ⊂ CokerρEs+/K+ , by theorem 6.5.17.Thus we can replaceE by Es and suppose thatE is separably closed, henceΓE divisible, byexample 6.1.21(ii). By corollary 6.4.17(i) we have a commutative diagram with exact rows :

0 // E+ ⊗K+ ΩK+/Z

α

// E+ ⊗K+ ΩK+/Z(log Γ+) //

ρE+/K+

E+ ⊗K+ Coker βK+//

γ

0

0 // ΩE+/Z// ΩE+/Z(log Γ+

E) // Coker βE+ // 0.

By theorem 6.5.8(ii), the mapα is injective. The same holds forγ, in view of claim 6.5.16. Itfollows already thatρE+/K+ is injective. Moreover, by theorem 6.5.15,Cokerα is a torsion-freeE+-module. Since bothΓ andΓE are divisible, it follows easily from corollary 6.4.17(ii)thatCokerβK andCokerβE areQ-vector spaces (cp. the proof of theorem 6.5.17), hence thesame holds forCoker γ. Consequently,CokerρE+/K+ is a torsion-freeZ-module, and thus weare reduced to show thatQ ⊗Z CokerρE+/K+ is a torsion-freeE+-module. However,Q ⊗Z

CokerρE+/K+ ≃ CokerρE+Q/K+

Q, whereE+

Q := E+ ⊗Z Q andK+Q := K+

Q are valuation ringswith residue fields of characteristic zero (or else they vanish, in which case we are done). Butthe assertion to prove is already known in this case, by lemma6.5.12.

Corollary 6.5.21. Let (K, | · |) be a valued field, andk a perfect field such thatk ⊂ K+. ThenΩK+/k andCokerρK+/k are torsion-freeK+-modules.

Proof. We haveka ⊂ Ksh+; letE := ka ·K ⊂ Ksh and denote byj : SpecE+ → SpecK+ themorphism induced by the imbeddingK ⊂ E. By lemma 6.2.5 the natural map :j∗Γ+ → Γ+

E

is an isomorphism of log structures; moreoverΩka/k = 0, sincek is perfect. HenceΩK+/k ⊂ΩE+/ka , and furthermore, by lemma 6.4.12(iii) we haveCokerρK+/k ⊂ CokerρE+/ka . Then theassertion follows from theorems 6.5.15 and 6.5.20.

Remark 6.5.22. Notice that corollary 6.5.21 is a straightforward consequence of a standard (asyet unproven) conjecture on the existence of resolution of singularities over perfect fields.

6.6. Deeply ramified extensions.We keep the notation of section 6.3. We borrow the notionof deeply ramified extension of valuation rings from the paper [21], even though our definitionapplies more generally to valuations of arbitrary rank.

Definition 6.6.1. Let (K, | · |) be a valued field,| · |Ks a valuation onKs which extends| · |. Wesay that(K, | · |) is deeply ramifiedif ΩKs+/K+ = 0. Notice that the definition does not dependon the choice of the extension| · |Ks.

Proposition 6.6.2. Let (K, | · |) be a valued field whose valuation has rank one,| · |Ks anextension of| · | toKs. Then the following conditions are equivalent :

(i) (K, | · |) is deeply ramified;(ii) The morphism of almost algebras(K+)a → (Ks+)a is weaklyetale;

(iii) (ΩKs+/K+)a = 0.

Moreover, the above equivalent conditions imply that the valuation ofK is not discrete.

Proof. We leave to the reader the verification that (iii) (and,a fortiori, (i)) can hold only in casethe valuation ofK is not discrete.

Let K+sh be a strict henselization ofK+ contained inKs+, andKsh its fraction field. It iseasy to check that(K, | · |) is deeply ramified if and only if(Ksh, | · |Ksh) is. Moreover, in view


of lemma 3.1.2(iv), condition (ii) holds for(K, | · |) if and only if it holds for(Ksh, | · |Ksh),and similarly for condition (iii). Hence we can assume thatK+ is strictly henselian. It is alsoclear that (i)⇒(iii)⇐(ii). To show that (iii)⇒(ii), let E ⊂ Ks be a finite separable extension ofK and setE+ := Ks+ ∩ E; it follows from theorem 6.3.20 that the natural mapΩE+/K+ ⊗E+

Ks+ → ΩKs+/K+ is injective; thus, if (iii) holds, we deduce that(ΩE+/K+)a = 0 for everyfinite separable extensionE of K. Again by theorem 6.3.20 we derive thatDE+/K+ = E+a forevery suchE. Finally, lemmata 6.3.10(i) and 4.1.29 show thatE+a is etale overK+a, whence(ii). Suppose next that the residue characteristic ofK+ is zero; then every finite extensionEof K factors as a tower of Kummer extensions of prime degree, thereforeΩE+/K+ = 0 bycorollary 6.3.18(i), which implies that (i)⇔(iii) in this case. Finally, suppose that the residuecharacteristic isp > 0. Let us choosea ∈ K+ such that|a| ≥ |p|. It follows easily fromexample 6.1.21(iii) that every elementx ∈ Ks+ can be written in the formx = yp + a · z forsomey, z ∈ Ks+. Hencedx = p · dyp−1 + a · dz, which means thatΩKs+/K+ = a · ΩKs+/K+.Therefore, even in this case we deduce (iii)⇒(i).

6.6.3. Let us say that aK+a-moduleM isK+a-divisible if, for everyx ∈ K+ \ 0 we haveM = x ·M .

Lemma 6.6.4. Let (K, | · |) be a valued field such thatQ ⊂ K, and letp := char(κ) > 0. Let(Ks, | · |Ks) be an extension of the valuation| · | to a separable closure ofK. Denote byT theKs+-torsion submodule ofΩKs+/Z. ThenT ≃ Ks/Ks+.

Proof. Let (Qa, | · |Qa) be the restriction of| · |Ks to the algebraic closure ofQ in Ks. Fromtheorem 6.5.20 it follows easily thatT ≃ ΩQa+/Z⊗Qa+Ks+, hence we can suppose thatK = Q.From theorem 6.3.20 we deduce that the natural mapQa+ ⊗E+ ΩE+/Z → ΩQa+/Z is injectivefor every subextensionE ⊂ Qa. For everyn ∈ N and every subextensionE ⊂ Qa, letEn := E(ζpn), whereζpn is any primitivepn-th root of1 and setE∞ :=

⋃n>0En.

Claim6.6.5. For every finite subextensionE ⊂ Qa, there existsn ∈ N such that the image ofE+n ⊗E+ ΩE+/Z in ΩE+

n /Zis included in the image ofE+

n ⊗Q+n

ΩQ+n /Z

.

Proof of the claim:For everyn ∈ N, E+n is a discrete valuation ring andκ(En) is a finite sepa-

rable extension ofκ(Q) = Fp; from the exact sequencemEn/m2En→ ΩE+

n /Z→ Ωκ(En)/Fp = 0

we deduce thatΩE+n /Z

is a (torsion) cyclicE+n -module. By comparing the annihilators of the

modules under consideration, one obtains easily the claim.

A standard calculation shows thatΩQ+∞/Z ≃ Q∞/Q+

∞. This, together with claim 6.6.5 impliesthe lemma.

Proposition 6.6.6. Keep the notation and assumptions of proposition6.6.2and suppose more-over that the characteristicp of the residue fieldκ ofK+ is positive and that the valuation onK is not discrete. Let(K∧, | · |∧) be the completion of(K, | · |) for the valuation topology. Thenthe following conditions are equivalent:

(i) (K, | · |) is deeply ramified;(ii) The Frobenius endomorphism ofK∧+/pK∧+ is surjective;(iii) For someb ∈ K+ \ 0 such that1 > |b| ≥ |p|, the Frobenius endomorphism on

(K+/bK+)a is an epimorphism;(iv) ΩK+/Z(log Γ+) is aK+-divisibleK+-module;(v) ΩK+/Z is aK+-divisibleK+-module;(vi) (ΩK+/Z)a is aK+a-divisibleK+a-module;

(vii) Coker ρKs+/K+ = 0 (notation of (6.5.1));(viii) Coker(ρKs+/K+)a = 0.


Proof. Suppose that (i) holds; then by proposition 6.6.2 it followsthat the morphism(K+)a →(Ks+)a is weakly etale, so the same holds for the morphism(K+/bK+)a → (Ks+/bKs+)a,for every b ∈ K+ with |b| ≥ |p|. In view of example 6.1.21(iii), one sees that the Frobe-nius endomorphism onKs+/bKs+ is an epimorphism. Using theorem 3.5.13(ii) we deducethat the Frobenius endomorphism on(K+/bK+)a is an epimorphism as well. This shows that(i)⇒(iii). To show that (iii)⇒(ii), let us chooseε ∈ m \ 0 such that|εp| > |b|; by hypoth-esis, for everyx ∈ K+ there existsy ∈ K+ such thatεp · x − yp ∈ bK+. It follows easilythat the Frobenius endomorphism is surjective onK+/(b · ε−p)K+. Replacingb by b · ε−pwe can assume that the Frobenius endomorphism is surjectiveon K+/bK+. Let b1 ∈ K+

such that1 > |bp1| ≥ |b|; we letFil•1(K+/pK+) (resp. Fil•2(K

+/pK+)) be theb1-adic (resp.bp1-adic) filtration onK+/pK+. The group topology onK+/pK+ defined by the filtrationsFil•i (K

+/pK+) (i = 1, 2) is the same as the one induced by the valuation topology ofK+;moreover, one verifies easily that the Frobenius endomorphism defines a morphism of filteredabelian groupsFil•1(K

+/pK+)→ Fil•2(K+/pK+) and that the associated morphism of graded

abelian groups is surjective. It then follows from [16, Ch.III, §2, n.8, Cor.2] that (ii) holds.Next suppose that (ii) holds; chooseb ∈ K+ such that1 > |b| > |bp| ≥ |p|; by hypothesis,the Frobenius endomorphism onK+/bpK+ is surjective; the same holds for the Frobenius maponKs+/bpKs+, in view of example 6.1.21(iii). Hence, the assumptions of lemma 6.5.9(ii) arefulfilled, and we deduce thatLKs+/K+ ≃ LKs+[b−1]/K+[b−1]. Now, if char(K) = p, this impliesalready thatΩKs+/K+ ≃ ΩKs/K = 0, which is (i). In casechar(K) = 0, we only deduce thatΩKs+/K+ ≃ ΩKs+[1/p]/K+[1/p]; however,R := K+[1/p] is a valuation ring of residue character-istic 0. We are therefore reduced to showing thatR is deeply ramified. Arguing as in the proofof proposition 6.6.2 we can even assume thatR is strictly henselian, in which case the assertionfollows from corollary 6.3.18(i).

Furthermore, (ii) implies easily thatΩ(K+/bpK+)/Z = 0 (sincedxp = p · dxp−1 = 0). LetI := bpK+; it follows that the natural mapI/I2 → (K+/bpK+) ⊗Z ΩK+/Z is surjective,i.e.ΩK+/Z = bp ·ΩK+/Z +K+ · dbp ⊂ bp ·ΩK+/Z, which implies (v). Next, by corollary 6.4.17(ii),we haveΩK+/Z(log Γ+)/ΩK+/Z ≃ κ ⊗Z Γ, and this last term vanishes since (ii) implies thatΓ = Γp. This shows that (ii)⇒(iv) as well. Clearly (v)⇒(vi). Suppose that (vi) holds. We willneed the following :

Claim 6.6.7. ΩKs+/Z is aKs+-divisible module andC := Coker(ρKs+/K+) is aK+-torsionmodule (notation of (6.5.1)). Furthermore,Ca ≃ (ΩKs+/K+)a.

Proof of the claim:In view of example 6.1.21(iii),(Ks, | · |Ks) satisfies condition (iii), hence thefirst assertion follows from the implications (iii)⇔(ii)⇒(v), which have already been shown.Furthermore, it is clear thatΩKs+/K+ is a torsionK+-module and therefore the second asser-tion follows easily from corollary 6.4.17. The latter corollary also implies that(ΩK+/Z)a ≃ΩK+/Z(log Γ+)a, and similarly forΩKs+/Z, whence the third assertion.

Now, suppose first thatFp ⊂ K+; in this caseΩKs+/Z is a torsion-freeKs+-module accordingto corollary 6.5.21. Letb ∈ K+ be any element; by theorem 6.5.17 and snake lemma we deducethat theb-torsion submoduleC[b]a := Ker(Ca → Ca : x 7→ b · x) is isomorphic to the cokernelof the scalar multiplication byb on the moduleKs+ ⊗K+ ΩK+/Z(log Γ+); the latter vanishesby assumption, and by claim 6.6.7 we haveCa =

⋃b∈K+ C[b]a, whenceCa = 0, which is

equivalent to (i) by claim 6.6.7 and proposition 6.6.2.Finally, in caseK+ is of mixed characteristic, denote byT (resp. T ′) theK+-torsion sub-

module ofΩKs+/Z (resp. ofKs+ ⊗K+ ΩK+/Z) and defineT [b] (resp. T ′[b]) as itsb-torsionsubmodule, for anyb ∈ K+. The foregoing argument shows thatT a is isomorphic to theK+a-torsion submodule of(ΩKs+/Z(log Γ+

Ks))a, and similarly for(T ′)a; moreover, by snake lemmawe obtain a short exact sequence0→ T ′[b]a → T [b]a → C[b]a → 0 for everyb ∈ K+, whencea short exact sequence0 → (T ′)a → T a → Ca → 0. Under (vi),(T ′)a is a divisible module;


however, it is clear from lemma 6.6.4 that the only divisible(Ks+)a-submodules ofT a are0 andT a. Consequently, in light of claim 6.6.7 and proposition 6.6.2, in order to prove that (vi)⇒(i),it suffices to show that(T ′)a 6= 0. In turns, this is implied by the following :

Claim6.6.8. The image inΩK+/Z(log Γ) of d log(p) ∈ ΩQ+/Z(log Γ+Q), has annihilatorpK+.

Proof of the claim:(Of course,Q+ := Q+ ∩ Q). By theorem 6.5.17 it suffices to consider the

image ofd log(p) in ΩKs+/Z(log Γ+Ks). Then, by theorem 6.5.20, we reduce to consider the case

Ks = Q. Then, once more by theorem 6.5.17, it suffices to look at the annihilator ofd log(p) inΩQ+/Z(log Γ+

Q) itself, and the claim follows.Since (vi) is implied by both (iv) and (v), we deduce at once that all the conditions (i)-(vi) are

equivalent. Furthermore, it is clear from claim 6.6.7 and proposition 6.6.2 that both (vii) and(viii) are equivalent to (i), so the proposition follows.

Remark 6.6.9. By inspecting the proof of proposition 6.6.6, we see that theargument for(ii)⇒(i) still goes through for valued fields(K, | · |) of arbitrary rank and characteristicp > 0.

Lemma 6.6.10.Let (K, | · |) be a valued field andb ∈ K an element with0 < |b| < 1. Denoteby q(b) the radical of the idealbK+ and setp(b) :=

⋂r>0 b

r · K+. Thenp(b) and q(b) areconsecutiveprime ideals,i.e. there are no prime ideals strictly contained betweenp(b) andq(b). Equivalently, the ringW (b) := (K+/p(b))q(b) is a valuation ring of rank one and theimage ofb is topologically nilpotent in the valuation topology ofW (b).

Proof. It is easy to verify thatp(b) andq(b) are prime ideals, and using (6.1.22) one deducesthatW (b) is a valuation ring of rank one, which means thatp(b) andq(b) are consecutive,

Theorem 6.6.11.Let (K, | · |) be a valued field,(K∧, | · |∧) its completion. The followingconditions are equivalent :

(i) (K, | · |) is deeply ramified.(ii) For every valued extension(E, | · |E) of (K, | · |), for everyb ∈ K+ \ 0 and for every

i > 0 we haveHi(L(E+/bE+)/(K+/bK+)) = 0.(iii) For every pair of consecutive prime idealsp ⊂ q ⊂ K+, the valuation ring(K+/p)q is

deeply ramified.(iv) For every pair of convex subgroupsH1 ⊂ H2 ⊂ Γ, the quotientH2/H1 is not isomorphic

to Z, and moreover, ifp := char(κ) > 0, the Frobenius endomorphism onK∧+/pK∧+ issurjective.

Proof. To show that (ii)⇒(i), we takeE := Ks and we choose a valuation onKs extending| · |. Then, by arguing as in the proof of lemma 6.5.9(ii), we deduce from (ii) that the scalarmultiplication byb onΩKs+/K+ is injective. Since the latter is a torsionKs+-module, we deduce(i). To show (i)⇒(ii), we reduce first to the case whereE = Ea; indeed, let| · |Ea be avaluation onEa extending|·|E and suppose that the sought vanishing is known for the extension(K, | · |) ⊂ (Ea, | · |Ea); by transitivity, it then suffices to show :

Claim6.6.12. Hi(L(Ea+/bEa+)/(E+/bE+)) = 0 for everyi > 1.

Proof of the claim:By [46, II.2.2.1] we haveL(Ea+/bEa+)/(E+/bE+) ≃ LEa+/E+

L

⊗Ea+Ea+/bEa+,whence a spectral sequence

E2pq := TorE

a+

p (Hq(LEa+/E+), Ea+/bEa+)⇒ Hp+q(L(Ea+/bEa+)/(E+/bE+)).

SinceEa+/bEa+ is anEa+-module of Tor-dimension≤ 1, we see thatE2pq = 0 for every

p > 1; furthermore, by theorem 6.3.28(i), it follows thatE2pq = 0 wheneverp, q > 0, so the

claim follows.


Thus, we can suppose thatE is algebraically closed. A spectral sequence analogous to theforegoing computesHi(L(E+/bE+)/(Ka+/bKa+)), and using theorems 6.5.8(ii) and 6.5.15, wefind that the latter vanishes fori > 0. Consequently, by applying transitivity to the towerof extensionsK+/bK+ ⊂ Ka+/bKa+ ⊂ E+/bE+, we reduce to show the assertion for thecaseE = Ka. However, by example 6.1.21(iii) we haveKa+/bKa+ ≃ Ks+/bKs+ for everyb ∈ K+ \ 0, so we can further reduce to the caseE = Ks. In this case, one concludesthe proof by another spectral sequence argument, this time using assumption (i) and theorem6.3.28(ii) to show that the relevant termsE2

pq vanish.To show that (iii)⇒(iv), we consider two subgroupsH1 ⊂ H2 as in (iv); if c.rk(H2/H1) > 1,

then clearlyH2/H1 cannot be isomorphic toZ, so we can assume thatH1 andH2 are con-secutive, so that the corresponding prime ideals are too (see (6.1.22)). In this case, (iii) andproposition 6.6.2 show thatH2/H1 is not isomorphic toZ, which is the first assertion of (iv).To prove the second assertion, it will suffice to show the following :

Claim 6.6.13. Suppose thatp := char(κ) > 0 and that (iii) holds. Then, for everyb ∈ K+\0with 1 > |b| ≥ |p|, the Frobenius endomorphism onK+/bK+ is surjective.

Proof of the claim:For such ab as above, definep(b), q(b) andW (b) as in lemma 6.6.10; thenW (b) is a valuation ring of rank one, so it is deeply ramified by assumption (iii). Then, byproposition 6.6.6 it follows that the Frobenius endomorphism is surjective onW (b)/b ·W (b) ≃K+

q(b)/bK+q(b). We remark thatbK+

q(b) ⊂ K+; there follows a natural imbedding:K+/bK+q(b) ⊂

W (b)/b ·W (b), commuting with the Frobenius maps. It is then easy to deducethat the Frobe-nius endomorphism is surjective onK+/bK+

q(b). Moreover, by proposition 6.6.2, the valuationof W (b) is not discrete, hence its value group is isomorphic to a dense subgroup of(R,≥) (seeexample 6.1.21(vi)); therefore, by (6.1.22) and example 6.1.21(v), we deduce that there existsan elementc ∈ K+ such that|b| > |c3p| and|b| < |c2p|. These inequalities have been chosenso thatcpK+

q(b) ⊂ K+ and bK+q(b) ⊂ c2pK+

q(b), whencebK+q(b) ⊂ cpK+, and finally we con-

clude that the Frobenius endomorphism induces a surjection: K+/cK+ → K+/cpK+. We letFil•1(K

+/bK+) (resp.Fil•2(K+/bK+)) be thec-adic (resp.cp-adic) filtration onK+/bK+. The

foregoing implies that the Frobenius endomorphism inducesa morphism of filtered modulesFil•1(K

+/bK+) → Fil•2(K+/bK+) which is surjective on the associated graded modules; by

[16, Ch.III, §2, n.8, Cor.2] the claim follows.

Next, assume (iv) and letW := (K+/p)q, for two consecutive prime idealsp ⊂ q ⊂ K+. Byassumption the Frobenius map is surjective onK+/bK+, wheneverb ∈ K+ \0 and|b| ≥ |p|;we deduce easily that the Frobenius endomorphism is surjective onW/bW , which implies (iii),in view of proposition 6.6.6.

(i)⇒(iii): indeed, letp ⊂ q be as in (iii); we need to show that(K+/p)q is deeply ramified.After replacingK+ byK+

q we can assume thatq is the maximal ideal ofK+. The ringk+ :=K+/p is a valuation ring; letk := Frac(k+), | · |k the valuation onk corresponding tok+, and(k′, | · |k′) a finite separable valued extension of(k, | · |k). It suffices to show thatΩk′+/k+ = 0.

We havek′ ≃ k[X]/(f(X)) for some irreducible monic polynomialf(X) ∈ k[X]; let f(X) ∈K+

p [X] be a lifting off(X) to a monic polynomial. ThenE+ := K+p [X]/(f(X)) is the integral

closure ofK+ in the finite separable extensionE := Frac(E+) of K, andE+/pE+ ≃ k′, sothatE+ is a valuation ring, by lemma 6.1.13. Furthermore, the preimage ofk′+ in E+ is avaluation ringR of E with R∩K = K+ andR/pR ≃ k′+. From (i) and theorem 6.3.28(ii) wededuce thatΩR/K+ = 0, whenceΩk′+/k+ = 0 as required.

Finally we show that (iv) implies (i). We distinguish several cases. The case whenp :=char(K) > 0 has already been dealt with, in view of remark 6.6.9. Next suppose thatchar(κ) =0; we will adapt the argument given for the rank one case to prove corollary 6.3.18(i). As usual,we reduce to the case whereK is strictly henselian; it suffices to show thatΩE+/K+ = 0 for


every finite extension(E, | · |E) of K. ThenE factors as a tower of subextensionsE0 := K ⊂E1 ⊂ E2 ⊂ ... ⊂ En := E such thatEi+1 = Ei[b

1/lii ] for every i = 0, ..., n − 1, where

li := [Ei+1 : Ei] is a prime number andbi ∈ Ei such that|bi| /∈ li · ΓEi. It is easy to see that

assumption (iv) is inherited by every finite algebraic extension ofK, hence we can reduce tothe caseE = K[b1/l], with b := b1, l := l1. One verifies as in the proof of proposition 6.3.11(i)thatE+ consists of the elements of the form

∑l−1i=0 xi · bi/l such thatxi ∈ K and|xi · bi/l|E ≤ 1

for everyi = 0, ..., l − 1 and we have to show thatd(xi · bi/l) = 0 for everyi ≤ l − 1. We mayassume thati > 0, and up to replacingb by bi · xli, we can obtain thatb ∈ E+; we have then toverify thatdb1/l = 0. Definep(b), q(b) as in lemma 6.6.10, so thatp(b) andq(b) are consecutiveprime ideals, therefore(E+/p(b))q(b) is a deeply ramified rank one valuation ring; in particular,its value group is not discrete. Then, using (6.1.22) and example 6.1.21(v), we deduce that thereexists an elementc ∈ K+ such that|b| > |cl+1| and|b| < |cl|. We can writeb = x · cl for somex ∈ K+, whencedb1/l = c · dx1/l. However,|c| ≤ |b|(l−1)/l(l+1) ≤ |a1/l · b−1|l−1 = |x|(l−1)/l.Sincex(l−1)/l · dx1/l = 0, the claim follows. Finally, suppose thatp := char(κ) > 0 andchar(K) = 0. Arguing as in the previous case, we produce an elementb ∈ K+ such that|bp| > |p| and|bp+1| < |p|. The Frobenius map is surjective onK+/bpK+ by assumption, andonKs+/bpKs+ by example 6.1.21(iii), henceΩKs+/K+ ≃ ΩKs+[1/p]/K+[1/p], by lemma 6.5.9(ii).Now it suffices to remark thatK+[1/p] is a valuation ring with residue field of characteristiczero, so we are reduced to the previous case, and the proof is concluded.

Remark 6.6.14. By inspection of the proof, it is easy to check that condition(ii) of theorem6.6.11 is equivalent to the following. There exists a subsetS ⊂ K+ \ 0 such that the convexsubgroup generated by|S| := |s| | s ∈ S equalsΓK andHi(LE+/K+ ⊗K+ K+/s ·K+) = 0for every valued field extension(E, | · |E) of (K, | · |), everys ∈ S and everyi > 0.


7. ANALYTIC GEOMETRY

In this final chapter we bring into the picturep-adic analytic geometry and formal schemes.The first three sections develop a theory of theanalytic cotangent complex: we show how toattach a complexLan

X/Y to any morphism of locally finite typeφ : X → Y of formal schemesor of R.Huber’s adic spaces. This complex is obtained viaderived completionfrom the usualcotangent complex of the morphism of ringed spaces underlyingφ. We prove thatLan

X/Y con-trols theanalyticdeformation theory of the morphismφ, in the same way as the usual cotangentcomplex computes the deformations of the map of ringed spaces underlyingφ. We hope thereader will agree with us that these sections - though largely independent from the rest of themonograph - are not misplaced, in view of the prominence of the cotangent complex construc-tion throughout our work. Some of what we do here had been already anticipated in an appendixof Andre’s treatise [2, Suppl.(c)].

The main result of the remaining sections 7.4 and 7.5 is a kindof weak purity statementvalid for affinoid varieties over a deeply ramified valued field of rank one (theorem 7.4.17).The occurence of analytic geometry in purity issues (and inp-adic Hodge theory at large) israther natural; indeed, the literature on the subject is littered with indications of the relevanceof analytic varieties, and already in [64], Tate explicitlyasked for ap-adic Hodge theory in theframework of rigid varieties. We elect instead to use the language of adic spaces, introducedby R.Huber in [44]. Adic spaces are generalizations of Zariski-Riemann spaces, that had al-ready made a few cameo appearences in earlier works on rigid analytic geometry. Recall thatZariski-Riemann spaces were introduced originally by Zariski in his quest for the resolution ofsingularities of algebraic varieties. Zariski’s idea was to attach to any (singular, reduced andirreducible) varietyX defined over a fieldk, the ringed space(X,OX) defined as the projectivelimit of the cofiltered system of all blow-up mapsXα → X; so a point ofX is a compatiblesystemx := (xα ∈ Xα | Xα → X). It is easy to verify that, for any suchx, the stalkOX,x isa valuation ring dominating the image ofx in X. The strategy to construct a regular model forX was broken up in two stages : first one sought to show that for any point x ∈ X one can findsomexα underx which is non-singular in the blow-upXα. This first stage goes under the nameof local uniformization; translated in algebraic terms, this means that for every valuation ringvof the fieldk(X) of rational functions onX, there is a model ofk(X) on which the center ofv is a non-singular point (a model is a reduced irreduciblek-schemeY of finite type such thatk(Y ) = k(X)). Local uniformization and the quasi-compactness of the Zariski-Riemann spaceimply that there is a “finite resolving system”,i.e. a finite number of models such that everyvaluation ofk(X) has a non-singular center on one of them (notice that the local uniformizationof a pointxα “spreads around” to an open neighborhoodUα of xα in Xα, so we achieve uni-formization not just forx, but for all the valuations contained in the preimageU of Uα, whichis open inX).

The second step is to try to reduce the number of models in a finite resolving system; restrict-ing to open subvarieties one reduces this to the question of going from a resolving system ofcardinality two to a non-singular model. This is what Zariski called the “fundamental theorem”(see [68,§2.5]). Zariski showed local uniformization for all valuation rings containing the fieldQ, but for the fundamental theorem he could only find proofs in dimensions≤ 3.

Our theorem 7.4.17 is directly inspired by Zariski’s strategy : rather than looking at thesingularities of a variety, we try to resolve the singularities of an etale coveringY → X ofsmooth affinoid adic spaces over a deeply ramified non-archimedean fieldK (so the map issingular only on the special fibre of a given integral model defined on the valuation ring ofK). We assume thatX admits generically etale coordinatest1, ..., td ∈ OX(X), and the roleof Zariski’s X is played by the projective system of all finite coverings of the formXn :=


X[t1/n1 , ..., t

1/nd ], wheren ranges over the positive integers. We have to show that the etale

coveringYn := Xn ×X Y → Xn becomes “less and less” singular (that is, on its special fibre)asn grows; so we need a numerical invariant that quantifies the singularity of the covering :this is thediscriminantdY/X . The analogue of local uniformization is our proposition 7.4.13,whose proof uses the results of section 6.6. Next, in order toexploit the quasi-compactnessof the affinoid adic spacesYn, we have to show that the estimates furnished by proposition7.4.13 “spread around” in an open subset; to this aim we provethat the discriminant function issemicontinuous : this is the purpose of section 7.5. Notice that in our case we do not need ananalogue of Zariski’s difficult fundamental theorem; this is because the discriminant functionalways decreases whenn grows : what makes the problem of resolution of singularities muchharder is that a non-singular pointxα on a blow up modelXα may be dominated by singularpointsxβ on some further blow upXβ → Xα (so, for instance one cannot trivially reducethe cardinality of a resolving system by forming joins of thevarious models of the system).Besides, our spaces are not varieties, but adic spaces,i.e. morally we work only “at the levelof the Zariski-Riemann space” and we do not need - as for Zariski’s problem - to descend to(integral) models of our spaces; so we are dealing exclusively with valuation rings (or mildextensions thereof), rather than more complicated local rings.

In essence, this is the complete outline of the method; however, our theorem is - alas - muchweaker than Faltings’ and does not yield by itself the kind ofGalois cohomology vanishingsthat are required to deduce comparison theorems for the cohomology of algebraic varieties; weexplain more precisely the current status of the question in(7.5.27).

Throughout this chapter we fix a valued field(K, | · |) with valuation of rank one, completefor its valuation topology. As usual,m denotes the maximal ideal ofK+. We also leta be atopologically nilpotent element inK×.

7.1. Derived completion functor. Let A be a completeK+-algebra of topologically finitepresentation. For anyA-moduleM , we denote byM∧ the (separated)a-adic completion ofM .

Proposition 7.1.1. LetA be as in(7.1).

(i) Every finitely generatedA-module which is torsion-free as aK+-module, is finitely pre-sented.

(ii) A is a coherent ring.(iii) LetN be a finitely presentedA-module,N ′ ⊂ N a submodule. Then there exists an integer

c ≥ 0 such that

akN ∩N ′ ⊂ ak−cN ′(7.1.2)

for everyk ≥ c. In particular, the topology onN ′ induced by thea-adic topology onN ,agrees with thea-adic topology ofN ′.

(iv) Every finitely generatedA-module isa-adically complete and separated.(v) Every submodule of a freeA-moduleF of finite type is closed for thea-adic topology of

F .(vi) EveryA-algebra of topologically finite type is separated.

Proof. (i) is an easy consequence of [13, Lemma 1.2]. To show (ii), one chooses a presen-tationA := K+〈T1, ..., Tn〉/I for some finitely generated idealI, and then reduces to provethe statement forK+〈T1, ..., Tn〉, in which case it follows from (i). Next, letN , N ′ be asin (iii) and defineT to be theK+-torsion submodule ofN ′′ := N/N ′; clearly T is anA-submodule, and theA-moduleN ′′/T isK+-torsion-free, therefore is finitely presented by (i).SinceN is finitely generated, this implies thatM := Ker(N → N ′′/T ) is finitely gener-ated. Hence, there exists an integerc ≥ 0 such thatacM ⊂ N ′. If now k ≥ c, we haveakN ∩N ′ ⊂ akN ∩M = akM ⊂ ak−cN ′, which shows (iii). Next let us show:


Claim 7.1.3. Assertion (iv) holds for every finitely presentedA-module.

Proof of the claim:Let N be a finitely presentedA-module and choose a presentation0 →K → An → N → 0. By (ii), K is again finitely presented, and by (iii), the topology onKinduced by thea-adic topology onAn coincides with thea-adic topology ofK. Hence, aftertakinga-adic completion, we obtain a short exact sequence :0 → K∧ → An → N∧ → 0 (see[54, Th.8.1]). It follows that the natural mapK → K∧ is injective, which shows that the mapN → N∧ is surjective for every finitely presentedA-moduleN . In particular, this holds forK,whenceK ≃ K∧, andN ≃ N∧, as claimed.

Finally, letM be a submodule ofAn. By (iii), the topology onM induced byAn coincideswith thea-adic topology. Consequently, ifM is finitely presented, thenM is complete for thea-adic topology by claim 7.1.3, hence complete as a subspace of An, hence closed inAn. For anarbitraryM , defineM :=

⋃n>0(M : an); thenM is a submodule ofAn andA/M is torsion-

free as aK+-module, so it is finitely presented by (i), thereforeM is finitely presented by (ii).It follows thatacM ⊂ M for somec ≥ 0, whenceakAn ∩M = akAn ∩M for everyk ≥ c. Bythe foregoing,M is complete, soakAn ∩M is also complete, and finallyM is complete, henceclosed. This settles (v) and (iv) follows as well. (vi) is an immediate consequence of (v).

Lemma 7.1.4. LetA→ B be a map ofK+-algebras of topologically finite presentation. ThenB is of topologically finite presentation as anA-algebra. More precisely, ifφ : A〈T1, ..., Tn〉 →B is any surjective map,Kerφ is finitely generated.

Proof. By proposition 7.1.1(vi),B is complete and separated, hence we can find a surjectivemapφ : A〈T1, ..., Tn〉 → B. It remains to show thatKerφ is finitely generated for any suchφ.We can writeA := K+〈Tn+1, ..., Tm〉/I for some finitely generated idealI, and thus reduce tothe case whereA = K+ andφ : K+〈T1, ..., Tn〉 → B. We will need the following :

Claim 7.1.5. Let α : K+〈Y1, ..., Yr+s〉 → B be a surjective map andβ : K+〈Y1, ..., Yr〉 →K+〈Y1, ..., Yr+s〉 the natural imbedding. Suppose thatγ := α β is surjective as well. ThenKerα is finitely generated if and only ifKer γ is finitely generated.

Proof of the claim:For i = r+1, ..., r+ s, choosefi ∈ K+〈Y1, ..., Yr〉 such thatγ(fi) = α(Yi).We define a surjective mapδ : K+〈Y1, ..., Yr+s〉 → K+〈Y1, ..., Yr〉 by settingδ(Yi) := Yi fori ≤ r andδ(Yi) := fi for i > r. Clearlyγ δ = α. There follows a short exact sequence0 → Ker δ → Kerα → Ker γ → 0. However,Ker δ is the closure of the idealI generated byYi−fi for i = r+1, ..., r+s. By proposition 7.1.1(v), we deduce thatKer δ = I, and the claimfollows easily.

By hypothesis there is at least one surjectionψ : K+〈Y1, ..., Yr〉 → B with finitely gener-ated kernel. Letµ : B⊗K+B → B be the multiplication map and setθ := µ (φ⊗K+ψ) :K+〈T1, ..., Tn, X1, ..., Xk〉 → B. Applying twice claim 7.1 we deduce first thatKer θ is finitelygenerated, and then thatKerφ is too, as required.

Lemma 7.1.6. LetF be a flatA-module. Then:(i) F∧ is a flatA-module.(ii) For every finitely presentedA-moduleM , the natural map

M ⊗A F∧ → (M ⊗A F )∧(7.1.7)

is an isomorphism.

Proof. To begin with, we claim that the functorN 7→ (N ⊗A F )∧ is exact on the abeliancategory of finitely presentedA-modules. Indeed, letE := (0 → N ′ → N → N ′′ → 0) be anexact sequence of finitely presentedA-modules; we have to show that(E ⊗A F )∧ is still exact.ObviouslyE ⊗A F is exact, so the assertion will follow by [54, Th.8.1(ii)], once we know:


Claim7.1.8. The topology onN ′⊗A F induced by the imbedding intoN ⊗A F agrees with thea-adic topology.

Proof of the claim:By proposition 7.1.1(iii), we can findc ≥ 0 such that (7.1.2) holds. SinceFis flat, we derive

ak(N ⊗A F ) ∩ (N ′ ⊗A F ) ⊂ ak−c(N ′ ⊗A F )

which implies the claim.

(ii): clearly (7.1.7) is an isomorphism in caseM is a free module of finite type. For a generalM , one chooses a resolutionR := (An → Am → M → 0); by the foregoing, the sequence(R ⊗A F )∧ is still exact, so one concludes by applying the 5-lemma to the map of complexesR⊗A F∧ → (R ⊗A F )∧.

(i): we have to show that, for every injective map ofA-modulesf : N ′ → N , f ⊗A 1F∧ isstill injective. By the usual reductions, we can assume thatbothN andN ′ are finitely presented.In view of (ii), this is equivalent to showing that the induced map(N ′ ⊗A F )∧ → (N ⊗A F )∧

is injective, which is already known.

7.1.9. We will need to consider the left derived functor of thea-adic completion functor, whichwe denote:

D−(A-Mod)→ D−(A-Mod) : (K•) 7→ (K•)∧.(7.1.10)

As usual, it can be defined by completing termwise bounded above complexes of projectiveA-modules. However, the following lemma shows that it can also be computed by arbitrary flatresolutions.

Lemma 7.1.11.Letφ : K•1 → K•2 be a quasi-isomorphism of bounded above complexes of flatA-modules and denote by(K•i )

∧ the termwisea-adic completion ofK•i (i = 1, 2). Then theinduced morphism

(K•1 )∧ → (K•2)∧(7.1.12)

is a quasi-isomorphism.

Proof. SinceK•1 andK•2 are termwise flat, we deduce quasi-isomorphisms

φn : K•1,n := K•1 ⊗A A/anA→ K2,n := K•2 ⊗A A/anAfor every n ∈ N. The map of inverse system of complexes(K•1,n)n∈N → (K•2,n)n∈N canbe viewed as a morphism of complexes of objects of the abeliancategory(A-Mod)N of in-verse systems ofA-modules. As such, it induces a morphism(φn)n∈N in the derived categoryD((A-Mod)N), and it is clear that(φn)n∈N is a quasi-isomorphism. Let

R lim : D((A-Mod)N)→ D(A-Mod)

be the right derived functor of the inverse limit functorlim : (A-Mod)N → A-Mod. Weremark that, for everyj ∈ Z, the inverse systems(Kj

i,n)n∈N (i = 1, 2) are acyclic for the functorlim, since their transition maps are surjective. We derive thatR lim(K•i,n)n∈N ≃ (K•i )

∧, and,under this identification, the morphism (7.1.12) is the sameasR lim(φn)n∈N. Since the latterpreserves quasi-isomorphisms, the claim follows.

7.1.13. We denote byD−(A-Mod)∧ the essential image of the functor (7.1.10).

Corollary 7.1.14. (i) For any objectK• of D−(A-Mod), the natural morphism

(K•)∧ → ((K•)∧)∧


(ii) D−(A-Mod)∧ is a full triangulated subcategory ofD−(A-Mod).


Proof. Notice first that there are two natural morphisms as in (i), which coincide : namely,for any complexE in D−(A-Mod) one has a natural morphismuE : E → E∧; then onecan take either(uK)∧ or uK∧. Now, (i) is an immediate consequence of lemmata 7.1.11 and7.1.6. ClearlyD−(A-Mod)∧ is preserved by shift and by taking cones of arbitrary morphisms;furthermore, it follows from (i) that it is a full subcategory of D−(A-Mod).

We will need some generalities on pseudo-coherent complexes ofR-modules (for an arbitraryring R), which we borrow from [11, Exp.I]. In our situation, the definitions can be simplifiedsomewhat, since we are only concerned with sheaves over the one-point site that are pseudo-coherent relative to the subcategory of freeA-modules of finite type.

7.1.15. For givenn ∈ Z, one says that a complexK• of R-modules isn-pseudo-coherentifthere exists a quasi-isomorphismE• → K• whereE• is a complex bounded above such thatEi is a freeR-module of finite type for everyi ≥ n. One says thatK• is pseudo-coherentif itis n-pseudo-coherent for everyn ∈ Z.

7.1.16. LetK• be an-pseudo-coherent (resp. pseudo-coherent) complex ofR-modules, andF • → K• a quasi-isomorphism. ThenF • is n-pseudo-coherent (resp. pseudo-coherent) ([11,Exp.I, Prop.2.2(b)]). It follows that that the pseudo-coherent complexes form a (full) subcate-gory D(R-Mod)coh of D(R-Mod).

7.1.17. Furthermore, letX → Y → Z → X[1] be a distinguished triangle inD−(R-Mod).If X andZ aren-pseudo-coherent (resp. pseudo-coherent), then the same holds forY ([11,Exp.I, Prop.2.5(b)]).

Lemma 7.1.18.Let n, p ∈ N, K• a n-pseudo-coherent complexes inD≤0(R-Mod), andFp

one of the functors⊗pR, SympR, Λp

R, ΓpR defined in[46, I.4.2.2.6]. ThenLFp(K•) is an n-

pseudo-coherent complex.

Proof. It is well known thatFp sends freeR-modules of finite type to freeR-modules of finitetype. It follows easily that the assertion of the lemma can bechecked by inspecting the definitionof the unnormalized chain complex associated to a simplicial complex, and of the simplicialcomplex associated to a chain complex via the Dold-Kan correspondence. We omit the details.

7.1.19. LetK• be a pseudo-coherent complex. By ([11, Exp.I, Prop.2.7]) there exists a quasi-isomorphismE• → K• whereE• is a bounded above complex of freeR-modules of finitetype.

7.1.20. Suppose now thatR is coherent; then we deduce easily that a complexK• of R-modules is pseudo-coherent if and only ifH i(K•) is a coherentR-module for everyi ∈ Z andH i(K•) = 0 for every sufficiently largei ∈ Z ([11, Exp.I, Cor.3.5]). By proposition 7.1.1(iv) itfollows also thatD−(A-Mod)coh ⊂ D−(A-Mod)∧ for everyK+-algebraA as in (7.1).

7.1.21. LetA be as in (7.1) andM anA-module of finite presentation. We denote byM [0] thecomplex consisting of the moduleM placed in degree zero. Any finite presentation ofM canbe extended to a quasi-isomorphismE• → M [0], whereE• is a complex of freeA-modules offinite type andEi = 0 for i > 0 ([11, Exp.I, Cor.3.5(a)]). Together with proposition 7.1.1(iv),it follows easily that the natural morphismM [0]→ M [0]∧ is a quasi-isomorphism.

Lemma 7.1.22.LetA→ B be a map of completeK+-algebras of topologically finite presen-tation. Then, for every objectK• of D−(A-Mod), the natural morphism

(K•L

⊗A B)∧ → (K•∧L

⊗A B)∧



Proof. We can suppose thatK• is a complex of freeA-modules. Then we are reduced toshowing that, for every freeA-moduleF , the natural map(F ⊗A B)∧ → (F∧ ⊗A B)∧ is anisomorphism. We leave this task to the reader.

Definition 7.1.23. Let φ : A → B be a map of completeK+-algebras of topologically finitepresentation. TheB-module ofanalytic differentials relative toφ is defined asΩan

B/A := Ω∧B/A.Theanalytic cotangent complexof φ is the complexLan

B/A := (LB/A)∧. Directly on the defini-tion we derive a natural isomorphism

H0(LanB/A) ≃ Ωan

B/A(7.1.24)

Notice thatLanB/A is defined here via the standard resolutionPA(B)→ B, and it is therefore well

defined as acomplexofB-modules, not just as an object in the derived categoryD−(B-Mod)∧.This will be essential in order to globalize the construction to formal schemes, in (7.2.2), and toadic spaces, in definition 7.2.31.

The following lemma will be useful in section 7.4.

Lemma 7.1.25.LetA → B be a continuous map ofK+-algebras of topologically finite pre-sentation. The natural mapφB/A : ΩB/A → Ωan

B/A is surjective withK+-divisible kernel.

Proof. One writesB = B0/I with B0 := A〈T1, ..., Tn〉 andI ⊂ B0. Directly from the con-struction ofΩan

B0/Aone checks thatφB0/A is onto. Then, a little diagram chasing shows that

φB/A is onto as well, and yields a surjective mapB ⊗B0 KerφB0/A → KerφB/A. This allowsto reduce to the case whereB = B0. In this case,KerφB/A is generated by the terms of theform δ(f) := df −∑n

i=1(∂f/∂Ti) · dTi, wheref ranges over all the elements ofB0. For givenf ∈ B0, we can writef = f0 + af1, with f0 ∈ A[T1, ..., Tn], f1 ∈ B0. It follows easily thatδ(f) = δ(af1) = a · δ(f1), whence the claim.

Remark 7.1.26. In view of (7.1.24), lemma 7.1.25 is also implied by the following more gen-

eral observation. LetKB/A := Cone(ψB/A : LB/A → LanB/A)[1]; one has:KB/A

L

⊗B B/aB ≃ 0.

Indeed, directly on the definition ofLanB/A one sees thatψB/A

L

⊗B 1B/aB is an isomorphism.

Proposition 7.1.27.Letφ : A → B be a map of completeK+-algebras of topologically finitepresentation, and suppose thatφ is formally smooth for thea-adic topology. Then there is anatural quasi-isomorphism

LanB/A ≃ Ωan

B/A[0].

Proof. For everyn ∈ N, setAn := A/an ·A andBn := B/an ·B. The hypothesis onφ impliesthatφn := φ ⊗A 1An is of finite presentation and formally smooth for the discrete topology,therefore

LBn/An ≃ ΩBn/An[0] ≃ ΩB/A ⊗A An[0](7.1.28)

for everyn ∈ N. Moreover,φ is flat by [13, Lemma 1.6], henceLBn/An ≃ LB/A ⊗A An. Onthe other hand, for everyi ∈ Z there is a short exact sequence (cp. [67, Th.3.5.8])

0→ limn∈N

1H i−1(LB/A ⊗A An)→ H i(LanB/A)→ lim

n∈NH i(LB/A ⊗A An)→ 0.

In view of (7.1.28), the inverse system(H i−1(LB/A ⊗A An))n∈N vanishes fori 6= 1 and hassurjective transition maps fori = 1, hence itslim1 vanishes for everyi ∈ Z, and the claimfollows easily.


Proposition 7.1.29.Let φ : A → B be a surjective map of completeK+-algebras of topo-logically finite presentation. ThenLB/A is a pseudo-coherent complex, in particular it lies inD−(B-Mod)∧ andLB/A ≃ Lan

B/A.

Proof. First of all, notice that by lemma 7.1.4,B is of finite presentation, hence it is coherent asanA-module. LetP := PA(B) be the standard simplicial resolution ofB by freeA-algebras.We obtain a morphism of simplicialB-algebrasφ : B ⊗A P → B by tensoring withB theaugmentationP → B (hereB is regarded as a constant simplicial algebra). By the foregoing,P is pseudo-coherent, henceP ⊗A B lies in D(B-Mod)coh. Let J := Kerφ. The short exactsequence of complexes0 → J → P ⊗A B → B → 0 is split, thereforeJ is also pseudo-coherent. Recall that we have natural isomorphisms:J i/J i+1 ∼→ Symi

B(LB/A) for everyi ∈ N(whereJ0 := B ⊗A P andSym0

B(LB/A) := B) ([46, Ch.III,§3.3]). Furthermore, we have (seeloc.cit.) :

Hn(Ji) = 0 for everyn, i ∈ N such thati > n.(7.1.30)

We prove by induction onn thatLB/A is n-pseudo-coherent for everyn ≤ 1. If n = 1 there isnothing to prove. Suppose that the claim is known for the integern. It then follows by lemma7.1.18 thatJ i/J i+1 is n-pseudo-coherent for everyi > 0. However, it follows from (7.1.30)thatJ i isn-pseudo-coherent as soon asi > −n. Hence, by (7.1.17) (and an easy induction), wededuce thatJ i is n-pseudo-coherent for everyi ∈ N. HenceJ2[1] is (n− 1)-pseudo-coherent;if we now apply (7.1.17) to the distinguished triangleJ → LB/A → J2[1] → J [1], we deducethatLB/A is (n− 1)-pseudo-coherent.

Theorem 7.1.31.LetA → B → C be maps of completeK+-algebras of topologically finitepresentation. Then:

(i) LanB/A lies in D−(B-Mod)coh.

(ii) There is a natural distinguished triangle inD−(C-Mod) :

C ⊗B LanB/A → Lan

C/A → LanC/B → C ⊗B Lan

B/A[1].(7.1.32)

Proof. (i): by lemma 7.1.4 we can find a surjectionB0 := A〈T1, ..., Tn〉 → B from a topolog-ically freeA-algebra ontoB. If we apply transitivity to the sequence of mapsA → B0 → Band take the (derived) completion of the resulting distinguished triangle, we end up with thetriangle:

(B ⊗B0 LB0/A)∧ → LanB/A → Lan

B/B0→ (B ⊗B0 LB0/A)∧[1].

We know already from proposition 7.1.29 thatLB/B0 is pseudo-coherent, hence it coincides withLanB/B0

. Lemma 7.1.22 yields a quasi-isomorphism:(B ⊗B0 LB0/A)∧∼→ (B ⊗B0 Lan

B0/A)∧; in

view of proposition 7.1.27,LanB0/A

is a freeB0-module of finite rank in degree zero, in particularit is pseudo-coherent, so the same holds for(B ⊗B0 LB0/A)∧, and taking into account (7.1.17),the claim follows.

(ii): if we apply transitivity to the sequence of mapsA→ B → C, and then we complete thedistinguished triangle thus obtained, we obtain (7.1.32),except that the first term is replaced by(C ⊗B LB/A)∧, which we can also write as(C ⊗B Lan

B/A)∧, in view of lemma 7.1.22. However,by (i), Lan

B/A is pseudo-coherent, so it remains such after tensoring byC; in particularC⊗BLanB/A

is already complete, and the claim follows.

7.2. Cotangent complex for formal schemes and adic spaces.In this section we show howto globalize the definition of the analytic cotangent complex introduced in section 7.1. Weconsider two kinds of globalization : first we define the cotangent complex of a morphismf : X → Y of formal schemes locally of finite presentation overSpf K+; then we will definethe cotangent complex for a morphism of adic spaces locally of finite type overSpa(K,K+).


Lemma 7.2.1. Let X := SpfA be an affine formal scheme finitely presented overSpf K+. Foreveryf ∈ A, let D(f) := x ∈ X | f /∈ mx. The natural mapA→ Γ(D(f),OX) is flat.

Proof. SinceΓ(D(f),OX) is the a-adic completion ofAf , the lemma follows from lemma7.1.6(i).

7.2.2. Letf : X → Y be a morphism of formal schemes locally of finite presentation overSpfK+, and suppose thatY is separated. For every affine open subsetU ⊂ X, the smallcategoryFU of all affine open subsetsV ⊂ Y with f(U) ⊂ V , is cofiltered under inclusion (orelse it is empty). For everyV ∈ FU , OY(V ) is aK+-algebra of topologically finite presentation,hence the induced morphismOY(V ) → OX(U) is of the kind considered in definition 7.1.23.We set

L(U/Y) := colimV ∈F o

U

LanOX (U)/OY (V ).

Definition 7.2.3. The mappingU 7→ L(U/Y) defines a complex of presheaves on a cofinalfamily of affine open subsets ofX. By applying degreewise the construction of [26, Ch.0,§3.2.1], we can extend the latter to a complex of presheaves ofOX-modules onX. We definetheanalytic cotangent complexLan

X/Y of the morphismf : X → Y as the complex of sheavesassociated to this complex of presheaves (this means that weform degreewise the associatedsheaf, and we consider the resulting complex).

7.2.4. More generally, ifY is not necessarily separated, we can choose an affinoid coveringY =

⋃i∈I Ui and the construction above applies to the restrictionsVi := f−1(Ui) → Ui; since

the definition ofLanVi/Ui

is local onVi, one can then glue them into a single cotangent complexLan

X/Y.

Lemma 7.2.5. LetLX/Y denote the (usual) cotangent complex of the morphism

(f, f ♯) : (X,OX)→ (Y,OY)

of ringed spaces; there exists a natural map of complexes

LX/Y→ LanX/Y(7.2.6)

inducing an isomorphism

LX/Y⊗OXOX/a

nOX∼→ Lan

X/Y⊗OXOX/a

nOX(7.2.7)

for everyn ∈ N.

Proof. It suffices to construct (7.2.6) in caseY is affine. According to [46, Ch.II, (1.2.3.6)]and [46, Ch.II, (1.2.3.4)], the complexLX/Y is naturally isomorphic to the sheafification of thecomplex of presheaves defined by the rule:

U 7→ colimV ∈F o

U

LOX (U)/OY (V )

and then it is clear how to define 7.2.6. From the constructionit is obvious that (7.2.7) is anisomorphism.

It is occasionally important to know that bothLanX/Y and the morphism (7.2.6) are well defined

in the category of complexes ofOX-modules (not just in its derived category).


7.2.8. Furthermore, denote byΩanX/Y the sheaf ofanalytic relative differentialsfor the mor-

phismf, which is defined as(I /I 2)|X, whereI is the ideal defining the diagonal imbeddingX→ X×Y X. We see easily that there is a natural isomorphism

H0(LanX/Y) ≃ Ωan

X/Y.(7.2.9)

Proposition 7.2.10.Let f : X→ Y be a morphism of formal schemes locally of finite type overSpf(K+). Then:

(i) LanX/Y is a pseudo-coherent complex ofOX-modules.

(ii) If f is a closed imbedding, then(7.2.6)is a quasi-isomorphism.(iii) If f is formally smooth, then(7.2.9)induces a quasi-isomorphism:Lan

X/Y

∼→ ΩanX/Y[0].

Proof. (i): according to [11, Exp.I, Prop.2.1(b)], it suffices to show thatHi(LanX/Y) is a coherent

sheaf ofOX-modules for everyi ∈ N. To this aim, letU ⊂ X be an affine open subset such thatthe familyFU (notation of (7.2.2)) is not empty; pick anyV ∈ FU . After replacingX byU , wecan suppose thatU = X. SetA := Γ(X,OX) and letLi be the sheaf of coherentOX-modulesassociated to the coherentA-moduleLi := Hi(Lan

OX (X)/OY (V )) (cp. [26, Ch.I,§10.10.1], wherethis concept is discussed in the case of locally noetherian formal schemes). The assertion willbe an immediate consequence of theorem 7.1.31(i) and the following :

Claim 7.2.11. There is a natural isomorphism ofOX-modules:Li∼→ Hi(Lan

X/Y).

Proof of the claim:By the definition ofLanX/Y we deduce a natural morphism ofOX-modules:

α : Li → Hi(LanX/Y). It therefore suffices to show thatα induces an isomorphism on the stalks.

To this aim, we remark first that the natural mapLi → Hi(L(X/Y)) is an isomorphism. Indeed,it suffices to consider another open subsetV ′ ∈ FX with V ′ ⊂ V ; we haveLan

OY (V ′)/OY (V ) ≃0 by proposition 7.1.27, and then it follows by transitivity (theorem 7.1.31(ii)) that the mapLi → Hi(Lan

OX (X)/OY (V ′)) is an isomorphism. More generally, this argument shows that, forevery affine open subsetU ′ ⊂ X, the natural mapHi(Lan

OX (U ′)/OY (V )) → Hi(L(U ′/Y)) is an

isomorphism. However, on one hand we have(Li )x ≃ Li ⊗A OX,x. On the other hand, wehave ([26, Ch.0,§3.2.4]) :

Hi(LanX/Y)x ≃ colim

x∈U ′Hi(L(U ′/Y))(7.2.12)

where the colimit ranges over the setS of all affine open neighborhoods ofx in X. We canreplaceS by the cofinal subset of all open neighborhoods of the formD(f) (for f ∈ A suchthatf /∈ mx). Then, lemma 7.2.1, together with another easy application of transitivity allowsto identify the right-hand side of (7.2.12) withHi(L(X/Y))⊗AOX,x, so (i) follows. (ii) and (iii)are immediate consequences of proposition 7.1.29 and respectively proposition 7.1.27.

Proposition 7.2.13.Let Xf→ Y

g→ Z be two morphisms of formal schemes locally of finitepresentation overSpfK+. There is a natural distinguished triangle inD−(OX-Mod)

Lf∗LanY/Z→ Lan

X/Z→ LanX/Y→ Lf∗Lan

Y/Z[1].(7.2.14)

Proof. As explained in [46, Ch.II,§2.1], for every sequence of ring homomorphismsA→ B →C, the transitivity triangle is induced by a functorial exactsequence of complexesLC/B/A of flatC-modules (a ”true triangle” inloc. cit.). Suppose now thatA,B,C are completeK+-algebrasof topologically finite type; upona-adic completion, one deduces a true triangleL∧C/B/A. Then,to every sequence of affine open subsetsU ⊂ X, V ⊂ Y, W ⊂ Z such thatf(U) ⊂ V andg(V ) ⊂ W , one can associate the true triangleL∧OX (U)/OY (V )/OZ(W ); Since the construction isfunctorial in all arguments, one derives a presheaf of true triangles on a cofinal family of open


subsets ofX, which we can then sheafify in the usual manner. The resultingdistinguishedtriangle inD−(OX-Mod) gives rise to (7.2.14).

In the following we wish to define the cotangent complex of a morphism of adic spaces(studied in [44] and [45]). For simplicity, we will restrictto adic spaces of topologically finitetype overSpa(K,K+), which suffice for our applications. For the convenience of the readerwe recall a few basic definitions from [45].

7.2.15. Anf-adic ring is a topological ringA that admits an open subringA0 such that theinduced topology onA0 is pre-adic and defined by a finitely generated idealI ⊂ A0. As anexample, everyK-algebra of topologically finite type is an f-adic ring. A subringA0 with theabove properties is called aring of definitionfor A, andI is anideal of definition. One denotesbyA the open subring of power-bounded elements ofA.

7.2.16. LetA → B be complete f-adic rings andφ : A → B a ring homomorphism. Onesays thatφ is of topologically finite typeif there exist rings of definitionA0 ⊂ A andB0 ⊂ Bsuch thatφ(A0) ⊂ B0, the restrictionA0 → B0 factors through a quotient map (i.e. open andsurjective)A0〈T1, ..., Tn〉 → B0 andB is finitely generated overA · B0.

7.2.17. Anaffinoid ring is a pairA = (A⊲, A+) consisting of an f-adic ringA⊲ and a subringA+ ⊂ A⊲ which is open, integrally closed inA⊲ and contained in the subringA. A+ is calledthesubring of integral elementsof A.

7.2.18. ThecompletionA∧ of an affinoid ringA = (A⊲, A+) is the pair((A⊲)∧, (A+)∧) (itturns out that(A+)∧ is integrally closed in(A⊲)∧).

A homomorphismφ : (A⊲, A+) → (B⊲, B+) of affinoid rings is a ring homomorphismφ⊲ : A⊲ → B⊲ such thatφ(A+) ⊂ B+. One says thatφ is of topologically finite typeif φ⊲ is oftopologically finite type and there exists an open subringC ⊂ B+ such thatB+ is the integralclosure ofC, φ(A+) ⊂ C and the induced mapA+ → C is of topologically finite type. (cp.(7.2.16)).

7.2.19. For instance,(K,K+) is an affinoid ring, complete for its valuation topology; noticethat in this case we haveK+ = K. A complete f-adic ring of topologically finite type overK is the same as aK-algebra of topologically finite type. Furthermore, suppose thatA andB are complete f-adic rings of topologically finite type overK, and letf : A → B be acontinuous ring homomorphism. Then there is a unique subring of integral elementsB+ suchthatf : (A,A)→ (B,B+) is a morphism of affinoid rings of topologically finite type; namelyone must takeB+ := B. Especially,A+ := A is the only ring of integral elements ofA suchthat the affinoid ring(A,A+) is of topologically finite type over(K,K+) ([44, Prop.2.4.15]).

7.2.20. Given an arbitrary ringA, a valuationonA is a map| · | : A → Γ ∪ 0 whereΓ isan ordered abelian group whose composition law we denote multiplicatively, and the orderingis extended toΓ ∪ 0 as usual. Then| · | is required to satisfy the usual conditions, namely:|x · y| = |x| · |y| and|x+ y| ≤ max(|x|, |y|) for everyx, y ∈ A, and|0| = 0, |1| = 1.

7.2.21. Now, letA be an f-adic ring, and| · | : A → Γ ∪ 0 a valuation onA. For everyγ ∈ Γ, letUγ := α ∈ Γ | α < γ ∪ 0. We endowΓ ∪ 0 with the topology which restrictsto the discrete topology onΓ, and which admits(Uγ | γ ∈ Γ) as a fundamental system of openneighborhoods of0. We say that| · | is continuousif it is continuous with respect to the abovetopology onΓ∪0. One denotes byCont(A) the set of all (equivalence classes of) continuousvaluations onA. Givena, b ∈ A, let U(a/b) ⊂ Cont(A) be the subset of all valuations| · |such that|a| ≤ |b| 6= 0. Cont(A) is endowed with the topology which admits the collection(U(a/b) | a, b ∈ A) as a sub-basis. With this topology,Cont(A) is a spectral topological space(see [45, 1.1.13] for the definition of spectral space). In particular, this implies thatCont(A)


admits a basis of quasi-compact open subsets. Such a basis isprovided by therational subsets,defined as follows. A subsetU ⊂ Cont(A) is called rational if there existf1, ..., fn, g ∈ A suchthat the idealJ := f1A + ...+ fnA is open inA andU consists of all| · | ∈ Cont(A) such that|fi| ≤ |g| 6= 0 for everyi = 1, ..., n. (Notice that, since we have chosen to restrict to f-adic ringscontainingK, asking forJ to be an open ideal is the same as requiring thatJ = A). Givenf1, ..., fn, g ∈ A with the above property, we denote byR(f1/g, ..., fn/g) the correspondingrational subset.

7.2.22. IfA := (A⊲, A+), then one defines the subsetSpaA := | · | ∈ Cont(A⊲) | |a| ≤1 for everya ∈ A+ ⊂ Cont(A⊲). SpaA, endowed with the subspace topology, is called theadic spectrumof the affinoid ringA. SpaA is a pro-constructible subset ofCont(A), hence itis a spectral space too. Any continuous mapA → B of affinoid rings induces in the obviousway a continuous map on adic spectra:SpaB → SpaA.

7.2.23. For any affinoid ringA, one can endowX := SpaAwith a presheafOX of topologicalrings, as follows. First of all, for anyf1, ..., fn, g ∈ A⊲ as in (7.2.21), one defines an affinoidring A(f1/g, ..., fn/g), such thatA(f1/g, ..., fn/g)

⊲ := (A⊲)g andA(f1/g, ..., fn/g)+ is the

integral closure of the subringA[f1/g, ..., fn/g] in A(f1/g, ..., fn/g)⊲. If B ⊂ A⊲ is a ring

of definition andI ⊂ B an ideal of definition, letB(f1/g, ..., fn/g) be the subring of(A⊲)ggenerated byB andf1/g ,..., fn/g; we endowB(f1/g, ..., fn/g) with the pre-adic topologydefined by the idealI · B(f1/g, ..., fn/g); then the f-adic topology onA(f1/g, ..., fn/g)

⊲ isdefined to be the unique ring topology for whichB(f1/g, ..., fn/g) is a ring of definition. Next,letA〈f1/g, ..., fn/g〉 := A(f1/g, ..., fn/g)

∧ (cp. (7.2.18)). With this preliminaries, one sets:

OX(R(f1/g, ..., fn/g)) := A〈f1/g, ..., fn/g〉⊲.

In this way,OX is well defined on every rational subset. One can then extend the definition toan arbitrary open subset ofSpaA, following [26, Ch.0,§3.2.1]. It is not difficult to check that,for every open subsetU ⊂ SpaA, and everyx ∈ U , any valuation| · |x in the equivalence classx extends to the whole ofOX(U), hence to the stalkOX,x. One denotes byO+

X the sub-presheafdefined by the rule:O+

X(U) := f ∈ OX(U) | |f |x ≤ 1 for everyx ∈ U. In the cases ofinterest, the presheafOX is a sheaf (andO+

X is therefore a subsheaf). In such cases, one canshow that, for every rational subsetR(f1/g, ..., fn/g), the natural mapA〈f1/g, ..., fn/g〉+ →O+X(R(f1/g, ..., fn/g)) is an isomorphism of topological rings.This holds notably whenA⊲ is aK-algebra of topologically finite type. One calls the datum

(SpaA,OSpaA,O+SpaA) an affinoid adic space. General adic spaces are obtained as usual, by

gluing affinoids. Adic spaces form a category, whose morphismsf : X → Y are the morphismsof topologically locally ringed spaces(X,OX)→ (Y,OY ) which induce morphisms of sheavesf ∗O+

Y → O+X .

7.2.24. Letf : X → Y be a morphism of adic spaces. One says thatf is locally of finitetypeif for every x ∈ X there exist open affinoid subspacesU ⊂ X, V ⊂ Y such thatx ∈ U ,f(U) ⊂ V and the induced morphism of affinoid rings(OY (V ),O+

Y (V )) → (OX(U),O+X(U))

is of topologically finite type.

7.2.25. A morphismf : X → Y between adic spaces (defined overSpa(K,K+)) is calledsmooth(resp. unramified, resp. etale) if f is locally of finite type and if, for any affinoidring A, any idealI of A⊲ with I2 = 0 and any morphismSpaA → Y , the mappingHomY (SpaA,X)→ HomY (SpaA/I,X) is surjective (resp. injective, resp. bijective).


7.2.26. In [45,§1.9] it is shown how to associate functorially to every formal schemeX (saylocally of finite presentation overSpf K+) an adic spaced(X), together with a morphism oftopologically ringed spacesλ : d(X)→ X, characterized by a certain universal property whichwe won’t spell out here, but that includes the condition thatIm(OX → λ∗Od(X)) ⊂ O+

d(X). IfX = SpfA0 for aK+-algebraA0 of topologically finite type, thend(X) = SpaA, whereA isthe affinoid ring(A0 ⊗K+ K,A+), with A+ defined as the integral closure of the image ofA0

in A0 ⊗K+ K. Moreover,X is quasi-compact if and only ifd(X) is.

7.2.27. LetX be a formal scheme of finite presentation overSpf K+. The collectionCX ofall morphismsf : X′ → X of formal schemes of finite presentation overSpf K+ such thatd(f) is an isomorphism, forms a small cofiltered category (with morphisms given as usual bythe commutative diagrams). It is shown in [44,§3.9] that there is a natural isomorphism oftopologically ringed spaces

(d(X),O+d(X))

∼→ lim(X′→X)∈C

(X′,OX′).(7.2.28)

(Actually, the argument inloc.cit. is worked out only in the case of noetherian formal schemes,but it is not difficult to adapt it to the present situation).

7.2.29. Letf : A → B be a morphism of affinoid rings of topologically finite type over(K,K+) (especiallyA+ = A andB+ = B, see (7.2.19)). We letCf be the filtered familyconsisting of all the pairs(A0, B0) of K+-algebras of topologically finite presentation, suchthatA0 (resp.B0) is an open subalgebra ofA (resp. ofB) andf(A0) ⊂ B0. Theanalyticcotangent complexof the morphismf is the complex ofB+-modules

L+B/A := colim

(A0,B0)∈Cf

LanB0/A0

.

7.2.30. Letf : X → Y be a morphism of adic spaces locally of finite type overSpa(K,K+),and suppose thatY is separated. For every affinoid open subsetU ⊂ X, the small categoryFUof all affinoid open subsetsV ⊂ Y with f(U) ⊂ V , is cofiltered under inclusion (or else it isempty). For everyV ∈ FU , (OY (V ),OY (U)+) is an affinoid(K,K+)-algebra of topologicallyfinite type, hence the induced morphismOY (V )→ OX(U) is of the kind considered in (7.2.29).We set

L(U/Y ) := colimV ∈F o

U

L+OX(U)/OY (V ).

Definition 7.2.31. The mappingU 7→ L(U/Y ) defines a complex of presheaves on a cofinalfamily of affinoid open subsets ofX. By applying degreewise the construction of [26, Ch.0,§3.2.1], we can extend the latter to a complex of presheaves ofO+

X-modules onX. We define theanalytic cotangent complexL+

X/Y of the morphismf : X → Y as the complex of sheaves asso-ciated to this complex of presheaves (cp. definition (7.2.3)). We also setLan

X/Y := L+X/Y ⊗K+K.

The definition can be extended to the case of a morphismf : X → Y whereY is not necessarilyseparated: one argues as in (7.2.4).

7.2.32. LetX+ denote the ringed space(X,O+X) and define likewiseY +. Just as in the proof

of lemma 7.2.5, by inspecting the construction we obtain natural maps of complexes

LX+/Y + → L+X/Y LX/Y → Lan

X/Y .(7.2.33)


7.2.34. LetA,B be complete f-adic rings andA→ B a ring homomorphism of topologicallyfinite presentation. We refer to [45,§1.6] for the construction of auniversalA-derivation ofB, which is a continuousA-derivationd : B → Ωan

B/A from B to a complete topologicalB-moduleΩan

B/A, universal forA-derivationsB → M to complete topologicalB-modulesM .The construction ofΩan

B/A can be globalized to a sheaf of relative differentialsΩanX/Y for any

morphism of adic spacesX → Y locally of finite presentation. Then one checks easily that:

H0(LanX/Y ) ≃ Ωan

X/Y .(7.2.35)

Proposition 7.2.36.LetXf→ Y

g→ Z be two morphisms of adic spaces locally of finite typeoverSpa(K,K+). There is a natural distinguished triangle inD−(OX-Mod)

Lf ∗LanY/Z → Lan

X/Z → LanX/Y → Lf ∗Lan

Y/Z [1].(7.2.37)

Proof. Mutatis mutandis, this is the same as the proof of proposition 7.2.13, so we canleave thedetails to the reader.

Remark 7.2.38. In fact, the proof of proposition 7.2.36 shows that (7.2.37)is represented bya functorial true triangle (see [46, Ch.I,§3.2.4]). This will be important in the sequel, whenwe will need to compute the truncationτ[−1LX/Z of the analytic cotangent in the situationcontemplated in proposition 7.2.44.

Theorem 7.2.39.Let f : X → Y (resp. f : X → Y ) be a morphism of formal schemes(resp. of adic spaces) locally of finite presentation overSpf K+ (resp. locally of finite type overSpa(K,K+)) such that the induced morphismd(f) : d(X) → d(Y) is smooth (resp. such thatf is smooth). Then:

(i) LanX/Y⊗K+ K ≃ Ωan

X/Y[0]⊗K+ K in D−((OX⊗K+ K)-Mod).

(ii) LanX/Y ≃ Ωan

X/Y [0] in D−(OX-Mod).

Proof. After the usual reductions, both assertions come down to thefollowing situation. Wehave a map ofK+-algebras of topologically finite presentationφ : A0 → B0, such thatd(Spf φ) is a smooth morphism of affinoid adic spaces. We have to show thatLan

B0/A0⊗K+ K ≃

ΩanB0/A0

⊗K+ K[0] in D(B-Mod). We can writeB0 = C0/I0, whereC0 := A0〈T1, ..., Tn〉 andI0 is some finitely generated ideal. SetI := I0⊗K+ K,A := A0⊗K+ K and letn be a maximalideal ofC := C0 ⊗K+ K with I ⊂ n.

Claim 7.2.40. In is generated by a regular sequence of elements of the local ringCn.

Proof of the claim:Let p := n ∩A; p is a maximal ideal inA, and its residue fieldK ′ is a finiteextension ofK. Let n be the image ofn in C ⊗A K ′ andI ⊂ Od(Spf C0) the sheaf of idealscorresponding toI; the maximal idealn yields a point ind(Spf C0), which we denote byx(n).We have an isomorphism on then-adic completions:

(Ix(n))∧ ≃ (In)

∧.(7.2.41)

Moreover, there are natural maps :

Ix(n)/I2x(n) → n/n2 → Ωan

C/A ⊗C C/nand, by [14, Prop.2.5], there exists a set of generatorsg1, ..., gk for Ix(n) such that the imagesdg1, ..., dgk in Ωan

C/A ⊗C C/n are linearly independent; it follows that the imagesg1, ..., gk inn/n2 are also linearly independent. Due to (7.2.41), we can assume thatg1, ..., gk ∈ In, and thenit follows thatg1 ⊗ 1, ..., gk ⊗ 1 are the firstk elements of a regular system of parameters forthe regular local ringCn⊗A K ′. From lemma 7.1.6 it follows thatCn is a flatAp-module; then


by [28, Ch.0, Prop.15.1.16] we deduce thatg1, ..., gn is a regular sequence of elements ofCn, asrequired.

SetB := B0 ⊗K+ K; it follows from claim 7.2.40 and [46, Ch.III, Prop.3.2.4] thatCn ⊗CLC/B ≃ LCn/Bn

≃ In/I2n[1] for every maximal idealn, henceLC/B ≃ I/I2[1]. By proposition

7.1.29, we deriveLanC0/B0

⊗K+ K ≃ I/I2[1]. Finally, by theorem 7.1.31(ii) and proposition

7.1.27 we deduce an isomorphism inD−(B-Mod) :

LanB0/A0

⊗K+ K ≃ (0→ I/I2 → B ⊗C0 ΩanC0/A0

→ 0)

and the latter complex is quasi-isomorphic toΩanB/A[0] by [14, Prop.2.5].

Lemma 7.2.42.Letf : X → Y be a closed imbedding of adic spaces locally of finite type overSpa(K,K+). Then the natural morphism

LX/Y → LanX/Y

is an isomorphism inD(OX-Mod).

Proof. The question is local onX, so we can reduce to the case whereY = Spa(A,A) forsomeK-algebra of topologically finite type andX = Spa(B,B), whereB = A/I for someidealI ⊂ A. Let C be the filtered family of all triples of the form(U,A0, B0) whereU ⊂ Y isan affinoid open neighborhood ofX, A0 ⊂ OY (U), B0 ⊂ B are two openK+-subalgebrasof topologically finite presentation withIm(A0 → B) ⊂ B0.

Claim7.2.43. The familyC ′ ⊂ Cπ of all (U,A0, B0) such thatIm(A0 → B) = B0 is cofinal.

Proof of the claim:Let (U,A0, B0) be any triple inC . We can find finitely many elementsb1, ..., bk ∈ B0 such thatA0[b1, ..., bk] is dense inB0. Choose elementsb1, ..., bk ∈ A that liftthebi; the subsetU ′ := x ∈ U | |bi(x)| ≤ 1; i = 1, ..., k is an affinoid open neighborhoodof X and the topological closureA′0 of A0[b1, ..., bk] is aK+-algebra of topologically finitepresentation contained inOX(U ′); moreoverIm(A′0 → B) = B0, hence(U ′, A′0, B0) ∈ C .

Finally, let (U,A0, B0) ∈ C ′; by proposition 7.1.29 we haveLanB0/A0

≃ LB0/A0. After taking

colimits we deduce:colimX⊂U⊂Y

LanB/OY (U) ≃ colim

X⊂U⊂YLB/OY (U)

whereU runs over the cofiltered family of all open affinoid neighborhoods ofX in Y . Thelemma follows easily.

Proposition 7.2.44.LetXf→ Y

g→ Z be morphisms of adic spaces locally of finite type overSpa(K,K+), wheref is a closed imbedding with defining idealI ⊂ OY , andg is smooth.Then there is a natural isomorphism inD(OX-Mod)

τ[−1LanX/Z

∼→ (0→ (I /I 2)|Xd→ f ∗Ωan

Y/Z → 0)(7.2.45)

whered is the natural map.

Proof. From lemma 7.2.42 and [46, Ch.III, Cor.1.2.8.1] we deduce a natural isomorphism

τ[−1LanX/Y ≃ (I /I 2)|X [1].

Taking into account proposition 7.2.36, theorem 7.2.39, one can repeat the proof of [46, Ch.III,Cor.1.2.9.1], which yields a distinguished triangle:

f ∗H0(LY/Z)[0]→ τ[−1LX/Z → H1(LX/Y )[1]→(7.2.46)

such that the connecting morphism(I /I 2)|X ≃ H1(LX/Y ) → f ∗H0(LY/Z) ≃ f ∗ΩanY/Z is

naturally identified (up to sign) with the differential mapf 7→ df . There follows an isomor-phism such as (7.2.45); however, the naturality of (7.2.45)is not explicitly verified inloc.cit. :


in general, this kind of manipulations, when carried out in the derived category, do not lead tofunctorial identifications. The problem is that we can have morphisms of distinguished trian-gles:

A• //

f

B• //

g

C• //

h

A•[1]

f [1]

A′• // B′• // C ′• // A′•[1]

such that bothf andg are the zero maps, and yeth is not. The issue can be resolved if oneremarks that (7.2.46) is deduced from a transitivity triangle via proposition 7.2.36, and thus it isactually well defined in the derived category of true trianglesT(OX-Mod) (cp. remark 7.2.38and [46, Ch.I,§3.2.4]). It suffices then to apply the following

Claim 7.2.47. Let C be any abelian category,T := (0→ A• → B• → C• → 0) a true triangleof T(C ) andn ∈ Z an integer such thatH i+1(A•) = 0 = H i(C•) for everyi 6= n. Then thereis a natural isomorphism inT(C ):

T ≃ (0→ Hn+1(A•)[n+ 1]→ Cone(Hn(C•)[n]d→ Hn+1(A•)[n+ 1])→ Hn(C•)[n]→ 0)

whered is the connecting morphism of the long exact homology sequence associated toT .

Proof of the claim:Under the stated assumptions, the natural maps of complexesφ : τ[nC• → C

andψ : A• → τn+1]A• are quasi-isomorphisms; we setT ′ := ψ ∗ T ∗ φ (notation of (2.5.5).

Clearly the pullback and push out maps define a natural isomorphismT ′∼→ T in T(C ). Say

thatT ′ = (0 → A′• → B′• → C ′• → 0); sinceHn+1(C ′•) = 0, it follows that the complexT ′′ := τ[n+1T

′ := (0 → τ[n+1A′• → τ[n+1B

′• → τ[n+1C′• → 0) is again a true triangle of

T(C ), naturally isomorphic toT ′. Likewise,T ′′ is naturally isomorphic to the true triangleτn+1]T

′′, and by inspection, one sees that the latter has the expectedshape.

7.3. Deformations of formal schemes and adic spaces.

Lemma 7.3.1. Let (X,OX) be a formal scheme locally of finite presentation overSpf K+, letI be a coherentOX-module and(0 → I → O1 → OX → 0) an extension of sheaves ofK+-algebras onX. Then(X,O1) is a formal scheme locally of finite presentation overSpf K+.

Proof. We may and do assume thatX is an affine formal scheme overSpf K+; then the assertionfollows from claims 7.3.2 and 7.3.3 below.

Claim 7.3.2. O1(X) is a completeK+-algebra of topologically finite presentation.

Proof of the claim:SinceX is affine, we derive a short exact sequence0→ I (X)→ O1(X)→OX(X) → 0. SinceI is coherent, theOX(X)-moduleI (X) is finitely presented. LetT ⊂I (X) be theK+-torsion submodule; by proposition 7.1.1(i), it follows thatI (X)/T is finitelypresented, henceT is finitely generated overOX(X); likewise, theK+-torsion submoduleT ′ ofOX(X) is a finitely generatedOX(X)-module, and then theK+-torsion submoduleT1 of O1(X)is a finitely generatedO1(X)-module. LetN := Ker(O1(X) → OX(X)/T ′); in the usual waywe derive that there existsk0 ∈ N such that

akO1(X) ∩I (X) = akN ∩I (X) ⊂ ak−k0I (X) for everyk ≥ k0.

We deduce by [54, Th.8.1(ii)] a short exact sequence of completeK+-algebras0 → I (X) →O1(X)∧ → OX(X) → 0, which shows thatO1(X) is complete. SinceI (X) andOX(X) aretopologically of finite type, we can then find a continuous surjectionA := K+〈T1, ..., Tn〉 →O1(X); by lemma 7.1.4 it follows thatO1(X) is a finitely presentedA-algebra, henceI (X) isa finitely presentedA-module, and then the same holds forO1(X), so the claim is proved.


Claim7.3.3. For f ∈ OX(X), let D(f) ⊂ X be as in lemma 7.2.1. Then the natural map(O1(X)f)

∧ → O1(D(f)) is an isomorphism.

Proof of the claim:The existence of the said map is a consequence of claim 7.3.2;however, wehave as well a natural short exact sequence0→ I (X)∧f → O1(D(f))→ OX(X)∧f → 0, so theassertion is immediate.

The meaning of lemma 7.3.1 is that the square zero deformations ofX by I in the categoryof ringed spaces are the same as those in the category of formal SpfK+-schemes locally offinite presentation. Especially, the latter are classified by the appropriateExt-group of the(usual) cotangent complex of the map of ringed spacesX → Spf K+; we aim to show that thesame computation can be carried out with the analytic cotangent complexLan

X/K+ introduced indefinition 7.2.3.

7.3.4. LetT be a topos,I a small category; the categoryT I of all functors fromI to T is atopos in a natural way, and we define a functorc : T → T I by assigning to an objectX of T theconstant functorcX : I → T of valueX (socX(i) = X for every objecti of I, andcX(φ) = 1Xfor every morphismφ of I). The functorc admits a right adjointlim

I: T I → T , and the adjoint

pair (c, limI

) defines a morphism of topoiπT : T I → T . If φ : S → T is any morphism of topoi,

we obtain a commutative diagram of topoi:

SIπS //

φI

S

φ

T I

πT // T

whence two spectral sequences:

Epq2 := Rpφ∗ lim

i∈I

qFi ⇒ Rp+q(φ πS)∗(Fi | i ∈ I)F pq

2 := limi∈I

pRqφ∗Fi ⇒ Rp+q(φ πS)∗(Fi | i ∈ I)

for every abelian sheaf(Fi | i ∈ I) onSI .

Lemma 7.3.5. LetX be an adic formal scheme overSpf K+, F a coherentOX-module and setFn := F/anF for everyn ∈ N. Then(Fn | n ∈ N) defines an abelian sheaf onXN, and wehavelim

n∈N

qFn = 0 onX, for everyq > 0.

Proof. We apply the spectral sequences of (7.3.4) to the indexing categoryN and the morphismof topoiφ : U → SpfK+, whereU ⊂ X is any affine open subset. HenceF pq

2 = 0 wheneverq > 0, therefore the abutment is isomorphic toF p0

2 = limn∈N

p Fn(U); however, the inverse system

(Fn(U) | n ∈ N) is surjective, henceF p0 = 0 for p > 0, and it follows thatEpq∞ = 0 whenever

p+ q > 0 andE00∞ ≃ lim

n∈NFn(U) ≃ F (U). SinceU is arbitrary, the claim follows easily.

7.3.6. With the notation of (7.3.4), letO• := (Oi | i ∈ I) be a ring object of the toposT I

(briefly: aT I-ring), or which is the same, a functorO from I to the category ofT -rings. LetalsoOT be aT -ring, andπ♯T : OT → πT∗O• a morphism ofT -rings. Then the pair(πT , π

♯T )

defines a morphism of ringed topoiπT : (T I ,O•) → (T,OT ). The corresponding functorπT∗ = lim

I: O•-Mod→ OT -Mod admits a left adjointπ∗T , defined by the rule:

F 7→ π−1T F ⊗π−1

T OTO• = (F ⊗OT

Oi | i ∈ I).


(Of course,π−1T is the same as the functorc : T → T I of (7.3.4)). The adjoint pair(πT∗, π∗T )

extends to an adjoint pair of derived functors(RπT∗, Lπ∗T ); more precisely, for every complex

K•1 ∈ D−(OT -Mod) andK•2 ∈ D+(O•-Mod) there is a natural isomorphism (“trivial duality”)

HomD(O•-Mod)(π−1T K•1

L

⊗π−1T OT

O•, K•2) ≃ HomD(OT -Mod)(K

•1 , Rlim

IK•2 )

(see [46, Ch.III, Prop.4.6]).

Proposition 7.3.7. Let X be formal scheme locally of finite presentation overSpf K+ andFany coherentOX-module. Then the natural morphismLX/K+ → Lan

X/K+ induces isomorphisms

ExtiOX(Lan

X/K+,F )∼→ ExtiOX

(LX/K+,F ) for everyi ∈ N.

Proof. We endow the toposXN with the ring objectO• := (OX/anOX | n ∈ N); the natural

morphismOX → limN

O• determines a morphism of ringed topoiπ : (XN,O•) → (X,OX)

as in (7.3.6). By lemma 7.3.5, the natural mapF → RlimNπ∗F is an isomorphism and by

lemma 7.1.7, bothLX/K+ andLanX/K+ are complexes of flatOX-modules, hence the trivial duality

isomorphism reads

ExtiO•(π∗Lan

X/K+ , π∗F ) ≃ ExtiOX(Lan

X/K+,F ) for everyi ∈ N

and likewise forLX/K+. To conclude it remains only to remark thatπ∗LX/K+ ≃ π∗LanX/K+.

Lemma 7.3.1 and proposition 7.3.7 enable one to derive the standard results on nilpotentdeformations, along the lines of section 3.2 : the statements are unchanged, except that thetopos-theoretic cotangent complex is replaced by the (muchmore manageable) analytic one.The details will be left to the reader.

7.3.8. Letf : X → Y be a morphism of adic spaces locally of finite type overSpa(K,K+).We would like to show that the infinitesimal deformation theory of f is captured by the ana-lytic cotangent complexLan

X/Y , in analogy with the usual topos-theoretic situation, and with thetreatment for formal schemes already presented. This turnsout to be indeed the case, howeverthe proofs are rather more delicate than those for formal schemes, due to the existence of squarezero deformations of affinoid algebras that are not themselves affinoid. In homological terms,this reflects the fact that the natural map

Ext1OX(Lan

X/Y ,F )→ Ext1OX(LX/Y ,F )(7.3.9)

is not in general an isomorphism for arbitrary coherentOX-modulesF . Nevertheless, we havethe following:

Proposition 7.3.10.Letf : X → Y be as in(7.3.8). Then the map(7.3.9)is injective for everycoherentOX-moduleF .

Proof. Using the long exactExt sequence, we reduce to showing:

Claim 7.3.11. Ext0OX(Cone (LX/Y → Lan

X/Y ),F ) = 0.

Proof of the claim:Indeed, we have

Ext0OX(Cone (LX/Y → Lan

X/Y ),F )≃Ext0OX

(H0(Cone (LX/Y → LanX/Y )),F )

≃Ext0OX

(Coker(H0(LX/Y )→ H0(LanX/Y )),F )

≃Ext0OX

(Coker(ΩX/Y → ΩanX/Y ),F )

and the latter group vanishes, due to the universal propertyof ΩanX/Y (see (7.2.34)).


7.3.12. Letf : X → Y be a morphism of adic spaces locally of finite type overSpa(K,K+).An analytic deformationof X overY is a datum of the form(j,F , β) consisting of :(a) a closed imbeddingj : X → X ′ of Y -adic spaces, such that the idealI ⊂ OX′ defining

j satisfiesI 2 = 0, and(b) a coherentOX-moduleF with an isomorphism ofOX-modulesβ : j∗I

∼→ F .One defines in the obvious way a morphism of analytic deformations, and we letExanY (X,F )denote the set of isomorphism classes of analytic deformations ofX by F overY .

Proposition 7.3.13.LetX be an affinoid adic space of finite type overSpa(K,K+) and (j :X → X ′,F , β) an analytic deformation. ThenX ′ is affinoid of finite type overSpa(K,K+).

Proof. To start out, we notice thatj is a homeomorphism on the topological spaces underlyingX andX ′, henceOX′(X ′) = Γ(X, j∗OX′); we deduce a short exact sequence of continuousmaps:

0→ F (X)→ OX′(X ′)→ OX(X)→ 0.

Claim7.3.14. Let φ : B → A be a surjective map of completeK-algebras of topologicallyfinite type, such thatKerφ is a square zero ideal ofB. ThenB = φ−1(A).

Proof of the claim:Quite generally, letC be anyK-algebra of topologically finite type; accord-ing to [12,§6.2.3, Prop.1] one hasC = f ∈ C | |f(x)| ≤ 1 for everyx ∈ MaxC. In oursituation, it is clear thatMaxA = MaxB, whence the claim.

Claim7.3.15. OX′(X ′) is a completeK-algebra of topologically finite type overK.

Proof of the claim:Let g1, ..., gm be a set of topological generators forOX(X), and choosearbitrary liftingsgi, ..., gm ∈ OX′(X ′). Pick also a finite set of generatorsgm+1, ..., gn for theOX(X)-moduleF (X). We define a mapφ : K[T1, ..., Tn] → OX′(X ′) by the ruleTi 7→ gi(i = 1, ..., n). LetX ′ =

⋃k Uk be a covering ofX ′ by finitely many of its affinoid domains. We

deduce a mapψ : K[T1, ..., Tn] →∏

k OX′(Uk). By viewing0 → j∗F → OX′ → j∗OX → 0as a short exact sequence of coherentOX′-modules onX ′, we deduce short exact sequences0→ F (Uk)→ OX′(Uk)

πk→ OX(Uk)→ 0 for everyk. By the open mapping theorem (see [12,§2.8.1]) one deduces easily that the topology ofOX(Uk) is the same as the quotient topologydeduced from the surjectionπk, especiallyOX(Uk) is aK-algebra of topologically finite type.Since the images ofg1, ..., gm in OX(Uk) are power bounded for everyk, it then follows fromclaim 7.3.14 that the images ofg1, ..., gm in OX′(Uk) are power bounded for everyk. Henceψextends to a mapψ∧ : K〈T1, ..., Tn〉 →

∏k OX′(Uk), and by construction,ψ∧ factors through a

continuous mapφ∧ : K〈T1, ..., Tn〉 → OX′(X ′). It is easy to check thatφ∧ is surjective, so theclaim follows, again by the open mapping theorem.

SetA := OX′(X ′); by claim 7.3.15 we have the affinoid adic spaceX ′′ := Spa(A,A) of

finite type overSpa(K,K+). By construction we get morphismsXj→ X ′

α→ X ′′ of adic spacesinducing homeomorphisms on the underlying topologies. To conclude it suffices to show:

Claim7.3.16. The morphismα♯ : OX′′ → α∗OX′ is an isomorphism of sheaves of topologicalalgebras.

Proof of the claim:Using claim 7.3.22 we see that it suffices to show thatα♯ is an isomorphismof sheaves ofK-algebras. However, for any affinoid open domain inX ′′ we have a commutativediagram with exact rows:

0 // F (U) // OX′′(U) //

OX(U) // 0

0 // F (U) // OX′(U) // OX(U) // 0.


Since the affinoid open subsets ofX ′′ form a basis of the topology for bothX ′′ andX ′, theclaim follows.

7.3.17. In the situation of (7.3.12), let(j : X → X ′,F , β) be an analytic deformation ofXover Y . Clearly j is a homeomorphism on the underlying topological spaces, hence we canview the given deformation as the datum of a map off−1OY -algebrasOX′ → OX , whence atransitivity distinguished triangle (proposition 7.2.36)

Lj∗LanX′/S → Lan

X/S → LanX/X′ → Lj∗Lan

X′/S[1]

which in turns yields a distinguished triangle:

RHomOX(Lan

X/X′ ,F )→ RHomOX(Lan

X/S ,F )→ RHomOX′ (LanX′/S, j∗F )→ .

Especially, we get a map

Ext1OX

(LanX/X′ ,F )→ Ext1

OX(Lan

X/S ,F ).(7.3.18)

Let I ⊂ OX′ be the ideal that defines the imbeddingj; by proposition 7.2.44 we have a naturalisomorphismLan

X/X′

∼→ j∗I [1], whence an isomorphism

Ext1OX

(LanX/X′ ,F )

∼→ HomOX(j∗I ,F ).(7.3.19)

Combining (7.3.18) and (7.3.19) we see that the given isomorphismβ : j∗I → F determinesa unique elementean(X ′, β) ∈ Ext1

OX(Lan

X/S,F ). One verifies easily thatean(X ′, β) dependsonly on the isomorphism class of the analytic deformation(j,F , β), therefore it defines a map

ean : ExanY (X,F )→ Ext1OX

(LanX/S ,F ).

By inspecting the construction, it is easy to check thatean fits into a commutative diagram

ExanY (X,F )ean //

Ext1OX(Lan

X/S ,F )

Exalf−1OY(OX ,F )

e // Ext1OX(LX/Y ,F )

(7.3.20)

wheree is the isomorphism of [46, Ch.III, Th.1.2.3] and the right vertical arrow is (7.3.9).

Theorem 7.3.21.For every coherentOX-moduleF , the mapean is a natural bijection.

Proof. In view of (7.3.20), in order to show thatean is injective, it suffices to verify that if(ji : X → Xi, βi) for i = 1, 2 are two analytic deformations ofX by F over Y , and(j1, β1)

∼→ (j2, β2) is an isomorphism in the category of extensions off−1OY -algebras, thenthe corresponding mapOX1 → OX2 is continuous, i.e. it is an isomorphism of sheaves of topo-logical algebras (since in this case the map restricts to an isomorphismO+

X1

∼→ O+X2

). Thiscan be checked locally, so we may assume thatX is affinoid; in this case bothX1 andX2 areaffinoid as well, by proposition 7.3.13. Then the assertion is a straightforward consequence ofthe following well known:

Claim 7.3.22. Every mapA → B from a noetherianK-Banach algebraA to K-algebraB oftopologically finite type, is continuous.

Proof of the claim:This is [12,§6.1.3, Th.1].

The surjectivity is a local issue as well; indeed, any class in the target ofean represents anextension off−1OY -algebras0 → F → E → OX → 0, and the question amounts to showingthat E represents an analytic deformation, which can be checked locally. Thus, suppose thatX = Spa(B,B), Y = Spa(A,A). We can writeB = P/I, whereP := A〈T1, ..., Tn〉, andby proposition 7.2.44, the complexτ[−1Lan

X/Y is naturally isomorphic to the complex of sheaves


associated to the complex ofB-modules0 → I/I2 → ΩanP/A ⊗P B → 0. Whence, a natural

isomorphism

Ext1OX(Lan

X/Y ,F ) ≃ Coker(HomOX(j∗Ωan

Z/Y ,F )→ HomOX((I/I2)∼,F ))

≃ Coker(HomB(ΩanP/A ⊗P B,F (X))→ HomB(I/I2,F (X)))

(where we have denoted byj : X → Z := Spa(P, P ) the induced closed imbedding). How-ever, givenα : I/I2 → F (X), the correspondingf−1OY algebraE is the sheaf of algebrasassociated to theA-algebraE defined as follows. Letβ : I/I2 → E0 := (P/I2) ⊕ F (X)be the map given by the rule:x 7→ (x, α(x)) for everyx ∈ I/I2; E0 is endowed with anA-algebra structure given by the rule(x, f) · (y, g) = (xy, fy + gx) and one verifies easilythat the image ofβ is an ideal inE0; then setE := E0/Im β. It is clear thatE0 is anA-algebra of topologically finite type, and the projectionE → B determines a closed imbeddingj : X → X ′ := Spa(E,E) that identifiesj∗OX′ with E .

With the foregoing results, one can derive the usual resultson existence of deformationsand thereof obstructions; again we leave the details to the industrious reader. To conclude thissection we want to show that (7.3.9) fails to be surjective already in the simplest situations. Tocarry out this analysis requires the use of more refined commutative algebra : the followingproposition 7.3.23 collects all the information that we shall be needing.

Proposition 7.3.23.Let A be a completeK-algebra of topologically finite type,n ∈ N aninteger, and setP := A〈T1, ..., Tn〉. Then:

(i) the natural morphismSpecP → SpecA is regular.(ii) Hi(LP/A) = 0 for everyi > 0 andH0(LP/A) is a flatP -module.

Proof. We begin with the following observation:

Claim7.3.24. LetR→ S be a faithfully flat morphism of noetherian local rings. IfS is regular,thenR is regular.

Proof of the claim:One verifies with no trouble that the homological dimension of R does notexceed the homological dimension ofS.

Claim7.3.25. Let R → S be a local morphism of local noetherian rings; letn ⊂ S be themaximal ideal. Suppose thatR is quasi-excellent and thatS is formally smooth overR for itsn-adic topology. Then the induced morphismSpecS → SpecR is regular.

Proof of the claim:This is [3, Th.].In view of claim 7.3.25, the proof of (i) is reduced to the following:

Claim7.3.26. Let n ⊂ P be any maximal ideal and setq := n ∩A. Then:(i) Pn is formally smooth overAq for its n-adic topology.

(ii) Aq is an excellent local ring.

Proof of the claim:(i): It is well known thatq is a maximal ideal and the residue fieldK ′ := A/qis a finite extension ofK. It follows easily from lemma 7.1.6(i) thatPn is flat overA, hence by[28, Ch.IV, Th.19.7.1] it suffices to show thatPn⊗AK ′ is geometrically regular overK ′, whichallows to reduce to the case whereA = K. Then, in view of claim 7.3.24, it suffices to showthat, for every finite field extensionK ⊂ K ′ there exists a larger finite extensionK ′ ⊂ K ′′

such that the semilocal ringPn ⊗K K ′′ is regular. However, the residue fieldκ := P/n isa finite extension ofK, and ifK ⊂ K ′ is any finite normal extension withκ ⊂ K ′, everymaximal ideal ofP ⊗K K ′ containingn ⊗K K ′ is of the formp := (T1 − a1, ..., Tn − an) forcertaina1, ..., an ∈ K ′. We are therefore reduced to the case wheren = (T1 − a1, ..., Tn − an)for someai ∈ K, i = 1, ..., n; in this case then-adic completionP ∧n of Pn is isomorphic


to K[[X1, ..., Xn]], which is regular. (ii) is a theorem due to Kiehl, whose complete proof isreproduced in [22, Th.1.1.3].

Finally, (ii) follows from (i) and from the following:

Claim 7.3.27. LetR→ S be a map of noetherian rings. Then the following are equivalent:(a) Hi(LS/R) = 0 for everyi > 0 andH0(LS/R) is a flatS-module.(b) The induced morphismSpecS → SpecR is regular.

Proof of the claim:Taking into account [54, Th.28.7], this is seen to be a paraphrase of [2,Suppl.(c), Th.30, p.331].

7.3.28. Let us now specialize to the case whereP := K〈T 〉; it is well known thatP is aprincipal ideal domain; sinceΩP/K is a freeP -module, we can choose a splitting:

ΩP/K ≃ ΩanP/K ⊕ Ωna

P/K(7.3.29)

whereΩnaP/K := Ker(ΩP/K → Ωan

P/K) (“na” stays for “not analytic”). Our first aim is to showthatP admits nilpotent extensions (by finitely generatedP -modules) that are not affinoid al-gebras. In view of proposition 7.3.23(ii), the square zero extensions ofP by aP -moduleFare classified byExt1

P (ΩP/K , F ) (and then it is clear that any non-trivial element of this groupcannot represent an analytic deformation, sinceP admits none). Hence, we come down tocomputingExt1

P (ΩnaP/K , F ).

Lemma 7.3.30. (i) ΩnaP/K is a vector space over the fraction fieldE ofP .

(ii) If eitherchar(K) = 0 or char(K) = p > 0 and[K : Kp] =∞, thenΩnaP/K 6= 0.

Proof. (i): by proposition 7.3.23(ii) we know thatΩnaP/K is a flatP -module, especially it is

torsion-free. Letf ∈ P be any irreducible element; thenκ := P/fP is a finite field extensionof K, and we haveHomP (ΩP/K , κ) = HomP (Ωan

P/K , κ) by the universal property ofΩanP/K . In

other words,Ωna⊗P κ = 0, so multiplication byf is a bijection onΩnaP/K , and the claim follows.

(ii): suppose first thatchar(K) = 0; by (i) we have:ΩE/K ≃ E ⊗P ΩP/K ≃ ΩnaP/K ⊕ (E ⊗P

ΩanP/K). SinceΩan

P/K is a freeP -module of rank one with generatordT , and sinceE is a separableextension ofK, it suffices to show thattr.deg(E : K(T )) > 0. This can be checked explicitly;for instance, letπ ∈ K be a non-zero element such thatlog(1 + πT ) ∈ P ; it is well knownthat this power series is transcendental overK(T ). Finally, suppose thatchar(K) = p > 0 and[K : Kp] = ∞. We construct a splitting for the surjectionΩE/K → E ⊗P Ωan

P/K , as follows.The tower of field extensionsK ⊂ F := Ep ·K(T ) ⊂ E yields an exact sequence

E ⊗F ΩF/Kα→ ΩE/K

β→ ΩE/F → 0

and it is easy to check thatα factors through a surjectionE ⊗F ΩF/K → E ⊗P ΩanP/K and

the induced mapE ⊗P ΩanP/K → ΩE/K is the sought splitting. Hence, it suffices to exhibit an

elementf ∈ E such thatβ(df) 6= 0. To this aim, we will use the following general remark:

Claim 7.3.31. Let E be a field withchar(E) = p > 0 andF ⊂ E a subfield withEp ⊂ F .Then:Ker(d : E → ΩE/F ) = F .

Proof of the claim:Let (xi | i ∈ I) be ap-basis for the extensionF ⊂ E, and for everyi ∈ IsetEi := F [xi | i ∈ I \ i]. ClearlyE = Ei[xi] and the minimal polynomial ofxi overEiis mi(X) := Xp − xpi . A simple calculation shows thatKer(d : E → ΩE/Ei

) = Ei. SinceF =

⋂i∈I Ei, it follows that(d : E → ΩE/F ) =

⋂i∈I Ker(d : E → ΩE/Ei

) = F .In view of claim 7.3.31, we need to exhibit an elementf ∈ E such thatf /∈ Ep · K(T ).

However, under the standing assumptions, we can find a countable sequence(bn | n ∈ N)of elementsbn ∈ K that are linearly independent overKp. Setf :=

∑n∈N a

pnbnTn, so


that f ∈ P , and suppose by way of contradiction, thatf ∈ Ep · K(T ); then we could findg ∈ E · K1/p(T 1/p) such thatgp = f . In turns, this means thatg ∈ K ′ · E(T 1/p) for somefinite K-extensionK ′ ⊂ K1/p. However, any suchg can be written uniquely in the formg =

∑n∈N cnT

n/p for a sequence(cn | n ∈ N) of elementscn ∈ K ′. Thengp =∑

n∈N cpnT

n,

and consequentlycpn = apnbn for everyn ∈ N, in other words,K ′ contains all the elementsb1/pn

for everyn ∈ N, which is absurd.

7.3.32. We shall complete our calculation forF := P . To this aim, recall thatE is an injectiveP -module, therefore

Ext1P (Ωna

P/K , P ) ≃ HomP (ΩnaP/K , E/P )/HomP (Ωna

P/K , E).

Let AE be the (finite) adele ring ofE, i.e. the restricted product of all the completions

AE :=∏′

p∈MaxP

Ep.

By the strong approximation theorem (cp. [20, Ch.II,§15, Th.]) the natural imbeddingE ⊂ AE

has everywhere dense image, whence an isomorphism:

E/P ≃ ⊕p∈MaxP

Ep/Pp.

A simple calculation shows that the mapEp→ HomP (E,Ep/Pp) given by the rule:f 7→ (x 7→xf) is an isomorphism; whence a natural isomorphism:

AE∼→ HomP (E,E/P ) a 7→ (x 7→ ax).

Finally, Ext1P (E, P ) ≃ AE/E; sinceΩna

P/K is a direct sum of copies ofE, this achieves ouraim of producing non-trivial square zero extensions ofK-algebras:0 → P → E → P → 0,whenever the hypotheses of lemma 7.3.30(ii) are fulfilled.

7.3.33. Now we want to carry out the local counterpart of the calculations of (7.3.32). Namely,letX := Spa(P, P ); we will show that there exist non-trivial deformations of(X,OX) by theOX-moduleOX, in the category of locally ringed spaces. To this aim, choose anyK-rationalpoint p ∈ X, and letOX(∞) denote the quasi-coherentOX-module whose local sections onany open subsetU ⊂ X are the meromorphic functions onU with poles (of arbitrary finiteorder) only atp. We deduce an exact sequence

HomOX(Ωna

X/K ,OX(∞))f→ HomOX

(ΩnaX/K ,OX(∞)/OX)

g→ Ext1OX

(ΩnaX/K ,OX)

whereΩnaX/K denotes the quasi-coherent sheaf obtained by sheafifying the modulesΩna defined

as in (7.3.28). After choosing a global splitting (7.3.29),we can writeΩX/K ≃ ΩanX/K ⊕ Ωna

X/K .

Lemma 7.3.34.With the notation of(7.3.33), we have:HomOX(Ωna

X/K ,OX(∞)) = 0.

Proof. Indeed, letφ : ΩnaX/K → OX(∞) be anyOX-linear map; we extendφ to a mapφ′ :

ΩX/K → OX(∞) by prescribing thatφ′ restricts to the zero map on the direct factorΩanX/K . Then

φ′ corresponds to aK-linear derivation∂ : OX → OX(∞); the restriction of∂ toU := X \pis a derivation ofOU with values in the coherentOU -moduleOU ; hence the restriction ofφ′|Uto Ωna

U/K vanishes identically. It follows that∂|U vanishes identically. Letj : U → X be theimbedding; since the natural mapOX(∞)→ j∗OU is injective, it follows that∂ must vanish aswell, whenceφ = 0.

Lemma 7.3.35.With the notation of(7.3.33), let x ∈ X be aK-rational point and supposethat the hypotheses of lemma7.3.30(ii)hold forK. ThenΩna

X/K,x 6= 0.


Proof. By lemma 7.3.30(ii) we know already that the global sectionsof ΩnaX/K do not vanish. It

suffices therefore to show that, for every affinoid subdomainU := Spa(A,A) ⊂ X containingx, the natural mapΩP/K ⊗P A → ΩA/K is injective. However, the kernel of this map is aquotient ofH1(LA/P ), hence it suffices to show the following

Claim 7.3.36. Hi(LA/P ) = 0 for everyi > 0.

Proof of the claim:In light of claim 7.3.27, it suffices to show that the mapP → A is regular.Then, in view of claims 7.3.25 and 7.3.26(ii), it suffices to show that, for every maximal idealn ⊂ A, the ringA is formally smooth overP for its n-adic topology. Letq := n∩P ; sinceA isflat overP , [28, Ch.IV, Th.19.7.1] reduces to showing that the inducedmorphismP/q→ A/qis formally smooth, which is trivial, since the latter is an isomorphism.

7.3.37. It follows easily from proposition 7.3.23(ii) and lemma 7.2.1 thatLX/K ≃ ΩX/K [0],so the extensions0 → OX → E → OX → 0 are classified byExt1

OX(Ωna

X/K ,OX). In orderto show that the latter is not trivial, it suffices, in view of lemma 7.3.34, to exhibit a nonzeromapΩna

X/K → Q := OX(∞)/OX. However,Q is a skyscaper sheaf sitting at the pointp,with stalk equal toF/OX,p, whereF := Frac(OX,p). We are therefore reduced to showingthe existence of a nonzero mapΩna

X/K,p → F . However, we deduce easily from lemma 7.3.30thatΩna

X/K,p is anF -vector space, and by lemma 7.3.35 the latter does not vanish, provided thateitherchar(K) = 0 orK has characteristicp > 0 and[K : Kp] =∞.

7.3.38. The foregoing results notwithstanding, there is atleast one situation in which the ab-stract topos deformation theory of an adic space is the same as its analytic deformation theory.This is explained by the following final proposition.

Proposition 7.3.39.Suppose thatchar(K) = p > 0 and that[K : Kp] < ∞. Then, for everymorphismX → Y of adic spaces locally of finite type overSpa(K,K+), the natural mapLX/Y → Lan

X/Y is an isomorphism inD(OX-Mod).

Proof. It suffices to show the corresponding statement for maps of affinoid ringsA → B,hence choose a presentationB ≃ A〈T1, ..., Tn〉/I. We have to verify that the induced mapsHi(LB/A) → Hi(Lan

B/A) are isomorphisms for everyi ∈ N. The sequenceA → P :=

A〈T1, ..., Tn〉 → B yields by transitivity two distinguished triangles : one relative to theusual cotangent complex and one relative to the analytic cotangent complex. Hence, by thefive lemma, we are reduced to showing that the mapsHi(B ⊗P LP/A) → Hi(B ⊗P Lan

P/A)

andHi(LB/P ) → Hi(LanB/P ) are isomorphisms for everyi ∈ N. For the latter we appeal to

lemma 7.2.42. Also, it follows easily from proposition 7.3.23 and theorem 7.2.39 that bothHi(B ⊗P LP/A) andHi(B ⊗P Lan

P/A) vanish fori > 1, hence we are reduced to showing thatthe map

τ[−1LB/A → τ[−1LanB/A(7.3.40)

is an isomorphism inD(OX-Mod). By proposition 7.2.44 we have:τ[−1LanB/A ≃ (0→ I/I2 →

B ⊗P ΩanP/A → 0); on the other hand, proposition 7.3.23 and [46, Ch.III, Cor.1.2.9.1] give:

τ[−1LB/A ≃ (0 → I/I2 → B ⊗P ΩP/A → 0). Both these isomorphisms are natural, so(7.3.40) is represented by the obvious map between the latter complexes. Thus, we are reducedto showing that the natural mapΩP/A → Ωan

P/A is an isomorphism. LetC ⊂ P be theA-subalgebra generated by the image of the Frobenius endomorphismΦ : P → P ; a standardcalculation shows thatΩP/A ≃ ΩP/C . However, under the standing assumptions,P is finiteover its subalgebraC, thereforeΩP/A is a finiteP -module, and the claim follows.


7.4. Analytic geometry over a deeply ramified base.In this section we assume throughoutthat(K, | · |) is a deeply ramified complete valued field, with valuation of rank one. Recall thata ∈ K× denotes a topologically nilpotent element ofK.

If X is a formal scheme of finite type overSpf K+, we will sometimes writeLanX/K+ instead

of LanX/Spf K+. Similarly we defineL+

X/K for an adic spaceX of finite type overSpa(K,K+),and setΩ+

X/K := H0(L+X/K).

Theorem 7.4.1.LetX be a smooth adic space overSpa(K,K+). ThenL+X/K ≃ Ω+

X/K [0] in

D−(O+X-Mod), andΩ+

X/K is a flat sheaf ofO+X-modules.

Proof. Both assertions can be checked on the stalks, therefore letx ∈ X be any point. The stalkOX,x is a local ring and its residue fieldκ(x) carries a natural valuation; the preimage inOX,x ofthe corresponding valuation ringκ(x)+ is the subringO+

X,x. Let I :=⋂n∈N a

nO+X,x; it follows

from this description thatκ(x)+ = O+X,x/I. Especially, we have

O+X,x/aO

+X,x ≃ κ(x)+/a · κ(x)+.(7.4.2)

Claim7.4.3. LetM be anO+X,x-module, and suppose thata is regular onM . ThenM/IM is a

flat κ(x)+-module.

Proof of the claim:By snake lemma we deriveKer(M/IM·a→M/IM) ⊂ Coker(IM

·a→ IM).However, it is clear thatI = aI, soa is regular onM/IM , whence the latter is a torsion-freeκ(x)+-module and the claim follows.

Let U be the cofiltered system of all affinoid open neighborhoods ofx in X; for U ∈ U letFU be the filtered system of allK+-subalgebras ofO+

X(U) of topologically finite presentation.We derive

(L+X/K)x

L

⊗K+ K+/aK+ ≃ colimU∈U

colimA∈FU

LanA/K+

L

⊗K+ K+/aK+

≃ colimU∈U

colimA∈FU

LA/K+

L

⊗K+ K+/aK+

≃ LO+X,x/K

+

L

⊗K+ K+/aK+

≃ Lκ(x)+/K+

L

⊗K+ K+/aK+.

Together with theorem 6.6.11, this implies already that scalar multiplication bya is an auto-morphism ofHi(L

+X/K), for everyi > 0. However, according to theorem 7.2.39(ii),Hi(L

+X/K)

is aK+-torsion sheaf ofO+X-modules, fori > 0, whence the first assertion. It also follows that

(Ω+X/K)x is a torsion-free, hence flat,K+-module. To prove that(Ω+

X/K)x is a flatO+X,x-module,

we remark first that(ΩanX/K)x ≃ (Ω+

X/K)x ⊗K+ K, and the latter is a flatOX,x-module, sinceXis smooth overSpa(K,K+). By lemma 5.2.1 it suffices therefore to show

Claim7.4.4. (Ω+X/K)x ⊗K+ K+/aK+ is a flatO+

X,x ⊗K+ K+/aK+-module.

Proof of the claim:By claim 7.4.3 we know that(Ω+X/K)x ⊗O+

X,xκ(x)+ is a flatκ(x)+-module.

In view of (7.4.2), the claim follows after base change toκ(x)+/a · κ(x)+.

Definition 7.4.5. Let (Xα | α ∈ I) be a system of formal schemes of finite presentation overSpfK+, indexed by a small cofiltered categoryI.

(i) Let X∞ := limα∈I

Xα, where the limit is taken in the category of locally ringed spaces. For

everyα ∈ I, let πα : X∞ → Xα be the natural morphism of locally ringed spaces. WedefineΩan

X∞/K+ := colimα∈Io

π∗α(ΩanXα/K+), which is a sheaf ofOX∞-modules.


More generally, we letLanX∞/K+ := colim

α∈Ioπ∗α(L

anXα/K+).

(ii) We say that the cofiltered system(Xα | α ∈ I) is deeply ramifiedif the natural morphismΩan

X∞/K+ → ΩanX∞/K+ ⊗K+ K is an epimorphism.

Lemma 7.4.6. Let (Xα | α ∈ I) be a cofiltered system as in definition7.4.5. For any morphismβ → α of I, let fαβ : Xβ → Xα be the corresponding morphism of formalSpf K+-schemes.Moreover, for everyα ∈ I, letΩtf

Xα/K+ be the image of the morphismΩanXα/K+ → Ωan

Xα/K+⊗K+K(“ tf” stays for torsion-free). The following two conditions areequivalent :

(i) The system(Xα | α ∈ I) is deeply ramified.(ii) For everyα ∈ I there is a morphismβ → α of I, such that the image of the natural

morphismf∗αβ(ΩtfXα/K+)→ Ωtf

Xβ/K+ is contained in the subsheafa · ΩtfXβ/K+ .

Proof. It is clear that (ii)⇒(i). We show that (i)⇒(ii). Under the above assumptions, everyXα

is quasi-compact, hence we can cover it by finitely many affineformal schemesUi := Spf Ai(i = 1, ..., n) of finite type overSpf K+. Then, for everyi = 1, ..., n, the restriction ofΩan

Xα/K+

to Ui is the coherent sheaf(ΩanAi/K+) (notation of [26, Ch.I,§10.10.1]). HenceΩtf

Xα/K+ is acoherent sheaf ofOXα-modules. For every morphismβ → α, let

Uαβ := x ∈ Xβ | Im(f∗αβ(ΩtfXα/K+)x → (Ωtf

Xβ/K+)x) ⊂ a · (ΩtfXβ/K+)x.

Uαβ is therefore a constructible open subset ofXβ, and we denote its complement byZαβ. Byassumption (i) we know that

limβ→α

Zαβ =⋂

β→α

π−1α (Zαβ) = ∅.

If we retopologize the reduced schemesZαβ by their constructible topologies, we get an inversesystem of compact spaces, and deduce that someZαβ is empty by [18, Ch.I,§9, n.6, Prop.8(b)].

Example 7.4.7.The prototype of deeply ramified systems is given by the towerof morphisms

...→ BdK+(0, ρ1/pn

)φn→ Bd

K+(0, ρ1/pn−1

)φn−1→ ...

φ1→ BdK+(0, ρ)(7.4.8)

where, for anyr = (r1, ..., rd) ∈ (K×)d, we have denoted

BdK+(0, |r|) := SpfK+〈r−1

1 T1, ..., r−1d Td〉

(i.e., the formald-dimensional polydisc defined by the equations|Ti| ≤ |ri|, i = 1, ..., d). Themorphismsφn are induced by the ring homomorphismsTi 7→ T pi (i = 1, ..., d). Notice that thetower (7.4.8) is defined wheneverρi ∈ Γp

∞

K :=⋂n∈N Γp

n

K for everyi = 1, ..., d. We leave to thereader the verification that condition (ii) of lemma 7.4.6 isindeed satisfied.

Lemma 7.4.9. LetX := (Xα | α ∈ I) be a cofiltered system as in definition7.4.5.(i) If Y := (Yα | α ∈ I) → (Xα | α ∈ I) is a morphism of cofiltered systems such that the

induced morphisms of adic spacesd(Yα) → d(Xα) are unramified for everyα ∈ I (cp.(7.2.26)), thenY is deeply ramified ifX is.

(ii) Let Z := (Zβ | β ∈ J) be another such cofiltered system, and suppose thatX andZ areisomorphic as pro-objects of the category of formal schemes. ThenX is deeply ramified ifand only ifZ is.

(iii) If X and Z := (Zβ | β ∈ J) are two deeply ramified cofiltered systems, then the fibredproductX× Z := (Xα ×Spf(K+) Zβ | (α, β) ∈ I × J) is deeply ramified.

Proof. (i): by [14, Prop.2.2] the natural morphismΩanX∞/K+ ⊗K+ K → Ωan

Y∞/K+ ⊗K+ K is anepimorphism; the claim follows easily. (ii) and (iii) are easy and shall be left to the reader.


The counterpart of the above definitions for adic schemes is given in the following:

Definition 7.4.10. Let (Xα | α ∈ I) be a system of adic spaces of finite type overSpa(K,K+),indexed by a small cofiltered categoryI.

(i) Let (X∞,OX∞,O+X∞

) := limα∈I

(Xα,OXα,O+Xα

), where the limit is taken in the category

of locally ringed spaces. For everyα ∈ I, let πα : X∞ → Xα be the natural morphismof locally ringed spaces. We defineΩ+

X∞/K := colimα∈Io

π∗α(Ω+Xα/K

), which is a sheaf of

O+X∞

-modules. More generally, we letL+X∞/K := colim

α∈Ioπ∗α(L

+Xα/K

).

(ii) We say that the cofiltered system(Xα | α ∈ I) is deeply ramifiedif the natural morphismΩ+X∞/K → Ω+

X∞/K ⊗K+ K is an epimorphism.

7.4.11. Let(Xα | α ∈ I) be as in definition 7.4.10. For everyx ∈ X∞, we letκ(x)+ :=O+X∞,x/

⋂n∈N a

nO+X∞,x. The ringκ(x)+ is a filtered colimit of valuation rings, hence it is a

valuation ring. Moreover, the image ofa in κ(x)+ is topologically nilpotent for the valuationtopology ofκ(x)+.

7.4.12. Given a cofiltered systemX := (Xα | α ∈ I) of formal schemes, one obtains acofiltered system of adic spacesX := (Xα := d(Xα) | α ∈ I), and using (7.2.27) and lemma7.4.9(i) one sees easily thatX is deeply ramified wheneverX is. Together with example 7.4.7,this yields plenty of examples of deeply ramified systems of adic spaces.

Proposition 7.4.13.Let X := (Xα | α ∈ I) be a deeply ramified cofiltered system of adicspaces. Then, for every pointx ∈ X∞, the valuation ringκ(x)+ is deeply ramified.

Proof. For everyK+-moduleM , let us denote byTn(M) the submodule ofM annihilated byan. Furthermore, letT (M) :=

⋃n∈N Tn(M). Since the cofiltered systemX is deeply ramified,

we have:(Ω+X∞/K)x = T (Ω+

X∞/K)x+a·(Ω+X∞/K)x. To lighten notation, letO+

x := O+X∞,x. From

lemma 7.1.25 one deduces easily that the natural mapΩO+x /K+ → (Ω+

X∞/K)x is surjective witha-divisible kernel. Hence, by snake lemma, the induced mapTn(ΩO+

x /K+) → Tn(Ω+X∞/K)x is

surjective for everyn, anda fortiori the mapT (ΩO+x /K+)→ T (Ω+

X∞/K)x is onto. It follows eas-ily that ΩO+

x /K+ = T (ΩO+x /K+)+ a ·ΩO+

x /K+, and consequently:Ωκ(x)+/K+ = T (Ωκ(x)+/K+)+

a ·Ωκ(x)+/K+. However, it follows easily from theorem 6.6.11 thatT (Ωκ(x)+/K+) = 0, so finallyΩκ(x)+/K+ = a · Ωκ(x)+/K+ and

Lκ(x)+/K+

L

⊗K+ K+/aK+ ≃ 0.(7.4.14)

On the other hand, if(E, | · |E) is any valued field extension ofκ(x)+, we have

Hi(LE+/K+ ⊗K+ K+/aK+) = 0 for all i > 0(7.4.15)

by theorem 6.6.11. From (7.4.14), (7.4.15) and transitivity for the towerK+ ⊂ κ(x)+ ⊂ E+,we deriveHi(LE+/κ(x)+⊗K+K+/aK+) = 0 for all i > 0. Again by theorem 6.6.11 and remark6.6.14 we conclude.

7.4.16. LetX be a cofiltered system as in definition 7.4.10. LetA be a sheaf ofO+X∞

-algebras.We say thatA is aweaklyetaleO+

X∞-algebraif, for everyx ∈ X∞, the stalkA a

x is a weaklyetaleO+a

X∞,x-algebra.

Theorem 7.4.17.Suppose thatK is deeply ramified, and letX := (Xα | α ∈ I) be a deeplyramified cofiltered system. Let alsof : Y := (Yα | α ∈ I) → X be a morphism of cofilteredsystems, such that the morphismsYα → Xα are finiteetale for everyα ∈ I. Thenf∞∗O

+Y∞

is aweaklyetaleO+

X∞algebra.


Proof. To lighten notation, let us writeO+ (resp. A +) instead ofO+X∞

(resp. f∞∗O+Y∞

). Foreveryα ∈ I consider the cofiltered systemZ(α) := X ×Xα Yα indexed byI/α (the categoryof morphismsβ → α), which is defined by settingZ(α)β→α := Xβ ×Xα Yα. We have obviousmorphisms of cofiltered systemsf

/α: Z(α) → X; the sheafA + is the colimit of the sheaves

f/α∞∗O+Z(α)∞

. Hence it suffices to prove the assertion for the latter sheaves, and therefore inorder to show the theorem we can and do assume that there exists α ∈ I such that, for everyβ → α, the induced commutative diagram

Yβ //

Yα

Xβ

// Xα

is cartesian. Letx ∈ X∞.

Claim 7.4.18. A +x is a flatO+

x -algebra.

Proof of the claim:On one hand, by assumptionA +x [1/a] is a flatO+

x [1/a]-algebra; on theother hand,A +

x ⊗O+xκ(x)+ is a flatκ(x)+-module by claim 7.4.3, so thatA +

x /aA+x is a flat

O+x /aO

+x -module; thus the claim follows from lemma 5.2.1.

Let e ∈ C := A +x ⊗O+

xA +x [1/a] be the idempotent provided by lemma 3.1.4. In view of

claim 7.4.18, we only have to show thatε · e ∈ C+ := A +x ⊗O+

xA +x for everyε ∈ m. Let e be

the image ofe in C ⊗Ox κ(x). SetI :=⋂n≥0 a

nO+x .

Claim 7.4.19. A +x /IA

+x is the integral closure of ofκ(x)+ in Ax/IAx.

Proof of the claim:First of all, sinceI = aI, we haveIAx = IA +x , and thus the natural

mapA +x /IA

+x → Ax/IAx is injective. Next, we remark thatA +

x is the integral closure ofO+x in Ax; indeed, this follows from [12,§6.2.2, Lemma 3, Prop.2], after taking colimits. This

already shows thatA +x /IA

+x is a subalgebra ofAx/IAx integral overκ(x)+. To conclude,

suppose thatf ∈ Ax/IAx satisfies an integral equation:fn+b1 ·fn−1 + ...+bn = 0, for certainb1, ..., bn ∈ A +

x /IA+x ; pick arbitrary representativesf ∈ Ax, bi ∈ A +

x of these elements. Thenfn + b1 · fn−1 + ...+ bn ∈ IAx. SinceIAx = IA +

x , we deduce thatf is integral overA +x , so

f ∈ A +x and the claim follows.

Claim 7.4.20. (A +x /IA

+x )a is an etaleκ(x)+a-algebra.

Proof of the claim: In view of propositions 2.4.18(ii) and 2.4.19, it suffices toshow that(A +

x /IA+x )a is weakly etale overκ(x)+a. Let p be the height one prime ideal ofκ(x)+; then

κ(x)+p is a rank one valuation ring and the localization map inducesisomorphismsκ(x)+a ∼→

κ(x)+ap , (A +

x /IA+x )a

∼→ (A +x /IA

+x )ap (recall that the standing basic setup is the standard

setup ofK+). Taking claim 7.4.19 into account, it suffices then to show that (A +x /IA

+x )aq is

weakly etale overκ(x)+ap , for every prime idealq ⊂ A +

x /IA+x of height one. By proposition

7.4.13,κ(x)+ is deeply ramified, hence the same holds forκ(x)+p . Let κ(x)s be a separable

closure ofκ(x), andκ(x)s+ a valuation ring ofκ(x)s dominatingκ(x)+p ; we can assume that

(A +x /IA

+x )q ⊂ κ(x)s+. From claim 7.4.19 we deduce that(A +

x /IA+x )q is a valuation ring,

henceκ(x)s+ is a faithfully flat(A +x /IA

+x )q-algebra; then the claim follows in view of lemma

3.1.2(viii) and proposition 6.6.2.

From claim 7.4.20 we deduce thatε · e ∈ C+ ⊗O+xκ(x)+ for everyε ∈ m. To conclude, it

suffices therefore to remark:

Claim 7.4.21. For everyε ∈ m, we haveε · e ∈ C+ if and only if ε · e ∈ C+ ⊗O+xκ(x)+.


Proof of the claim:Let eε ∈ C+ be any lifting ofε · e; thenε · e − eε is in the kernel of theprojectionC → C ⊗O+

xκ(x). Let n ∈ N be a large enough integer, so thatan · e ∈ C+; it

follows thatan ·(ε ·e−eε) is in the kernel of the projectionC+ → C+⊗O+xκ(x)+, consequently

an · (ε · e − eε) = an · c for somec ∈ C+. Sincea is a non-zero-divisor inC+, it follows thatε · e = c+ eε ∈ C+, as required.

7.5. Semicontinuity of the discriminant.

Definition 7.5.1. Let (V,m) be a basic setup,A a V a-algebra andP an almost projectiveA-module of constant rankr ∈ N. Suppose moreover thatP is endowed with a bilinear formb : P ⊗A P → A. We let β : P → P ∗ be theA-linear morphism defined by the rule:β(x)(y) := b(x⊗ y) for everyx, y ∈ P∗. Thediscriminantof the pair(P, b) is the ideal

dA(P, b) := AnnA Coker(ΛrAβ : Λr

AP → ΛrAP∗).

7.5.2. As a special case, we can consider the pair(B, tB/A) consisting of anA-algebraBwhich is almost projective of constant rankr overA, and its trace formtB/A. In this situation,we letdB/A := dA(B, tB/A), and we call this ideal thediscriminant of theA-algebraB.

Lemma 7.5.3. LetB be an almost projectiveA-algebra of constant rankr as anA-module.ThenB is etale overA if and only if dB/A = A.

Proof. By theorem 4.1.14, it is clear thatdB/A = A whenB is etale overA. Suppose thereforethat dB/A = A; it follows that Λr

AτB/A is an epimorphism. However, by proposition 4.3.27,ΛrAB andΛr

AB∗ are invertibleA-modules. It then follows by lemma 4.1.5(iv) thatΛr

AτB/A isan isomorphism, henceτB/A is an isomorphism, by virtue of proposition 4.4.28. One concludesagain by theorem 4.1.14.

Lemma 7.5.4. Let (V,m) be the standard setup associated to a valued field(K, | · |) (cp.(6.1.15), especially,V := K+). Let P ′ ⊂ P be two almost projectiveV a-modules of con-stant rank equal tor. Let b : P ⊗V a P → V a be a bilinear form, such thatb ⊗V a 1Ka is aperfect pairing, and denote byb′ the restriction ofb toP ′ ⊗V a P ′. Then we have:

dV a(P ′, b′) = F0(P/P′)2 · dV a(P, b).

Proof. Let j : P ′ → P be the imbedding,β : P → P ∗ (resp. β ′ : P ′ → P ′∗) theV a-linearmorphism associated tob (resp. tob′). The assumptions implies thatΛr

V aj, ΛrV aj∗, Λr

V aβ andΛrV aβ ′ are all injective, and clearly we have

ΛrV aj∗ Λr

V aβ ΛrV aj = Λr

V aβ ′.

There follow short exact sequences:

0→ Coker ΛrV aβ → Coker Λr

V a(j∗ β)→ Coker ΛrV aj∗ → 0

0→ Coker ΛrV aj → Coker Λr

V aβ ′ → Coker ΛrV a(j∗ β)→ 0.

Using lemma 6.3.1 and remark 6.3.5(ii), we deduce:

F0(Coker ΛrV aβ ′) = F0(Coker Λr

V aj) · F0(Coker ΛrV aβ) · F0(Coker Λr

V aj∗).

Claim7.5.5. Let Q be an almost finitely generated projectiveA-module of constant rankr,which is also uniformly almost finitely generated. LetQ′ ⊂ Q any submodule. Then:

F0(Q/Q′) = AnnA(Coker(Λr

AQ′ → Λr

AQ)).


Proof of the claim:By lemma 2.4.6 and proposition 2.3.24(ii) we can check the identity after afaithfully flat base change, hence we can suppose thatQ ≃ Ar, in which case the identity holdsby definition of the Fitting idealF0.

Notice thatΛrV aP andΛr

V aP ′ are invertibleV a-modules by virtue of proposition 4.3.27, con-sequently claim 7.5.5 applies and yields:F0(Coker Λr

V aβ ′) = dV a(P ′, b′), F0(Coker ΛrV aβ) =

dV a(P, b) andF0(Coker ΛrV aj) = F0(Coker Λr

V aj∗) = F0(P/P′).

7.5.6. After these generalities, we return to the standard setup(K+,m) of this chapter, asso-ciated to a valued field(K, | · |) of rank one (cp. (6.1.15)). Consider an etaleK-algebraL;we denote byWL the integral closure ofK+ in L. L is the product of finitely many separablefield extensions ofK, thereforeW a

L is an almost projectiveK+a-module of constant rankn, byproposition 6.3.6. Hence, the discriminant ofW a

L overK+a is defined, and to lighten notation,we will denote it byd+

L/K . Furthermore, sinceL is etale overK, it is clear thatd+L/K is a frac-

tional ideal ofK+a (cp. (6.1.16)). Let| · | : Div(K+a)→ Γ∧K be the isomorphism provided bylemma 6.1.19. We obtain an element|d+

L/K | ∈ Γ∧K ; after choosing (cp. example (6.1.21)(vi)) anorder preserving isomorphism

((ΓK ⊗Z Q)∧,≤)∼→ (R>0,≤)(7.5.7)

on the multiplicative group of positive real numbers, we canthen view|d+L/K | ∈ (0, 1].

Lemma 7.5.8. LetK, L be as in(7.5.6)and denote byK∧ the completion ofK for the valua-tion topology. SetL∧ := K∧ ⊗K L. ThenK∧+ ⊗K+ d+

L/K = d+L∧/K∧.

Proof. Since the base changeK → K∧ is faithfully flat, everything is clear from the definitions,once we have established thatWL∧ ≃ K∧+ ⊗K+ WL. However, both rings can be identifiedwith thea-adic completion(WL)

∧ of WL, so the assertion follows.

7.5.9. LetX be an adic space locally of finite type overSpa(K,K+). X is a locally spectralspace, and every pointx ∈ X admits a unique maximal generisationr(x) ∈ X. The valuationring κ(r(x))+ has rank one, and admits a natural imbeddingK+ ⊂ κ(r(x))+, continuousfor the valuation topologies; especially, the image of the topologically nilpotent elementa istopologically nilpotent inκ(r(x))+. This imbedding induces therefore a natural isomorphismof completed value groups

(ΓK ⊗Z Q)∧∼→ (Γκ(r(x)) ⊗Z Q)∧.

In particular, our original choice of isomorphism (7.5.7) fixes univocally a similar isomorphismfor every pointr(x). We denote byM(X) the setr(X) endowed with the quotient topologyinduced by the mappingX → r(X) : x 7→ r(x). This topology is coarser than the subspacetopology induced by the imbedding intoX. The mappingx 7→ r(x) is a retraction ofX ontothe subsetM(X) of its maximal points. IfX is a quasi-separated quasi-compact adic space,M(X) is a compact Hausdorff topological space ([45, 8.1.8]).

7.5.10. LetX be as in (7.5.9), and letf : Y → X be a finite etale morphism of adic spaces.For every pointx ∈ X, the fibreE(x) := (f∗OY )x ⊗OX,x

κ(x) is a finite etaleκ(x)-algebra.If now x ∈ M(X), we can consider the discriminantd+

E(x)/κ(x) defined as in (7.5.6) (warning:notice that the definition makes sense when we choose the standard setup associated to thevaluation ringκ(x)+; since it may happen that the valuation ofK is discrete and that ofκ(x)is not discrete, the setups relative toK and toκ(x) may not agree in general). Upon passing toabsolute values, we finally obtain a real valued function:

d+Y/X :M(X)→ (0, 1] x 7→ |d+

E(x)/κ(x)|.


The study of the functiond+Y/X is reduced easily to the case whereX (henceY ) are affinoid. In

such case, one can state the main result in a more general form, as follows.

Definition 7.5.11. LetA be any (commutative unitary) ring.

(i) We denote byN (A) the set consisting of all multiplicative ultrametric seminorms| · | :A→ R. For everyx ∈ N (A) andf ∈ A we write usually|f(x)| in place ofx(f). N (A)is endowed with the coarsest topology such that, for everyf ∈ A, the real-valued map|f | : N (A)→ R given by the rule:x 7→ |f(x)|, is continuous.

(ii) For everyx ∈ N (A), we letSupp(x) := f ∈ A | |f(x)| = 0. ThenSupp(x) is a primeideal and we setκ(x) := Frac(A/Supp(x)). The seminormx induces a valuation on theresidue fieldκ(x), and as usual we denote byκ(x)+ its valuation ring.

(iii) Let A→ B be a finite etale morphism. For everyx ∈ N (A), we letE(x) := B ⊗A κ(x).ThenE(x) is an etaleκ(x)-algebra, so we can defined+

B/A(x) := d+E(x)/κ(x) (cp. the

warning in (7.5.10)). By settingx 7→ |d+B/A(x)| we obtain a well-defined function

|d+B/A| : N (A)→ (0, 1].

7.5.12. IfX = Spa(A,A+), with A a completeK-algebra of topologically finite type, thenM(X) is naturally homeomorphic to the subspaceM(A) ofN (A) consisting of the continuousseminorms that extend the absolute value ofK given by (7.5.7). It is shown in [10,§1.2] thatM(A) is a compact Hausdorff space, for every BanachK-algebraA.

Proposition 7.5.13.Let A be a ring,B a finite etaleA-algebra. Then the function|d+B/A| is

lower semi-continuous (i.e. it is continuous for the topology of(0, 1] whose open subsets are ofthe form(c, 1], c ∈ [0, 1]).

Proof. Let f ∈ A be any element; notice thatN (A[1/f ]) is naturally homeomorphic to theopen subsetU(f) := x ∈ N (A) | |f(x)| 6= 0. Hence, after replacingA by some localization,we can assume thatB is a freeA-module, say of rankn. For everyb ∈ B, let χ(b, T ) :=T n + s1(b) · T n−1 + ...+ sn(b) be the characteristic polynomial of theA-linear endomorphismB → B given by the ruleb′ 7→ b′ · b.Claim7.5.14. For every pointx ∈ N (A) and everyb ∈ B, the following are equivalent:

(i) b⊗ 1 ∈WE(x).(ii) |si(b)(x)| ≤ 1 for i = 1, ..., n.

Proof of the claim:Indeed, if (ii) holds, then the image ofχ(b, T ) in κ(x)[T ] is a monic poly-nomial with coefficients inκ(x)+ andb ⊗ 1 is one of its roots (Cayley-Hamilton), henceb ⊗ 1

is integral overκ(x)+, which is (i). Conversely, if (i) holds, letE(x) =∏k

j=1Ej be the de-composition ofE(x) as product of finite separable extensions ofκ(x), and letbj ∈ Ej be theimage ofb. It follows thatbj ∈ WEj

for everyj = 1, ..., k, and moreover the imageχ(b, T )

of χ(b, T ) in κ(x)[T ] decomposes as a product∏k

j=1 χ(bj , T ). It suffices therefore to showthat the coefficientssij(bj) satisfy (ii) for everyi ≤ n and j ≤ k, so we can assume thatE(x) is a field. Letmb(T ) ∈ κ(x)[T ] be the minimal polynomial ofb ⊗ 1; it is well knownthatχ(b, T ) dividesmb(T )n, hence the roots ofχ(b, T ) are conjugates ofb under the action ofG := Gal(κ(x)a/κ(x)). LetC be the integral closure ofκ(x)+ in a finite Galois extension ofκ(x) containingE(x); C is an integralκ(x)+-algebra and the Galois conjugates ofb⊗ 1 are allcontained inC. Since the latter are the roots ofχ(b, T ), the elementssi(b) ⊗ 1 are symmetricpolynomials of the elementsσ(b⊗ 1) (σ ∈ G), sosi(b)⊗ 1 ∈ C ∩κ(x) = κ(x)+, which is (ii).

Let tB/A be the trace morphism of theA-algebraB, and letx ∈ N (A). Then the tracemorphismtx := tE(x)/κ(x) equalstB/A ⊗A 1κ(x).


Claim 7.5.15. For every real numberε > 0 we can find a freeκ(x)+-submoduleWε of WE(x),such thatWε ⊗κ(x)+ κ(x) = E(x) and

|dκ(x)+(Wε, tx)|+ ε > |d+B/A(x)|.(7.5.16)

Proof of the claim:From claim 6.3.8, we derive that, for every positive real numberδ < 1 thereexists a free finitely generatedκ(x)+-submoduleWδ ⊂ WE(x) such that|F0(W

aE(x)/W

aδ )| > δ;

in view of lemma 7.5.4, the claim follows easily.Let w1, ..., wn be a basis ofWε; up to replacingA by a localizationA[1/g] for someg ∈ A,

we can writewi = bi ⊗ 1, for somebi ∈ B (i = 1, ..., n). Consequently:

|dκ(x)+(Wε, tx)| = | det(tB/A(bi ⊗ bj))(x)|.(7.5.17)

By claim 7.5.14 we have|sj(bi)(x)| ≤ 1 for i, j = 1, ..., n. Let 1 > δ > 0 be a real number; foreveryi, j ≤ n, we define an open neighborhoodUij of x inN (A) as follows. Suppose first that|sj(bi)(x)| < 1; since the real-valued functiony 7→ |sj(bi)(y)| is continuous onN (A) (for thestandard topology ofR), we can findUij such that|sj(bi)(y)| ≤ 1 for all y ∈ Uij .

Suppose next that|sj(bi)(x)| = 1; then, up to replacingA byA[1/sj(bi)], we can assume thatsj(bi) is invertible inA. We pickUij such that

|sj(bi)(y)| ≤ 1 + δ for everyy ∈ Uij.(7.5.18)

We setU :=⋂

1≤i,j≤n Uij . Next, we define, for everyy ∈ U , an elementcy ∈ A, as follows.Chooseα, β ≤ n such that|sβ(bα)(y)| = max1≤i,j≤n |sj(bi)(y)|. If |sβ(bα)(y)| ≤ 1, then setcy := 1; if |sβ(bα)(y)| > 1, setcy := sβ(bα)

−1. Then|sj(cy · bi)(y)| ≤ 1 for everyi, j = 1, ..., nand everyy ∈ U . Let W y be theκ(y)+-submodule ofB ⊗A κ(y) spanned by the images ofcy · b1, ..., cy · bn. It follows thatW y ⊂WE(y) for everyy ∈ U . We compute:

|d+Y/X(y)| ≥ |dκ(y)+(W y, ty)|= | det(tB/A(cy · bi ⊗ c(y) · bj))(y)|

= |cy(y)|2n · | det(tB/A(bi ⊗ bj))(y)|.However, the real-valued functiony 7→ | det(tB/A(bi ⊗ bj))(y)| is continuous onN (A), there-fore, combining (7.5.16) and (7.5.17), we see that, up to shrinking further the open neighbor-hoodU , we can assume that| det(tB/A(bi ⊗ bj))(y)|+ ε > |d+

B/A(x)| for all y ∈ U , so finally:

|d+B/A(y)| ≥ (1− δ)2n · (|d+

B/A(x)| − ε) for everyy ∈ Uwhich implies the claim.

Theorem 7.5.19.LetY → X be as in(7.5.10). Then the mapd+Y/X is lower semi-continuous.

Proof. We can assume thatX = Spa(A,A+), whereA is a completeK-algebra of topologicallyfinite type, and thereforeY = Spa(B,B+), for a finite etaleA-algebraB. Then there is a naturalhomeomorphismω :M(X)

∼→M(A), so the theorem follows from proposition 7.5.13 and :

Claim 7.5.20. d+Y/X = d+

B/A ω.

Proof of the claim:Let x ∈ M(X); x corresponds to a rank one valuation ofA, whose valuegroup we identify with (a subgroup of)R>0 according to (7.5.9). The resulting multiplicativeseminorm isω(x). We derive easily a natural imbeddingι : κ(ω(x)) ⊂ κ(x), compatible withthe identifications of value groups. One knows moreover thatι induces an isomorphism oncompletionsι∧ : κ(ω(x))∧

∼→ κ(x)∧, so the claim follows from lemma 7.5.8.

Lemma 7.5.21.Let(Kα, |·|α | α ∈ I) be a system of valued field extensions of(K, |·|), indexedby a filtered small categoryI and such thatK+

α is a valuation ring of rank one for everyα ∈ I.Let moreoverL be a finiteetaleKβ-algebra, for someβ ∈ I. SetLα := L ⊗Kβ

Kα for everymorphismβ → α in I. Then :


(i) (K∞, | · |∞) := colimα∈I

(Kα, | · |α) is a valued field extension of(K, | · |), with valuation ring

of rank one.(ii) SetL∞ := L⊗Kβ

K∞. Then, for every sequence of morphismsβ → γ → α in I, we have|d+Lα/Kα

| ≥ |d+Lγ/Kγ

|, and moreover: lim(β→α)∈I

|d+Lα/Kα

| = |d+L∞/K∞

|.

Proof. (i) is obvious. The proof of the first assertion in (ii) is easyand shall be left to the reader.For the second assertion in (ii) we remark that, due to claim 6.3.8, for everyε < 1 there existsa freeK+

∞-submoduleWε ⊂ WL∞ of finite type, such that|F0(WaL∞/W a

ε )| > ε. We can findα ∈ I such thatWε = W0 ⊗K+

αK+∞ for some freeK+

α submoduleW0 ⊂ L+α . It then follows

from lemma 7.5.4 that

|d+Lα/Kα

| ≥ |dK+α(W0, tLα/Kα)| = |dK+

∞(Wε, tL∞/K∞)| > ε2 · |d+

L∞/K∞|

for every morphismα→ β in I.

7.5.22. Suppose now that(K, | · |) is deeply ramified and letX := (Xα | α ∈ I) be a deeplyramified cofiltered system of adic spaces of finite type overSpa(K,K+). Suppose furthermorethat it is given, for someβ ∈ I, a finite etale morphismYβ → Xβ of adicSpa(K,K+)-spacesof finite type. For every morphismα → β of I we setYα := Yβ ×Xβ

Xα and denote byfα : Yα → Xα the induced morphism of adic spaces.

Theorem 7.5.23.In the situation of(7.5.22), for every positive real numberε < 1 there existsa morphismα→ β in I such that, for every morphismγ → α we have

|d+Yγ/Xγ

(x)| > ε for everyx ∈M(Xγ).

Proof. Notice thatY := (Yα | α → β) is a cofiltered system, hence we can defineX∞ andY∞as in (7.4.12), and we obtain a morphism of locally ringed spacesf∞ : Y∞ → X∞. For everyα ∈ I, letπα : X∞ → Xα be the natural morphism. Moreover, letM(X∞) := lim

α∈IM(Xα); as

a topological space, it is compact, by Tychonoff’s theorem and the fact thatM(Xα) is compact(7.5.9)); as a set, it admits an injective (usually non-continuous) mapM(X∞) → X∞, so wecan identify it as a subset of the latter.

Let x ∈ M(X∞); by proposition 7.4.13, the valuation ringκ(x)+ is deeply ramified. Setκ(x) := κ(x)+ ⊗K+ K; it is clear that the morphism

κ(x)→ E(x) := (f∞∗O+Y∞

)x ⊗κ(x)+ κ(x)

is finite and etale. LetWx be the integral closure ofκ(x)+ in E(x). By proposition 6.6.2 wededuce easily that the induced morphism ofK+a-algebrasκ(x)+a →W a

x is weakly etale, henceetale by proposition 6.3.6. Consequently|d+

E(x)/κ(x)| = 1, in light of lemma 7.5.3. For everyα → β, let xα := πα(x). Thenκ(x) is the colimit of the filtered system(κ(xα) | α → β), andsimilarly E(x) is the colimit of the finite etaleκ(xα)-algebras(fβ∗OYβ

)xβ⊗O+

Xβ,xβ

κ(xα) (for

all α → β). In this situation, lemma 7.5.21 applies and shows that, for everyε < 1 there existsα(ε, x) such that

|d+Yα/Xα

(xα)| > ε for everyα→ α(ε, x).(7.5.24)

In light of theorem 7.5.19, for everyα→ β, the subset

Xα(ε) := y ∈M(Xα) | |d+Yα/Xα

(y)| ≤ εis closed inM(Xα), hence compact. From (7.5.24) we see thatlim

α→βXα(ε) = ∅, therefore one

of theXα(ε) must be empty ([18, Ch.I,§9, n.6, Prop.8(b)]), and the claim follows.


7.5.25. Let us choose an imbeddingρ : (ΓK , ·,≤) → (R,+,≤) as in example 6.1.21(vi). Letf : Y → X be a finite etale morphism of adic spaces of finite type overSpa(K,K+). For everyx ∈ X, setA(x) := (f∗O

+Y )x ⊗O+

X,x(f∗O

+Y )x; we denote byex ∈ A(x) ⊗K+ K the unique

idempotent characterized by the conditions of proposition3.1.4. We define thedefectof themorphismf as the real number

def(f) := infr ∈ R≥0 | ε · ex ∈ A(x) for everyx ∈ X and everyε ∈ K+ with ρ(|ε|) ≤ −r .Clearlydef(f) ≥ 0 anddef(f) = 0 if and only if (f∗O

+Y )ax is an etaleO+a

X,x-algebra for everyx ∈ X. Furthermore we remark that, by proposition 2.4.19, the mapO+a

X,x → (f∗O+Y )ax is

weakly etale if and only if it is etale.

Corollary 7.5.26. In the situation of(7.5.22), for every real numberr > 0 there existsα ∈ Isuch that, for every morphismγ → α, we have:def(fγ) < r.

Proof. Let r > 0; according to theorem 7.5.23 and claim 7.4.21, there existsα ∈ I such thatε · ex ∈ A(x) for everyγ → α, everyx ∈M(Xγ) and everyε ∈ K+ with ρ(|ε|) ≤ −r. If nowy ∈ Xγ is any point, there is a unique generisationx of y inM(Xγ). Letφ : A(y)→ A(x) bethe induced specialisation map, and setφK := φ⊗K+ 1K . One verifies easily that

φ−1K (m · A(x)) ⊂ A(y).

SinceφK(ey) = ex, the claim follows easily.

7.5.27. To conclude, we want to explain briefly what kind of Galois cohomology calculationsare enabled by the results of this section. Letf : Y → X be a finite etaleGaloismorphismof Spa(K,K+)-adic spaces of finite type, and letG denote the group ofX-automorphisms ofY . Denote byf∗O

+Y [G]-Mod the category off∗O

+Y -modules onX, endowed with a semilin-

ear action ofG. Let ΓG : f∗O+Y [G]-Mod → O+

X-Mod be the functor that associates to anf∗O

+Y [G]-module the sheaf of itsG-invariant local sections. A standard argument shows that,

for everyf∗O+Y [G]-modulesF , the cone of the natural morphism inD(f∗O

+Y -Mod)

O+Y ⊗O+

XRΓGF → F(7.5.28)

is annihilated by allε ∈ m such thatρ(ε) < −def(f). However, for applications one is rathermore interested in understanding the Galois cohomology groupsH i := H i(G,H0(X,F )).One can try to studyH i via (7.5.28); indeed, a bridge between these two objects is providedby the higher derived functors of the related functorΓG : f∗O

+Y [G]-Mod → Γ(X,O+)-Mod,

defined byF 7→ Γ(X,ΓGF ) = Γ(X,F )G. We have two spectral sequences converging toRΓGF , namely

Epq2 : Hp(X,RqΓGF )⇒Rp+qΓGF

F pq2 : Hp(G,Hq(X,F ))⇒Rp+qΓGF .

Using (7.5.28) one deduces thatEpq2 degenerates up to some torsion, which can be estimated

precisely in terms of the defect of the morphismf . However, the spectral sequenceF pq2 contains

the termsHq(X,F ), about which not much is currently known. In this direction,the onlyresults that we could found in the literature concern the calculation ofH i(Y,O+

Y ), for an affinoidspace, under some very restrictive assumptions : in [7] these groups are shown to be almostzero modules fori > 0, in caseY admits a smooth formal model overK+; in [65] the case ofgeneralized polydiscs is taken up, and the same kind of almost vanishing is proven.


8. APPENDIX

In this appendix we have gathered a few miscellaneous results that were found in the courseof our investigation, and which may be useful for other applications.

8.1. Simplicial almost algebras. We need some preliminaries on simplicial objects : first ofall, a simplicial almost algebrais just an object in the categorys.(V a-Alg). Then for a givensimplicial almost algebraA we have the categoryA-Mod of A-modules : it consists of allsimplicial almostV -modulesM such thatM [n] is anA[n]-module and such that the face anddegeneracy morphismsdi : M [n] → M [n − 1] andsi : M [n] → M [n + 1] (i = 0, 1, ..., n) areA[n]-linear.

8.1.1. We will need also the derived category ofA-modules; it is defined as follows. A bitmore generally, letC be any abelian category. For an objectX of s.C let N(X) be thenormalized chain complex (defined as in [46, I.1.3]). By the theorem of Dold-Kan ([67, Th.8.4.1])X 7→ N(X) induces an equivalenceN : s.C → C•(C ) with the categoryC•(C ) ofchain complexes of object ofC that vanish in positive degrees. Now we say that a morphismX → Y in s.C is aquasi-isomorphismif the induced morphismN(X) → N(Y ) is a quasi-isomorphism of chain complexes.

8.1.2. In the following we fix a simplicial almost algebraA.

Definition 8.1.3. We say thatA is exactif the almost algebrasA[n] are exact for alln ∈ N. Amorphismφ : M → N of A-modules (orA-algebras) is aquasi-isomorphismif the morphismφ of underlying simplicial almostV -modules is a quasi-isomorphism. We define the categoryD(A-Mod) (resp. the categoryD(A-Alg)) as the localization of the categoryA-Mod (resp.A-Alg) with respect to the class of quasi-isomorphisms.

8.1.4. As usual, the morphisms inD(A-Mod) can be computed via a calculus of fractionon the categoryHot(A) of simplicial complexes up to homotopy. Moreover, ifA1 andA2 aretwo simplicial almost algebras, then the “extension of scalars” functors define equivalences ofcategories

D(A1 × A2-Mod)∼−→ D(A1-Mod)× D(A2-Mod)

D(A1 × A2-Alg)∼−→ D(A1-Alg)×D(A2-Alg).

Proposition 8.1.5. LetA be a simplicialV a-algebra.

(i) The functor onA-algebras given byB 7→ (s.V a×B)!! preserves quasi-isomorphisms andtherefore induces a functorD(A-Alg)→ D((s.V a × A)!!-Alg).

(ii) The localisation functorR 7→ Ra followed by “extension of scalars” vias.V a × A → Ainduces a functorD((s.V a ×A)!!-Alg)→ D(A-Alg) and the composition of this and theabove functor is naturally isomorphic to the identity functor onD(A-Alg).

Proof. (i) : let B → C be a quasi-isomorphism ofA-algebras. Clearly the induced morphisms.V a × B → s.V a × C is still a quasi-isomorphism ofV -algebras. But by remark 2.2.26,s.V a × B and s.V a × C are exact simplicial almostV -algebras; moreover, it follows fromcorollary 2.2.22 that(s.V a × B)! → (s.V a × C)! is a quasi-isomorphism ofV -modules. Thenthe claim follows easily from the exactness of the sequence (2.2.24). Now (ii) is clear.

Remark 8.1.6. In casem is flat, then allA-algebras are exact, and the same argument showsthat the functorB 7→ B!! induces a functorD(A-Alg)→ D(A!!-Alg). In this case, compositionwith localisation is naturally isomorphic to the identity functor onD(A-Alg).

Proposition 8.1.7. Let f : R→ S be a map ofV -algebras. We have:


(i) If fa : Ra → Sa is an isomorphism, thenLaS/R ≃ 0 in D(s.Sa-Mod).

(ii) The natural mapLaSa/Ra → La

S/R is an isomorphism inD(s.Sa-Mod).

Proof. (ii) is an easy consequence of (i) and of transitivity. To show (i) we prove by inductionon q that

VAN (q;S/R) Hq(LaS/R) = 0.

For q = 0 the claim follows immediately from [46, II.1.2.4.2]. Therefore suppose thatq > 0and thatVAN (j;D/C) is known for all almost isomorphisms ofV -algebrasC → D and allj < q. LetR := f(R). Then by transitivity ([46, II.2.1.2]) we have a distinguished triangle inD(s.Sa-Mod)

(S ⊗R LR/R)au→ La

S/Rv→ La

S/R→ σ(S ⊗R LR/R)a.

We deduce thatVAN (q;R/R) andVAN (q;S/R) imply VAN (q;S/R), thus we can assume thatf is either injective or surjective. LetS• → S be the simplicialV -algebra augmented overSdefined byS• := PV (S). It is a simplicial resolution ofS by freeV -algebras, in particularthe augmentation is a quasi-isomorphism of simplicialV -algebras. SetR• := S• ×S R. Thisis a simplicialV -algebra augmented overR via a quasi-isomorphism. Moreover, the inducedmorphismsR[n]a → S[n]a are isomorphisms. By [46, II.1.2.6.2] there is a quasi-isomorphismLS/R ≃ L∆

S•/R•. On the other hand we have a spectral sequence

E1ij := Hj(LS[i]/R[i])⇒ Hi+j(L

∆S•/R•

).

It follows easily thatVAN(j;S[i]/R[i]) for all i ≥ 0, j ≤ q impliesVAN(q;S/R). Thereforewe are reduced to the case whereS is a freeV -algebra andf is either injective or surjective.We examine separately these two cases. Iff : R→ V [T ] is surjective, then we can find a rightinverses : V [T ] → R for f . By applying transitivity to the sequenceV [T ] → R → V [T ] weget a distinguished triangle

(V [T ]⊗R LR/V [T ])a u→ La

V [T ]/V [T ]v→ La

V [T ]/R → σ(V [T ]⊗R LR/V [T ])a.

SinceLaV [T ]/V [T ] ≃ 0 there follows an isomorphism :Hq(LV [T ]/R)a ≃ Hq−1(V [T ]⊗RLR/V [T ])

a.Furthermore, sincefa is an isomorphism,sa is an isomorphism as well, hence by induction(and by a spectral sequence of the type [46, I.3.3.3.2])Hq−1(V [T ]⊗RLR/V [T ])

a ≃ 0. The claimfollows in this case.

Finally suppose thatf : R→ V [T ] is injective. WriteV [T ] = Sym(F ), for a freeV -moduleF and setF = m⊗V F ; sincefa is an isomorphism,Im(Sym(F )→ Sym(F )) ⊂ R. We applytransitivity to the sequenceSym(F ) → R → Sym(F ). By arguing as above we are reduced toshowing thatLa

Sym(F )/Sym(F )≃ 0.We know thatH0(La

Sym(F )/Sym(F )) ≃ 0 and we will show that

Hq(LaSym(F )/Sym(F )

) ≃ 0 for q > 0. To this purpose we apply transitivity to the sequenceV →Sym(F )→ Sym(F ). AsF andF are flatV -modules, [46, II.1.2.4.4] yieldsHq(LSym(F )/V ) ≃Hq(LSym(F )/V ) ≃ 0 for q > 0 andH0(LSym(F )/V ) is a flat Sym(F )-module. In particularHj(Sym(F ) ⊗Sym(F ) LSym(F )/V ) ≃ 0 for all j > 0. ConsequentlyHj+1(LSym(F )/Sym(F )) ≃ 0

for all j > 0 andH1(LSym(F )/Sym(F )) ≃ Ker(Sym(F ) ⊗Sym(F ) ΩSym(F )/V → ΩSym(F )/V ). Thelatter module is easily seen to be almost zero.

Theorem 8.1.8.Let φ : R → S be a map of simplicialV -algebras inducing an isomorphismRa ∼→ Sa in D(Ra-Mod). Then(L∆

S/R)a ≃ 0 in D(Sa-Mod).

Proof. Apply the base change theorem [46, II.2.2.1] to the (flat) projections ofs.V ×R ontoRand respectivelys.V to deduce that the natural mapL∆

s.V×S/s.V×R → L∆S/R ⊕ L∆

s.V/s.V → L∆S/R


is a quasi-isomorphism inD(s.V × S-Mod). By proposition 8.1.5 the induced morphism(s.V × R)a!! → (s.V × S)a!! is still a quasi-isomorphism. There are spectral sequences

E1ij := Hj(L(V ×R[i])/(V ×R[i])a

!!)⇒ Hi+j(L∆

(s.V×R)/(s.V ×R)a!!)

F 1ij := Hj(L(V ×S[i])/(V×S[i])a

!!)⇒ Hi+j(L∆

(s.V×S)/(s.V×S)a!!).

On the other hand, by proposition 8.1.7(i) we haveLa(V×R[i])/(V ×R[i])a

!!≃ 0 ≃ La

(V ×S[i])/(V×S[i])a!!

for all i ∈ N. Then the theorem follows directly from [46, II.1.2.6.2(b)] and transitivity.

8.1.9. In view of proposition 8.1.7(i) we haveLa(V a×A)!!/V×A!!

≃ 0 in D(V a × A-Mod). Bythis, transitivity and localisation ([46, II.2.3.1.1]) wederive thatLa

B/A → LaB!!/A!!

is a quasi-isomorphism for allA-algebrasB. If A andB are exact (e.g. if m is flat), we conclude fromproposition 2.5.42 that the natural mapLB/A → LB!!/A!!


8.1.10. Finally we want to discuss left derived functors of (the almost version of) some no-table non-additive functors that play a role in deformationtheory. LetR be a simplicialV -algebra. Then we have an obvious functorG : D(R-Mod) → D(Ra-Mod) obtained by ap-plying dimension-wise the localisation functor. LetΣ be the multiplicative set of morphisms ofD(R-Mod) that induce almost isomorphisms on the cohomology modules.An argument as insection 2.4 shows thatG induces an equivalence of categoriesΣ−1D(R-Mod)→ D(Ra-Mod).

8.1.11. Now letR be aV -algebra andFp one of the functors⊗p, Λp, Symp, Γp defined in[46, I.4.2.2.6].

Lemma 8.1.12.Let φ : M → N be an almost isomorphism ofR-modules. ThenFp(φ) :Fp(M)→ Fp(N) is an almost isomorphism.

Proof. Let ψ : m ⊗V N → M be the map corresponding to(φa)−1 under the bijection (2.2.4).By inspection, the compositionsφ ψ : m ⊗V N → N andψ (1m ⊗ φ) : m ⊗V M → Mare induced by scalar multiplication. Pick anys ∈ m and lift it to an elements ∈ m; defineψs : N →M by n 7→ ψ(s⊗ n) for all n ∈ N . Thenφ ψs = s · 1N andψs φ = s · 1M . Thiseasily implies thatsp annihilatesKer Fp(φ) andCoker Fp(φ). In light of proposition 2.1.7(ii),the claim follows.

8.1.13. LetB be an almostV -algebra. We define a functorF ap onB-Mod by lettingM 7→

(Fp(M!))a, whereM! is viewed as aB!!-module or aB∗-module (to show that these choices

define the same functor it suffices to observe thatB∗ ⊗B!!N ≃ N for all B∗-modulesN such

thatN = m ·N). For allp > 0 we have diagrams :

R-ModFp //

R-Mod

Ra-Mod

Fap //

OO

Ra-Mod

OO(8.1.14)

where the downward arrows are localisation and the upward arrows are the functorsM 7→ M!.Lemma 8.1.12 implies that the downward arrows in the diagramcommute (up to a naturalisomorphism) with the horizontal ones. It will follow from the following proposition 8.1.16that the diagram commutes also going upward.

8.1.15. For anyV -moduleN we have an exact sequenceΓ2N → ⊗2N → Λ2N → 0. Asobserved in the proof of proposition 2.1.7, the symmetric groupS2 acts trivially on⊗2m andΓ2m ≃ ⊗2m, soΛ2m = 0. Also we have natural isomorphismsΓpm ≃ m for all p > 0.


Proposition 8.1.16.LetR be a commutative ring andL a flatR-module withΛ2L = 0. Thenfor p > 0 and for allR-modulesN we have natural isomorphisms

Γp(L)⊗R Fp(N)∼→ Fp(L⊗R N).

Proof. Fix an elementx ∈ Fp(N). For eachR-algebraR′ and each elementl ∈ R′ ⊗R L weget a mapφl : R′⊗RN → R′⊗RL⊗RN by y 7→ l⊗y, hence a mapFp(φl) : R′⊗RFp(N) ≃Fp(R

′ ⊗R N) → Fp(R′ ⊗R L ⊗R N) ≃ R′ ⊗R Fp(L ⊗R N). For varyingl we obtain a

map of setsψR′,x : R′ ⊗R L → R′ ⊗R Fp(L ⊗R N) : l 7→ Fp(φl)(1 ⊗ x). According tothe terminology of [61], the system of mapsψR′,x for R′ ranging over allR-algebras forms ahomogeneous polynomial law of degreep from L to Fp(L ⊗R N), so it factors through theuniversal homogeneous degreep polynomial lawγp : L → Γp(L) . The resultingR-linearmapψx : Γp(L) → Fp(L ⊗R N) dependsR-linearly onx, hence we derive anR-linear mapψ : Γp(L)⊗RFp(N)→ Fp(L⊗RN). Next notice that by hypothesisS2 acts trivially on⊗2L soSp acts trivially on⊗pL and we get an isomorphismβ : Γp(L)

∼−→ ⊗pL. We deduce a naturalmap(⊗pL) ⊗R Fp(N) → Fp(L ⊗R N). Now, in order to prove the proposition for the caseFp = ⊗p, it suffices to show that this last map is just the natural isomorphism that “reordersthe factors”. Indeed, letx1, ..., xn ∈ L andq := (q1, ..., qn) ∈ Nn such that|q| := ∑

i qi := p;thenβ sends the generatorx[q1]

1 · ... ·x[qn]n to

(p

q1,...,qn

)·x⊗q11 ⊗ ...⊗x⊗qnn . On the other hand, pick

anyy ∈ ⊗pN and letR[T ] := R[T1, ..., Tn] be the polynomialR-algebra inn variables; write(T1 ⊗ x1 + ... + Tn ⊗ xn)⊗p ⊗ y = ψR[T ],y(T1 ⊗ x1 + ... + Tn ⊗ xn) =

∑r∈Nn T r ⊗ wr with

wr ∈ ⊗p(L ⊗R N). Thenψ((x[q1]1 · ... · x[qn]

n ) ⊗ y) = wq (see [61, pp.266-267]) and the claimfollows easily. Next notice thatΓp(L) is flat, so that tensoring withΓp(L) commutes with takingcoinvariants (resp. invariants) under the action of the symmetric group; this implies the assertionfor Fp := Symp (resp.Fp := TSp). To deal withFp := Λp recall that for anyV -moduleMandp > 0 we have the antisymmetrizer operatoraM :=

∑σ∈Sp

sgn(σ)·σ : ⊗pM → ⊗pM and asurjectionΛp(M)→ Im(aM) which is an isomorphism forM free, hence forM flat. The resultfor Fp = ⊗p (and again the flatness ofΓp(L)) then givesΓp(L)⊗Im(aN) ≃ Im(aL⊗RN), hence

the assertion forFp = Λp andN flat. For generalN let F1∂→ F0

ε→ N → 0 be a presentationwith Fi free. Definej0, j1 : F0 ⊕ F1 → F0 by j0(x, y) := x + ∂(y) andj1(x, y) := x. Byfunctoriality we derive an exact sequence

Λp(F0 ⊕ F1)//// Λp(F0) // Λp(N) // 0

which reduces the assertion to the flat case. ForFp := Γp the same reduction argument works aswell (cp. [61, p.284]) and for flat modules the assertion forΓp follows from the correspondingassertion forTSp.

Lemma 8.1.17.LetA be a simplicial almost algebra,L,E andF threeA-modules,f : E → Fa quasi-isomorphism. IfL is flat orE,F are flat, thenL⊗A f : L⊗A E → L⊗A F is a quasi-isomorphism.

Proof. It is deduced directly from [46, I.3.3.2.1] by applyingM 7→M!.

8.1.18. As usual, this allows one to show that⊗ : Hot(A)× Hot(A)→ Hot(A) admits a left

derived functorL

⊗ : D(A-Mod)× D(A-Mod) → D(A-Mod). If R is a simplicialV -algebrathen we have essentially commutative diagrams

D(R-Mod)× D(R-Mod)L

⊗ //

D(R-Mod)

D(Ra-Mod)× D(Ra-Mod)

L

⊗ //

OO

D(Ra-Mod)

OO


where again the downward (resp. upward) functors are induced by localisation (resp. byM 7→M!).

8.1.19. We mention the derived functors of the non-additivefunctorFp defined above in thesimplest case of modules over a constant simplicial ring. Let A be a (commutative)V a-algebra.

Lemma 8.1.20. If u : X → Y is a quasi-isomorphism of flats.A-modules thenF ap (u) :

F ap (X)→ F a

p (Y ) is a quasi-isomorphism.

Proof. This is deduced from [46, I.4.2.2.1] applied toN(X!) → N(Y!) which is a quasi-isomorphism of chain complexes of flatA!!-modules. We note thatloc. cit. deals with a moregeneral mixed simplicial construction ofFp which applies to bounded above complexes, butone can check that it reduces to the simplicial definition forcomplexes inC•(A!!).

8.1.21. Using the lemma one can constructLF ap : D(s.A-Mod) → D(s.A-Mod). If R is a

V -algebra we have the derived category version of the essentially commutative squares (8.1.14),relatingLFp : D(s.R-Mod)→ D(s.R-Mod) andLF a

p : D(s.Ra-Mod)→ D(s.Ra-Mod).

8.2. Fundamental group of an almost algebra.We will need some generalities from [40,Exp.V] and [5, Exp.VI]. In the following we fix a universeU and suppose that all our categoriesareU-categories and all our topoi areU-topoi in the sense of [4, Exp.I, Def.1.1 and Exp.IV,Def.1.1]. No further mention of universes will be necessary.

8.2.1. LetC be a site. Recall ([5, Exp.VI, Def.1.1]) that an objectX of C is calledquasi-compactif, for every covering family(Xi → X | i ∈ I) there is a finite subsetJ ⊂ I such thatthe subfamily(Xj → X | j ∈ J) is still covering.

8.2.2. LetE be a topos; in the following we always endowE with its canonical topology ([4,Exp.II, Def 2.5]), soE is a site in a natural way and the terminology of (8.2.1) applies to theobjects ofE. Moreover, ifC is any site andε : C → C ∼ the natural functor to the categoryC ∼ of sheaves onC , then an objectX of C is quasi-compact inC if and only if ε(X) isquasi-compact inC ∼ ([5, Exp.VI, Prop.1.2]).

Furthermore, since inE all finite limits are representable, we can make the following furtherdefinitions ([5, Exp.VI, Def.1.7]). A morphismf : X → Y in E is calledquasi-compactif, forevery morphismY ′ → Y in E with quasi-compactY ′, the objectX ×Y Y ′ is quasi-compact.We say thatf is quasi-separatedif the diagonal morphismX → X ×Y X is quasi-compact.We say thatf is coherentif it is quasi-compact and quasi-separated.

8.2.3. LetX be an object of a toposE. We say thatX is quasi-separatedif, for every quasi-compact objectS of E, every morphismS → X is quasi-compact. We say thatX is coherentif it is quasi-compact and quasi-separated ([5, Exp.VI, Def.1.13]). We denote byEcoh the fullsubcategory ofE consisting of all the coherent objects.

Suppose that the objectY of E is coherent and letf : X → Y be a coherent morphism; by[5, Exp.VI, Prop.1.14(ii)], it follows thatX is coherent.

Definition 8.2.4. (cp. [5, Exp.VI, Def.2.3]) We say that a toposE is coherentif it satisfies thefollowing conditions:

(i) E admits a full generating subcategoryC consisting of coherent objects.(ii) Every objectX of C is quasi-separated over the final object ofE, i.e. the diagonal mor-

phismX → X ×X is quasi-compact.(iii) The final object ofE is quasi-separated.


8.2.5. If E is a coherent topos, thenEcoh is stable under arbitrary finite limits (ofE) ([5,Exp.VI, 2.2.4]). Moreover, a toposE is coherent if and only if it is equivalent to a topos ofthe formC ∼, whereC is a site whose objects are quasi-compact and whose finite limits arerepresentable.

It is possible to characterize the categories of the formEcoh arising from a coherent topos :this leads to the following definition.

Definition 8.2.6. A pretoposis a small categoryC satisfying the following conditions ([5,Exp.VI, Exerc.3.11]).

(PT1) All finite limits are representable inC .(PT2) All finite sums are representable inC and they are universal and disjoint.(PT3) Every equivalence relation inC is effective, and every epimorphism is universally effec-

tive.

8.2.7. As in [5, Exp.VI, Exerc.3.11], we leave to the reader the verification that, for everycoherent toposE, the subcategoryEcoh is a pretopos, andE induces onEcoh theprecanonicaltopology,i.e. the topology whose covering families(Xi → X | i ∈ I) are those admitting afinite subfamily which is covering for the canonical topology of Ecoh. One deduces thatE isequivalent to(Ecoh)

∼, the topos of sheaves on the precanonical topology ofEcoh.Conversely, ifC is a pretopos, letE := C ∼ be the topos of sheaves on the precanonical

topology ofC ; thenE is a coherent topos and the natural functorε : C → E induces anequivalence ofC with Ecoh.

8.2.8. Furthermore, ifC is a pretopos (endowed with the precanonical topology), thenaturalfunctorC → C ∼ commutes with finite sums, with quotients under equivalencerelations, and itis left exact (i.e. commutes with finite limits) ([5, Exp.VI, Exerc.3.11]).

8.2.9. Recall ([4, Exp.IV, Def.6.1]) that apointof a toposE is a morphism of topoip : Set→E (where one views the categorySet as the topos of sheaves on the one-point topologicalspace). By [4, Exp.IV, Cor.1.5], the assignmentp 7→ p∗ defines an equivalence from the cate-gory of points ofE to the opposite of the category of all functorsF : E → Set that commutewith all colimits and are left exact. A functorF : E → Set with these properties is called afibre functorfor E. By [4, Exp.IV, Cor.1.7], a functorE → Set is a fibre functor if and only ifit is left exact and it takes covering families to surjectivefamilies.

8.2.10. By a theorem of Deligne ([5, Exp.VI, Prop.9.0]) every coherent non-empty topos ad-mits at least a fibre functor. (Actually Deligne’s result is both more precise and more general,but for our purposes, the foregoing statement will suffice).

In several contexts, it is useful to attach fibre functors to categories that are not quite topoi.These situations are axiomatized in the following definition.

Definition 8.2.11. A Galois category([40, Exp.V, §5]) is the datum of a categoryC and afunctorF from C to the categoryf .Set of finite sets, satisfying the following conditions:(G1) all finite limits exist inC (in particularC has a final object).(G2) Finite sums exist inC (in particularC has an initial object). Also, for every objectX of

C and every finite groupG of automorphisms ofX, the quotientX/G exists inC .

(G3) Every morphismu : X → Y in C factors as a compositionXu′→ Y ′

u′′→ Y , whereu′

is a strict epimorphism andu′′ is both a monomorphism and an isomorphism on a directsummand ofY .

(G4) The functorF is left exact.(G5) F commutes with finite direct sums and with quotients under actions of finite groups of

automorphisms. MoreoverF takes strict epimorphisms to epimorphisms.(G6) Letu : X → Y be a morphism ofC . Thenu is an isomorphism if and only ifF (u) is.


8.2.12. Given a Galois category(C , F ), one says thatF is afibre functorof C . It is shown in[40, Exp.V,§4] that for any Galois category(C , F ) the functorF is pro-representable and itsautomorphism group is a profinite groupπ in a natural way. Furthermore,F factors naturallythrough the forgetful functorπ-f .Set → f .Set from the categoryπ-f .Set of (discrete) finitesets with continuousπ-action, and the resulting functorC → π-f .Set is an equivalence.

8.2.13. The category of etale coveringsCov(A) of an almost algebraA (to be defined in(8.2.22)) is not directly presented as a Galois category, since it does not afford an a priori choiceof fibre functor; rather, the existence of a fibre functor is deduced from Deligne’s theorem. Theargument only appeals to some general properties of the categoryCov(A), which are abstractedin the following definition 8.2.14 and lemma 8.2.15.

Definition 8.2.14. A pregalois categoryis a categoryC satisfying the following conditions.

(PG1) Every monomorphismX → Y in C induces an isomorphism ofX onto a direct summandof Y .

(PG2) C admits a final objecte which is connected and non-empty (that is,e is not an initialobject).

(PG3) For every objectX of C , there existsn ∈ N such that, for every non-empty objectY of C ,

the productX×Y exists inC and is notY -isomorphic to an object of the form(n

∐Y )∐ Z(whereZ is any other object ofC ).

Lemma 8.2.15.Let C be a pregalois pretopos. Then there exists a functorF : C → f .Set

such that(C , F ) is a Galois category.

Proof. (G1) holds because it is the same as (PT1). (G2) follows easily from (PT2) and (PT3).In order to show (G3) we will need the following:

Claim8.2.16. A morphismu : X → Y of C is an isomorphism if and only if it is both amonomorphism and an epimorphism.

Proof of the claim:One direction is obvious, so we can suppose thatu is both a monomorphismand an epimorphism. By (PG1), it follows that, up to composing with an isomorphism,Y =X∐Z for some objectZ of C , andu can be identified with the natural morphismX → X∐Z.Let v : Z → X ∐ Z be the natural morphism; by (PT3) the induced morphismu ×Y Z :X ×Y Z → Z is an epimorphism and by (PT2) we haveX ×Y Z ≃ ∅, the initial object ofC .Since inC all epimorphisms are effective, one derives thatZ ≃ ∅, and the claim follows.

Now, letu : X → Y be a morphism inC ; the induced morphisms

pr1, pr2 : X ×Y X //// X

define an equivalence relation; by (PT3) there is a corresponding quotient morphismu′ : X →Y ′ and moreoveru′ is a strict epimorphism. Clearlyu factors via a morphismu′′ : Y ′ → Y . Weneed to show thatu′′ is a monomorphism, or equivalently, that the induced diagonal morphismδ : Y ′ → Y ′ ×Y Y ′ is an isomorphism. However, there is a natural commutative diagram

X ×Y ′ Xα //

X ×Y X

Y ′δ // Y ′ ×Y Y ′

whereα is an isomorphism by construction and both vertical arrows are epimorphisms. Itfollows thatδ is an epimorphism; since it is obviously a monomorphism as well, we deduce(G3) in view of (PG1) and claim 8.2.16. LetC ∼ be the topos of sheaves on the precanonicaltopology ofC ; by (8.2.7) and Deligne’s theorem (8.2.10), there exists a fibre functorC ∼ →


Set. Composing with the natural functorC → C ∼ we obtain a functorF : C → Set whichis left exact, commutes with finite sums and quotients under equivalence relations, in view of(8.2.8), so (G4) and (G5) hold forF .

Claim 8.2.17. LetX be a non-empty object ofC . ThenF (X) 6= ∅.

Proof of the claim:Since we know already that (G3) holds inC , we deduce using (PG2) thatthe unique morphismX → e is an epimorphism. Then (PT3) says thate can be written as thequotient ofX under the induced equivalence relationpr1, pr2 : X ×e X //

// X . SinceFcommutes with quotients under such equivalence relations,the claim follows after remarkingthatF (e) 6= ∅,

Claim 8.2.18. Let u : X → Y be a morphism inC such thatF (u) is surjective. Thenu is anepimorphism.

Proof of the claim:We use (G3) to factoru as an epimorphismu′ : X → Y ′ followed by amonomorphism of the formY ′ → Y ′ ∐ Z. We need to show thatZ = ∅ or equivalently, inview of claim 8.2.17, thatF (Z) = ∅. However, the assumption implies thatF (Y ′) maps ontoF (Y ′ ∐ Z); on the other hand,F commutes with finite sums, so the claim holds.

Claim 8.2.19. Let u : X → Y be a morphism inC such thatF (u) is injective. Thenu is amonomorphism.

Proof of the claim:The assumption means that the induced diagonal mapF (X)→ F (X)×F (Y )

F (X) is bijective. Then claim 8.2.18 implies that the diagonal morphismX → X ×Y X is anepimorphism. The latter is also obviously a monomorphism, hence an isomorphism, in view ofclaim 8.2.16; but this means thatu is a monomorphism.

Now, taking into account claims 8.2.16, 8.2.18 and 8.2.19 wededuce that (G6) holds forF .It remains only to show thatF takes values in finite sets. So, suppose by contradiction thatF (X) is an infinite set for some objectX of C . We define inductively a sequence of objects(Yi | i ∈ N), with morphismsφi+1 : Yi+1 → Yi for every i ∈ N, as follows. LetY0 := e,Y1 := X andφ1 the unique morphism. Let theni > 0 and suppose thatYi andφi have alreadybeen given. Using the diagonal morphism, we can writeYi×Yi−1

Yi ≃ Yi∐Z for some objectZ;we setYi+1 := Z and letφi+1 be the restriction of the projectionpr1 : Yi×Yi−1

Yi → Yi. NoticethatF (Yi) 6= ∅ for everyi ∈ N (indeed all the fibers of the induced mapF (Yi) → F (Yi) areinfinite wheneveri > 0); in particularYi 6= ∅ for everyi ∈ N. On the other hand, for every

n > 0,X ×Yn admits a decomposition of the form(n

∐Yn)∐Yn+1, which is against (PG3). Thecontradiction concludes the proof of the lemma.

8.2.20. LetE be a topos. Recall that an objectX of E is said to beconstantif it is a directsum of copies of the final objecte of E ([6, Exp.IX, §2.0]). The objectX is locally constantifthere exists a covering(Yi → e | i ∈ I) of e, such that, for everyi ∈ I, the restriction ofX × Yiis constant on the induced toposE/Yi

([4, Exp.III, §5.1]). If additionally there exists an integern, such that so that each(X × Yi)|Yi

is a direct sum of at mostn copies ofYi, then we say thatX is bounded.

Lemma 8.2.21.LetE be a topos. Denote byElcb the full subcategory of all locally constantbounded objects ofE (see(8.2.3)). Then:

(i) Elcb is a pretopos.(ii) If the final object ofe is connected and non-empty,Elcb is a pregalois pretopos.

Proof. Let X be an object ofElcb, and(Yi → e) a covering of the final object ofE by non-empty objects, such thatX × Yi is constant onE/Yi

for everyi ∈ I, say(X × Yi)|Yi≃ Yi× Si,

whereSi is some set. SinceX is bounded, there existsn ≥ 0 such that the cardinality of


everySi is no greater thann. Since all finite limits are representable inE, axiom (PT1) can bechecked locally onE, so we can reduce to the case of a finite inverse systemZ := (Zj | j ∈ J)of constant bounded objectsZj := e × Sj . Furthermore, thanks to [6, Exp.IX, Lemme 2.1(i)]we can assume that the inverse system is induced from an inverse systemS := (Sj | j ∈ J) ofmaps of sets, in which case it is easy to check thatlim

JZ ≃ lim

JS, which implies (PT1).

(PT2) is immediate. Similarly, ifR is a locally constant bounded equivalence relation onan objectX of Elcb, thenX/R is again inElcb; indeed, since equivalence relations inE areuniversally effective, this can be checked locally on a covering (Yi → e | i ∈ I). Then again,by [6, Exp.IX, Lemme 2.1(i)] we can reduce to the case of an equivalence relation on sets,where everything is obvious. This shows that (PT3) holds, and proves (i). Suppose next thate is connected and non-empty; sincee is in Elcb, it follows that (PG2) holds inElcb. To show(PG1), consider a morphismu : X → Y in Elcb. As in the foregoing, we can find a covering(Zi → e | i ∈ I) such that(X×Zi)|Zi

≃ Zi×Si, (Y ×Zi)|Zi≃ Zi×Ti for some setsSi, Ti, and

u|Ziis induced by a mapui : Si → Ti. LetS ′i := Ti \ ui(Si). Clearly we have an isomorphism

(Y ×Zi)|Zi≃ (X×Zi)|Zi

∐ (Zi×S ′i) for everyi ∈ I. Since the induced decompositions agreeonZi × Zj for everyi, j ∈ I, the constant objectsZi × S ′i glue to a locally constant objectX ′

of E, andu induces an isomorphismY ≃ X ∐X ′. Finally, if X is inElcb, find sets(Si | i ∈ I)and a covering(Yi → e | i ∈ I) such that(X × Yi)|Yi

≃ Si × Yi, and letm be the maximum ofthe cardinalities of the setsSi. Clearly (PG3) holds forX, if one choosesn := m+ 1.

8.2.22. LetA be an almost algebra. We consider the siteSA := (A-Alg)ofpqc obtained byendowing the category of affineA-schemes with the fpqc topology (see (4.4.23)). Moreover,the category ofetale coveringsof SpecA is defined as the full subcategoryCov(SpecA) of thecategory of affineA-schemes consisting of all etaleA-schemes of finite rank.

Proposition 8.2.23.The natural functorε : SA → S ∼A induces an equivalence of the category

Cov(SpecA) onto the category of locally constant bounded sheaves onSA.

Proof. Let B be an etaleA-algebra of rankr. By proposition 4.3.27 there is an isomorphismof almost algebrasA ≃ ∏r

i=0Ai such thatBi := B ⊗A Ai is anAi-algebra of constant ranki for every i = 0, ..., r. In particular,B0 = 0 andBi is faithfully flat etaleAi-algebra foreveryi > 0. We use the diagonal morphism to obtain a decompositionBi ⊗Ai

Bi ≃ Bi × Ci,whereCi is again an etaleBi-algebra of constant ranki − 1. Iterating this procedure we findfaithfully flat Ai-algebrasDi such thatB⊗ADi isDi-isomorphic to a direct product ofi copiesof Di, for everyi > 0. SettingD0 := A0, we obtain a covering(ε(SpecDi → SpecA) | i =0, ..., r) of ε(SpecA) in S ∼

A such that the restriction ofε(SpecB) to eachε(SpecDi) is abounded constant sheaf. This show that the restriction ofε toCov(SpecA) lands in the categoryof locally constant bounded sheaves. Since the fpqc topology is coarser than the canonicaltopology,ε is fully faithful. To show thatε is essentially surjective amounts to an exercise infaithfully flat descent : clearly every constant sheaf is represented by an etaleA-scheme; thenone uses [6, Exp.IX, Lemme 2.1(i)] to show that any descent datum of bounded constant sheavesis induced by a cocycle system of morphisms for the corresponding representing algebras, andone can descend the latter. We leave the details to the reader.

8.2.24. Let us say that an affine almost schemeS is connectedif ε(S) is a connected object ofthe categoryS ∼

OS, which simply means that the only non-zero idempotent ofOS∗ is the identity.

In this case, proposition 8.2.23 and lemma 8.2.21 show thatCov(S) is a pregalois pretopos,hence it admits a fibre functorF : Cov(S)→ f .Set by lemma 8.2.15.

Definition 8.2.25. Suppose that the affine almost schemeS is connected. Thefundamentalgroup of S is the groupπ1(S) defined as the automorphism group of any fibre functorF :Cov(S)→ f .Set, endowed with its natural profinite topology (see (8.2.12)).


8.2.26. It results from the general theory ([40, Exp.V,§5]) thatπ1(S) is independent (up toisomorphism) on the choice of fibre functor. We refer toloc.cit. for a general study of fun-damental groups of Galois categories. In essence, several of the standard results for schemesadmit adequate almost counterpart. We conclude this section with a sample of such statements.

Definition 8.2.27. Let A ⊂ B be a pair ofV a-algebras; theintegral closureof A in B is thesubalgebrai.c.(A,B) := i.c.(A∗, B∗)

a, where for a pair of ringsR ⊂ S we leti.c.(R, S) be theintegral closure ofR in S.

Lemma 8.2.28. If R ⊂ S is any pair ofV -algebras, theni.c.(Ra, Sa) = i.c.(R, S)a.

Proof. It suffices to show the following:

Claim 8.2.29. Given a commutative diagram ofV -algebras

R //

S

R1

// S1

whose vertical arrows are almost isomorphism, the induced mapi.c.(R, S)→ i.c.(R1, S1) is analmost isomorphism.

Proof of the claim:Clearly the kernel of the map is almost zero; it remains to show that foreveryb ∈ i.c.(R1, S1) andε ∈ m, the elementεb lifts to i.c.(R, S). By assumption we have arelationbN +

∑Ni=1 aib

N−i = 0, with ai ∈ R1, so(εb)N +∑N

i=1 εiai(εb)

N−i = 0. By liftingεiai to someai ∈ R, we deduce a monic polynomialP (T ) overR such thatP (εb) = 0, so if bis a lifting of εb, we havem · P (b) = 0. Since the restrictionmS → mS1 is surjective, we canchooseb ∈ mS, sob · P (b) = 0.

Remark 8.2.30. (i) If A ⊂ B areV a-algebras, thenA is integrally closed inB if and only ifA∗ is integrally closed inB∗. Indeed, by lemma 8.2.28 we know that the integral closure ofA∗in B∗ is almost isomorphic toA∗ and any suchV -algebra must be contained inA∗.

(ii) If (Ai ⊂ Bi | i ∈ I) is a (possibly infinite) family of pairs ofV a-algebras, then∏

i∈I Aiis integrally closed in

∏i∈I Bi if and only ifAi is integrally closed inBi for everyi ∈ I.

The following proposition is an analogue of [30, Prop.18.12.15].

Proposition 8.2.31.LetA ⊂ B be a pair ofV a-algebras such thatA = i.c.(A,B).

(i) For any etale almost finite projectiveA-algebraA1 of almost finite rank we haveA1 =i.c.(A1, A1 ⊗A B).

(ii) Suppose thatA and B are connected, and choose a fibre functorF for the categoryCov(B). Then the functorCov(A) → f .Set : C 7→ F (C ⊗A B) is a fibre functor,and the induced group homomorphism:π1(SpecB)→ π1(SpecA) is surjective.

Proof. (i): using remark 8.2.30(ii) we reduce to the case whereA1 has constant finite rankoverA. SetB1 := A1 ⊗A B, and suppose thatx ∈ B1∗ is integral overA1∗, or equivalently,overA∗. Consider the elemente ∈ (A1 ⊗A A1)∗ provided by proposition 3.1.4; for givenε ∈ m write ε · e =

∑ki ci ⊗ di for someci, di ∈ A1∗. According to remark 4.1.17 we have:∑k

i ci · TrB1/B(xdi) = ε · x for everyx ∈ B1∗. Corollary 4.4.31 implies thatTrB1/B(xdi) isintegral overA∗ for everyi ≤ k, hence it lies inA∗, by remark 8.2.30(i). Henceε · x ∈ A1∗, asclaimed.

(ii): is an immediate consequence of (i) and of the general theory of [40, Exp.V,§5].


8.2.32. As a special case, letR be aV -algebra whose spectrum is connected; we supposethat there exists an elementε ∈ m which is regular inR. Suppose moreover thatR is in-tegrally closed inR[ε−1], consequentlySpecR[ε−1] is connected andπ1(SpecR[ε−1]) is welldefined. It follows as well thatSpecRa is connected. Indeed, ifSpecRa were not connected,neither would beSpecRa

∗; but sinceRa∗ ⊂ R[ε−1], this is absurd. Thenπ1(SpecRa) is also

well defined and, after a fibre functor forCov(R[ε−1]) is chosen, the functorsCov(SpecR)→Cov(SpecRa) → Cov(SpecR[ε−1]) : B 7→ Ba 7→ Ba

∗ [ε−1] induce continuous group homo-

morphisms ([40, Exp.V, Cor.6.2])

π1(SpecR[ε−1])→ π1(SpecRa)→ π1(SpecR).(8.2.33)

Proposition 8.2.34.Under the assumptions of(8.2.32), we have:(i) Ra

∗ is integrally closed inR[ε−1].(ii) The maps(8.2.33)are surjective.

Proof. (i): let x ∈ R[ε−1] such thatxn+a1·xn−1+...+an = 0 for some elementsa1, ..., an ∈ Ra∗.

For everyδ ∈ m we have(δ ·x)n+(δ ·x)·(δ ·x)n−1+...+δn ·an = 0, which shows thatδ ·x ∈ R,since by assumptionR is integrally closed inR[ε−1] andδi · ai ∈ R for everyi = 1, ..., n. Theassertion follows.

(ii): it suffices to show the assertion for the leftmost map and for the composition of the twomaps. However, the composition of the two maps is actually a special case of the leftmost map(for the classical limitV = m), so we need only consider the leftmost map. Then the assertionfollows from (i) and proposition 8.2.31(ii).


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542.

INDEX

(V,m), m : Basic setup, 2.1.1(t, I)-adic topology on a module, 5.4.10(t, I)-adic topology on a scheme, 5.4.15A∞-Modtop : topological modules on a complete

almost algebra, 5.3.14B(m), 3.5.7IB/A, 2.5.22K+ : Valuation ring(s), 6.1.11K+h,K+sh,Ksh : (strict) henselization of a, 6.1.12,

6.2.16standard setup attached to a, 6.1.15total log structure on a, 6.4.14

Ma, 2.2.2R-Mod,R-Alg (for a ringR), 2.2.1V -Mon : category ofV -monoids, 3.5.2V a-Alg,A-Mod,A-Alg (for an almost algebraA)

, 2.2.7V a-Alg.Morph, V a-Alg.Mod, 2.5.22V a-Mod, (V,m)a-Mod, 2.2.2alHomA(M,N) : Almost homomorphisms, 2.2.10B!!, 2.2.23, 3.3.5Desc(C , Y/X), 3.4.23Idemp(R) : idempotent elements of a ring, 3.4.42I!∗, 2.5.19LG

X/Y,k, 52Lan

B/A : Analytic cotangent complex for topologicalalgebras, 7.1.23

M!, 2.2.19Ωna

P/K : non-analytic differentials, 7.3.28Ωan

B/A : module of analytic differentials for topolog-ical algebras, 7.1.23, 7.2.34

Proj(A) : projective almost scheme associated to agraded almost algebra, 5.7.11

Set : category of sets, 2.2.1TorA

i (M,N), alExtiA(M,N), 2.4.10ExalA,A†-Uni.Mod, 2.5.7, 2.5.8µ(F ), 6.2.3B : category of basic setups, 3.5B-monoid(s), 3.5.2

ΦA : Frobenius endomorphism of, 3.5.6B-Mon : category of, 3.5.2Frobenius nilpotent, 3.5.8invertible morphism up toΦm of , 3.5.8

B/Fp, B-Alg/Fp, B-Mon/Fp, Ba-Alg/Fp , 3.5.5B-Alg : category ofB-algebras, 3.5.2B-Mod : category ofB-modules, 3.5.1Ba-Mod, Ba-Alg, Ba-Mon, 3.5.3C : Category(ies)

C o : opposite of a, 2.21X : identity morphism of an object in a, 2.2D(C ), D+(C ), D−(C ) : derived categories of a,

2.2.1s.C : simplicial objects of a, 2.2Galois, 8.2.11

fibre functor of a, 8.2.12Pregalois, 8.2.14

tensor, 2.2.5X ∗φ,ψ∗X : push-out, pull-back of extensions

in a, 2.5.5ExmonC (B, I), ExalC (B, I) : isomorphism classes

of extensions in a, 2.5.5, 2.5.7ExmonC (B, I), ExunC , ExalC : extensions

of a monoid (or an algebra) in a , 2.5.2, 2.5.4µA : multiplication of a monoid in a, 2.2.5σM/A : scalar multiplication of a module in a,

2.2.5(left, right, unitary) module in a , 2.2.5θM|N : switch operator for two modules in a,

2.2.5monoid, algebra in a, 2.2.5

D-Mod : modules over a diagramD of almost al-gebras, 3.4.7

Gn, G∞, 4.2.6, 4.3.9IA(M),EM (m0),EM (m0), 2.3.1Oa

X -Mod, OaX -Alg : sheaves ofOa

X -modules, resp.Oa

X -algebras on a schemeX , 5.5.1Oa

X -Modqcoh, OaX -Algqcoh : quasi-coherent, 5.5.1

F!, F∗ (for anOaX -moduleF ), 5.5.3

χP (X), ψP (X), 4.3.7det(1P + φ), 4.3.1ηB,C , 4.1.13u.Et

o, 3.4.44

ExtpR(E,F ), 2.5.21HomOX

(F , ∗), 6.4.10MndX , Z-ModX , 6.4.1ω(R, u′) : obstruction class, 3.2.20ω(B, f0, u), ω(B, f0) : obstruction classes, 3.2.7,

3.2.10ωP/A, 2.4.26φ♯ (for a morphismφ of almost schemes), 3.3.3ρE+/K+ , 6.5.1D−(A-Mod)∧, 7.1.13HR(F, J), 5.4.1j : morphism of fpqc-sites, 5.7τe, τw : topologies onV a-Alg, 3.4.42ζP , 4.1.1uM , 4.4.32Bud(n, d,R) : n-buds of a formal group, 4.2.2

Gm,R(n) : n-bud ofGm, 4.2.1N (A) : Spectrum of seminorms of a ring, 7.5.11(n-)pseudo-coherent complex(es), 7.1.15

D(R-Mod)coh : derived category of, 7.1.16

Adic space(s), 7.2.23ExanY (X,F ) : analytic deformations of an, 7.3.12L+

X/Y , LanX/Y : analytic cotangent complex of a

morphism of, 7.2.31Ω+

X/K : sheaf of analytic differentials of an, 7.4d(X) : adic space associated to a formal scheme,

7.2.26etale morphism of, 7.2.25

232


d+Y/X : discriminant of an, 7.5.10

def(f) : defect of an, 7.5.25affinoid, 7.2.23deeply ramified system of, 7.4.10morphism locally of finite type of, 7.2.24smooth, unramified morphism of, 7.2.25

Affine almost group scheme(s), 3.3.3ℓG/S : colie complex of an, 3.3.33(right, left) action on an almost scheme by a, 3.3.6(right, left) action on an almost scheme of aG•

X : nerve of an, 3.3.6action on a quasi-coherent module, equivariant quasi-

coherent module, 3.3.6torsor over an, 3.3.28

Affinoid ring(s), 7.2.17SpaA : adic spectrum of an, 7.2.22completion of, 7.2.18

Almost algebra(s), 2.2.5, 2.2.7A-Etafp : etale almost finitely presented, 3.2.27Cov(SpecA) : etale covering of an, 8.2.22DerA(B,M) : derivations of an, 2.5.22LB/A : cotangent complex of an, 2.5.20

transitivity of the , 2.5.32ΩB/A : relative differentials of an , 2.5.22SpecA : spectrum of an, 3.3.3nil(A) : nilradical of an, 4.1.18rad(A) : Jacobson radical, 5.1.1(faithfully) flat, 3.1.1(tight) henselian pair, 5.1.8(uniformly) almost finite, 3.1.1(weakly) unramified, 3.1.1eB/A : idempotent associated to an, 3.1.4

(weakly) etale, 3.1.1A-w.Et,A-Et : category of, 3.2.17

almost finite projective, 3.1.1almost projective, 3.1.1

TrB/A : trace morphism of an , 4.1.7DB/A : different ideal of an , 4.1.22dB/A : discriminant of an, 7.5.2tB/A, τB/A : trace form of an, 4.1.12

connected, 8.2.24π1(SpecA) : fundamental group of a, 8.2.25

exact, 2.2.25henselization of, 5.1.13ideal in an, 2.2.5

tight, 5.1.5integral closure of an, 8.2.27projective topology on a complete linearly topolo-

gized, 5.3.4reduced, 4.1.18strictly finite, 3.1.1

Almost isomorphism, 2.1.3Almost module(s), 2.2.2A-Modafpr : almost finitely generated projective,

2.4.4M ⊗A N : tensor product of, 2.2.11M∗ : dual of an, 2.4.23M∗ : almost element(s) of an, 2.2.8, 2.2.9

M -regular, 4.1.21AnnA(M) : annihilator of an, 2.2.5Λ-nilpotent endomorphism of an , 4.3.1EM/A : evaluation ideal, 2.4.23EM/A(f) : evaluation ideal of an almost element

in an, 2.4.23Mc(A) : A-modules generated by at mostc al-

most elements, 2.3Un(A) : uniformly almost finitely generated, 2.3.8Fi(M) : Fitting ideals of a, 2.3.23

evM/A : evaluation morphism of an, 2.4.23(almost) finitely generated , 2.3.8(almost) finitely presented , 2.3.8(faithfully) flat, 2.4.4almost projective, 2.4.4

Split(A,P ) : splitting algebra of an, 4.4.1dA(P, b) : discriminant of a bilinear form on a,

7.5.1f.rkA(P ) : formal rank of an , 4.3.9f.rkA(P, p) : localization at a prime ideal of the

formal rank of an, 4.4.13trP/A : trace morphism of an , 4.1.1of almost finite rank , 4.3.9of constant rank , 4.3.9of finite rank , 4.3.9

invertible, 2.4.23linear topology on an, 5.3.1I-preadic,I-c-preadic,I-adic, 5.3.1complete, 5.3.1completion of a, 5.3.2continuous morphism of, 5.3.2generated by a family of submodules, 5.3.1ideal of definition of a, 5.3.1induced topology on a submodule, 5.3.2topological closure of a submodule in a, 5.3.1

reflexive, 2.4.23strictly invertible , 4.4.32that admits infinite splittings , 4.4.15

Almost scheme(s)Xsm : smooth locus of an, 5.6.1, 5.7.21LX/Y : cotangent complex of an, 3.3.13PRa(E), Grassr

Ra(E) : projective space, Grass-mannian, 5.7.7

ExalY (X/G,I ) : equivariant square zero defor-mation of an, 3.3.14

affine, morphism of affine, 3.3.3almost finite, etale, flat, almost projective morphism

of, 4.5.5almost finitely presented, 5.6.3closed imbedding of, 4.5.5open and closed imbedding of, 4.5.5quasi-affine, 5.7.1quasi-coherent module on an, 3.3.3square zero deformation of an, 3.3.13

Almost zero module, 2.1.3

bicovering morphism in a site, 3.5.16

Cauchy product, 2.3.4


Classical limit, 2.1.2Closed imbedding, difference of sheaves on(Ra-Alg)o

fpqc,5.7.5

ConditionsA andB, 2.1.6

Decomposition subgroup, 6.2.3

f-adic ring(s), 7.2.15Cont(A) : continuous valuations of an, 7.2.21topologically finite type map of, 7.2.16ideal of definition of an, 7.2.15ring of definition of an, 7.2.15

Formal scheme(s)Lan

X/Y : Cotangent complex of a morphism of, 7.2.3Ωan

X/Y : sheaf of analytic relative differentials of amorphism of, 7.2.8

Deeply ramified system of, 7.4.5

Groupoid(s), 4.5.1X0/G : quotient by an action of a, 4.5.5closed equivalence relation, flat, etale, almost fi-

nite, almost projective, of finite rank, 4.5.5with trivial automorphisms, 4.5.1

Inertia subgroup, 6.2.3

Log structure(s), 6.4.4logX : category of, 6.4.4regular, 6.4.9

Monoid(s), 6.1.26Mgp : group associated to an abelian, 6.1.29SpecM : spectrum of a, 6.1.27Mnd : category of commutative, 6.1.29convex submonoid of a, 6.1.26free, 6.1.28

basis of a, 6.1.28ideal, prime ideal in a, 6.1.27integral, 6.1.28

Ordered abelian group(s), 6.1.20Γ+ : submonoid of positive elements in an, 6.1.20Spec Γ : spectrum of an, 6.1.20c.rk(Γ), rk(Γ) : convex and rational rank of an,

6.1.20convex subgroup of an, 6.1.20

Pre-log structure(s), 6.4.2ΩX/Z(logM) : logarithmic differentials of a, 6.4.11pre-logX : category of, 6.4.2

Pretopos, 8.2.6Prufer Domain, 6.1.12

Simplicial almost algebra(s), 8.1exact, 8.1.2modules over a, 8.1

derived category of , 8.1.1quasi-isomorphism of , 8.1.2

Topos, 8.2coherent, 8.2.4

constant, locally constant, bounded object in a, 8.2.20fibred, 3.3.15

Top(X) : topos associated to a, 3.3.15point of a, 8.2.9quasi-compact object of a, 8.2.1quasi-compact, quasi-separated, coherent morphism

in a, 8.2.2quasi-separated, coherent object of a, 8.2.3

Valuation(s), 6.1.1Gauss, 6.1.4rank of a, 6.1.23

Valued field(s), 6.1.1Ks,Ka : separable, algebraic closure of a, 6.2.16Kt : maximal tame extension of a, 6.2.16ΓK : value group of a, 6.1.1Div(K+a) : fractional ideals of a, 6.1.16deeply ramified, 6.6.1extension of a, 6.1.3

d+L/K : discriminant of an algebraic, 7.5.6

valuation topology of a, 6.1.21

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