Bruno Vallette- A Koszul Duality for props

8/3/2019 Bruno Vallette- A Koszul Duality for props

1/78

A KOSZUL DUALITY FOR PROPS

BRUNO VALLETTE

Abstract. The notion of prop models the operations with multiple inputsand multiple outputs, acting on some algebraic structures like the bialgebrasor the Lie bialgebras. In this paper, we generalize the Koszul duality theoryof associative algebras and operads to props.

Introduction

The Koszul duality is a theory developed for the first time in 1970 by S. Priddy forassociative algebras in [Pr]. To every quadratic algebra A, it associates a dual coal-gebra A and a chain complex called Koszul complex. When this complex is acyclic,we say that A is a Koszul algebra. In this case, the algebra A and its represen-tations have many properties (cf. A. Beilinson, V. Ginzburg and W. Soergel [BGS]).

In 1994, this theory was generalized to algebraic operads by V. Ginzburg and M.M.Kapranov (cf. [GK]). An operad is an algebraic object that models the opera-tions with n inputs (and one output) An A acting on a type of algebras. Forinstance, there exists operads As, Com and Lie coding associative, commutativeand Lie algebras. The Koszul duality theory for operads has many applications:construction of a small chain complex to compute the homology groups of analgebra, minimal model of an operad, notion of algebra up to homotopy.

The discovery of quantum groups (cf. V. Drinfeld [Dr1, Dr2]) has popularizedalgebraic structures with products and coproducts, thats-to-say operations withmultiple inputs and multiple outputs An Am . It is the case of bialgebras,Hopf algebras and Lie bialgebras, for instance. The framework of operads is toonarrow to treat such structures. To model the operations with multiple inputs andoutputs, one has to use a more general algebraic object : the props. Following J.-P.Serre in [S], we call an algebra over a prop P, a P-gebra.

It is natural to try to generalize Koszul duality theory to props. Few works has beendone in that direction by M. Markl and A.A. Voronov in [MV] and W.L. Gan in[G]. Actually, M. Markl and A.A. Voronov proved Koszul duality theory for whatM. Kontsevich calls 12 -PROPs and W.L. Gan proved it for dioperads. One has to

bear in mind that an associative algebra is an operad, an operad is a 12

-PROP, a12

-PROP is a dioperad and a dioperad induces a prop.

2000 Mathematics Subject Classification. 18D50 (16W30, 17B26, 55P48).Keywords and phrases. Prop, Koszul Duality, Operad, Lie Bialgebra, Frobenius Algebra.

1

hal00581048,version1

30Mar2011

Author manuscript, published in "Transactions of the American Mathematical Society 359, 10 (2007) 4865-4943"DOI : 10.1090/S0002-9947-07-04182-7
http://dx.doi.org/10.1090/S0002-9947-07-04182-7http://hal.archives-ouvertes.fr/http://hal.archives-ouvertes.fr/hal-00581048/fr/http://dx.doi.org/10.1090/S0002-9947-07-04182-7


2/78

BRUNO VALLETTE

Associative algebras Operads 12

PROPs Dioperads ; Props.

In this article, we introduce the notion of properad, which is the connected partof a prop and contains all the information to code algebraic structure like bialge-bras, Lie bialgebras, infinitesimal Hopf algebras (cf. M. Aguiar [Ag1, Ag2, Ag3]).We prove the Koszul duality theory for properads which gives Koszul duality forquadratic props defined by connected relations (it is the case of the three previousexamples).

The first difficulty is to generalize the bar and cobar constructions to properads andprops. The differentials of the bar and cobar constructions for associative algebras,operads, 1

2-PROPs and dioperads are given by the notions of edge contraction

and vertex expansion introduced by M. Kontsevich in the context of graph ho-mology (cf. [Ko]). We define the differentials of the bar and cobar constructions ofproperads using conceptual properties. This generalizes the previous constructions.

The preceding proofs of Koszul duality theories are based on combinatorial prop-erties of trees or graphs of genus 0. Such proofs can not be used in the contextof props since we have to work with any kind of graphs. To solve this problem,we introduce an extra graduation induced by analytic functors and generalize thecomparison lemmas of B. Fresse [Fr] to properads.

To any quadratic properad P, we associated a Koszul dual coproperad P and aKoszul complex. The main theorem of this paper is a criterion which claims thatthe Koszul complex is acyclic if and only if the cobar construction of the Koszul

dual P is a resolution of P. In this case, we say that the properad is Koszul andthis resolution is the minimal model of P.

As a corollary, we get the deformation theories for gebras over Koszul properads.For instance, we prove that the properad of Lie bialgebras and the properad ofinfinitesimal Hopf algebras are Koszul, which gives the notions of Lie bialgebra upto homotopy and infinitesimal Hopf algebra up to homotopy. Structures of gebrasup to homotopy appear in the level of on some chain complexes. For instance,the Hochschild cohomology of an associative algebra with coefficients into itself hasa structure of Gerstenhaber algebra up to homotopy (cf. J.E. McClure and J.H.Smith [McCS] see also C. Berger and B. Fresse [BF]). This conjecture is know asDelignes conjecture. The same kind of conjectures exist for gebras with coprod-

ucts up to homotopy. One of them was formulated by M. Chas and D. Sullivan inthe context of String Topology (cf. [CS]). This conjecture says that the structureof involutive Lie bialgebra (a particular type of Lie bialgebras) on some homologygroups can be lifted to a structure of involutive Lie bialgebras up to homotopy.

Working with complexes of graphes of genus greater than 0 is not an easy task.This Koszul duality for props provides new methods for studying the homology ofthese chain complexes. One can interpret the bar and cobar constructions of Koszulproperads in terms of graph complexes. Therefore, it is possible to compute their(co)homology in Kontsevichs sense. For instance, the case of the properad of Liebialgebras leads to the computation of the cohomology of classical graphs and the

2


30Mar2011


3/78


4/78


5/78


We can now define a product in the category ofS-bimodules based on 2-level graphsG2. We denote by In() and Out() the sets of inputs and outputs of a vertex of a graph.

Definition (Composition product ). Given two S-bimodules P and Q, we definetheir product by the following formula

Q P :=

gG2

N2

Q(|Out()|, |In()|) k

N1

P(|Out()|, |In()|)

,where the equivalence relation is generated by

1 00bbbbbb

bb

2 3

(1) 55qqqqqq

qqq

(2) (3)zz

1

2

3

00aaa

aaaa

a 1 (1)

{{wwww

wwww

w(2)

(3)

66rrr

rrrr

rr

.

This product is given by the directed graphs on 2 levels where the vertices areindexed by elements of Q and P with respect to the inputs and outputs.

Remark. By concision, we will often omit the equivalence relation in the rest of thepaper. It will always be implicit in the following that a graph indexed by elementsof an S-bimodule is considered with the equivalence relation.

This product has an algebraic writing using symmetric groups.

Given two n-tuples and , we use the following conventions. The notation P(, )denotes the product P(j1, i1) k k P(jn, in) and the notation S denotes theimage of the direct product of the groups Si1 Sin in S||.

Theorem 1.1. Let P and Q be two S-bimodules. Their composition product isisomorphic to the S-bimodule given by the formula

QP(m, n) =NN

l, k, ,

k[Sm] Sl Q(l, k) Sk k[SN] S P(, ) S k[Sn]

,where the direct sum runs over the b-tuples l, k and the a-tuples , such that|l| = m, |k| = || = N, || = n and where the equivalence relation is defined by

q1 qb p1 pa

1l

q1(1) q1(b) k p(1) p(a) 1 ,

for Sm, Sn, SN and for Sb with k the corresponding permutationby block (cf. Conventions), Sa and the corresponding permutation by block.

Proof. To any element q1 qb p1 pa ofk[Sm] Q(l, k) k[SN]P(, )k[Sn], we associate a graph ( q1 qb p1 pa )such that its vertices are indexed by the q and the p, for 1 b and 1 a.To do that, we consider a geometric representation of the permutation . We gatherand index the outputs according to k and the inputs according to . We index the

5


30Mar2011


6/78

BRUNO VALLETTE

vertices by the q and the p. Then, for each q, we built l outgoing edges whoseroots are labelled by 1, . . . , l. We do the same with the p and the inputs of thegraph. Finally, we label the outputs of the graph according to and the inputsaccording to . For instance, the element (4123)q1 q2 (1324)p1 p2 (12435)gives the graph represented on Figure 2.

1

1 11dd

dddd

dd2

2

4

3~~~

~~~~

~3

1 11dd

dddd

dd5

2~~~

~~~~

~

p11

{{{

{{{{

{{2

p2

2

gggg

gggg

g1

nnnnnn

nnnn

nnnn

133g

gggg

gggg

2}}{{

{{{{

{{{

133g

gggg

gggg

2}}{{

{{{{

{{{

q11

q21

~~~~

~~~~2

3

11ddd

dddd

d

4 1 2 3

Figure 2. Image of (4123) q1 q2 (1324) p1 p2 (12435)under .

Let ( q1 qb p1 pa ) be the equivalence class of(q1 qb p1 pa ) for the relation . Therefore, the application naturally induces an application from the quotient k[Sm]Sl Q(l, k)Sk k[SN]SP(, ) S k[Sn]. The equivalence relation corresponds, via , to an (homeo-morphic) rearrangement in space of the graphs. Therefore, the non-planar graphcreated by is invariant on the equivalent class ofq1qbp1paunder the relation . This finally defines an application which is an isomorphismofS-bimodules.

Proposition 1.2. The composition product is associative. More precisely, letR, Q and P be three S-bimodules. There is a natural isomorphism of S-bimodules

(RQ) P = R (Q P).

Proof. Denote by G3 the set of 3-level graphs. The two S-bimodules (RQ)P

and R (Q P) are isomorphic to the S-bimodule given by the following formula gG3

N3

R(|Out()|, |In()|)

N2

Q(|Out()|, |In()|)

N1

P(|Out()|, |In()|)

.

1.3. Horizontal and connected vertical composition products ofS-bimodules.

6


30Mar2011


7/78


Definition (Concatenation product ). Let P and Q be two S-bimodules. Wedefine their concatenation product by the formula

PQ(m, n) :=

m+m=mn+n=n

k[Sm+m]SmSm P(m, n)kQ(m

, n)SnSnk[Sn+n ].

The product corresponds to the intuitive notion of concatenation of operations(cf. Figure 3).

1

1 ))XXX

XXXX

2

133ggg

gggg

g 3

2

{{{{

}}{{{

4

2

p

1

2

q

1

1 3 2

Figure 3. Concatenation of two operations.

The concatenation product is also called the horizontal product by contrast withthe composition product , which is called the vertical product.

Proposition 1.3. The concatenation product induces a symmetric monoidal cate-gory structure on the category of S-bimodules. The unit is given by the S-bimodulek defined by

k(0, 0) = k,

k(m, n) = 0 otherwise.Proof. The unit relation comes from

P k(m, n) = k[Sm] Sm P(m, n) k k Sn k[Sn] = P(m, n).

And the associativity relation comes fromP(m, n) Q(m, n)

R(m, n) = P(m, n)

Q(m, n) R(m, n)

=

k[Sm+m+m ] SmSmSm P(m, n) k Q(m, n) k R(m

, n)

SnSnSnk[Sn+n+n ].

The symmetry isomorphism is given by the following formula

P(m, n) Q(m, n) (1, 2)m, m

P(m, n) Q(m, n)

(1, 2)n, n

= Q(m, n) P(m, n).

Remark. The monoidal product is bilinear. (It means that the functors P and P are additive for every S-bimodule P).

The notions ofS-bimodules represents the operations acting on algebraic structures.The composition product models their compositions. For some gebras, one canrestrict to connected compositions, based on connected graphs, without loss ofinformation (cf. 2.9).

Definition (Connected graph). A connected graph g is a directed graph which isconnected as topological space. We denote the set of such graphs by Gc.

7


30Mar2011


8/78

BRUNO VALLETTE

Definition (Connected composition product c). Given two S-bimodules Q andP, we define their connected composition product by the following formula

Qc P :=

gG2c

N2

Q(|Out()|, |In()|) k

N1

P(|Out()|, |In()|)

.In this case, the algebraic writing is more complicated. It involves a special classof permutations of the symmetric group SN.

Definition (Connected permutations). Let N be an integer. Let k = (k1, . . . , kb)be a b-tuple and = (j1, . . . , ja) be a a-tuple such that |k| = k1 + + kb = || =j1 + + ja = N. A (k, )-connected permutation ofSN is a permutation ofSNsuch that the graph of a geometric representation of is connected if one gathersthe inputs labelled by j1 + +ji + 1, . . . , j1 + +ji +ji+1, for 0 i a 1, and

the outputs labelled by k1 + + ki + 1, . . . , k1 + + ki + ki+1, for 0 i b 1.The set of (k, )-connected permutations is denoted by Sc

k, .

Example. Consider the permutation (1324) in S4 and the following geometricrepresentation

1

2

TT

TTTT

3

4

1 2 3 4.

Take k = (2, 2) and = (2, 2). If one links the inputs 1, 2 and 3, 4 and the outputs1, 2 and 3, 4, it gives the following connected graph

ccccc

cc

Therefore, the permutation (1324) is ((2, 2), (2, 2))-connected.

Counterexample. Consider the same permutation (1324) of S4 but let k =(1, 1, 2) and = (2, 1, 1). One obtains the following non-connected graph

cccc

ccc

The next proposition will allow us the link the connected composition product withthe one defined in the theory of operads.

Proposition 1.4. If k = (N), all the permutations of SN are ((N), )-connected,for every . Otherwise stated, one has Sc(N), = SN.

If k is different from (N), one has Sck, (1, ..., 1) = .

Proof. The proof is obvious.

Proposition 1.5. Let P and Q be two S-bimodules. Their connected compositionproduct is isomorphic to the S-bimodule given by the formula Qc P(m, n) =

NN

l, k, ,

k[Sm] Sl Q(l, k) Sk k[Sck,

] S P(, ) S k[Sn]

,where the direct sum runs over the b-tuples l, k and the a-tuples , such that|l| = m, |k| = || = N, || = n.

8


30Mar2011


9/78


Proof. First, we have to prove that the right member is well defined. For aa-tuple and k a b-tuple such that || = |k| = N, we denote := (j1(1), . . .,j1(b)) and k

:= (k1(1), . . . , k1(b)). They also verify || = N and |k| = N.

Every (k, )-connected permutation gives by composition on the left with a blockpermutation of the type k and by composition on the right with a bock permutationof the type a (k, )-connected permutation. The geometric representations of SN and k are homeomorphic and one links the same inputs and outputs.Therefore, if Sc

k, , one has k S

ck,

.

The rest of the proof uses the same arguments as the proof of Theorem 1.1.

By opposition with the composition product , the connected composition productc is a monoidal product in the category ofS-bimodules. It remains to define theunit in this category. We denote

I := I(1, 1) = k,I(m, n) = 0 otherwise.

Proposition 1.6. The category (S-biMod, c, I) is a monoidal category.

Proof. To show the unit relation on Pc I(m, n), one has to study Sck, in the case

= (1, . . . , 1). If one does not gather all the outputs, this set is empty. Otherwise,for k = (n), one has Sc(n), (1, ..., 1) = Sn (cf. Proposition 1.4). This gives

Pc I(m, n) = P(m, n) Sn k[Sn] Sn1 kn = P(m, n) k.idn k

n = P(m, n).

(The case Ic P = P is symmetric.)Once again, the associativity relation comes from the following formula Rc (QcP) = (Rc Q)c P =

gGc3

N3

R(|Out()|, |In()|)

N2

Q(|Out()|, |In()|)

N1

P(|Out()|, |In()|)

.

Examples. The monoidal categories (k-Mod, k, k) and (S-Mod, , I) are two fullmonoidal subcategories of (S-biMod, c, I).

The first category is the category of modules over k endowed with the classical

tensor product. Every vector space V can be seen as the following S-bimoduleV(1, 1) := V,V(j, i) := 0 otherwise.

One has V c W(1, 1) = V k W. Since there exists no connected permuta-tion associated to a, b-tuples of the form (1, . . . , 1) (cf. Proposition 1.4), one hasV W(j, i) = 0 for (j, i) = (1, 1). And the morphisms between two S-bimodulesonly composed by an (S1, S1)-module are exactly the morphisms of k-modules.

The category of S-bimodules contains the category of S-modules related to thetheory of operads (cf. [GK, L3, May]). Given an S-module P, we consider the

9


30Mar2011


10/78

BRUNO VALLETTE

following S-bimodule P(1, n) := P(n) for n N,P(j, i) := 0 if j = 1.

We have seen in Proposition 1.4 that Sc(N), (1, ..., 1) = SN. It gives here

QcP(1, n) =

Nn

i1++iN=n

Q(1, N) SN k[SN] SN1P(1, ) S k[Sn]

,

where the equivalence relation is given by

q p1 pN q p(1) p(N).

It is equivalent to take the coinvariants under the action of SN in the expressionQ(1, N) k P(1, i1) k k P(1, iN). We find here the monoidal product defined

in the category of S-modules (cf. [GK, L3, May]), which is well known under thefollowing algebraic form (without the induced representations)

QcP(1, n) =

Nn

i1++iN=n

Q(1, N) P(1, i1) P (1, iN)

SN

= QP(n).

Notice that the restriction of the monoidal product c on S-modules correspondsto the description of the composition product defined by trees (cf. [GK, L3]).When j = 1, the composition Qc P(j, i) represents a concatenation of trees whichis a non-connected graph. Therefore, Qc P(j, i) = 0, for j = 1.

The default of the composition product to be a monoidal product, in the categoryof S-bimodules, comes from the fact that P I corresponds to concatenations ofelements of P and not only elements of P.

Definition (The S-bimodules T(P) and S(P)). Since the monoidal product is bilinear, the related free monoid on an S-bimodule P is given by every possibleconcatenations of elements of P

T(P) :=nN

Pn,

where P0 = k by convention.The monoidal product is symmetric. Therefore, we can consider the truncatedsymmetric algebra on P.

S(P) := S(P) =

nN

(Pn)Sn .

The functor S allows us to link the two composition products and c.

Proposition 1.7. Let P and Q be two S-bimodules. One has the following iso-morphism of S-bimodules

S(Qc P) = Q P.

Proof. The isomorphism lies on the fact that every graph of G2 is the concate-nation of connected graphs of G2c . And the order between these connected graphsdoes not matter.

Remark that P I = S(P), which is different from P, in general.

Definition (Saturated S-bimodules). A saturatedS-bimodule P is an S-bimodulesuch that P = S(P). We denote this category sat-S-biMod.

10


30Mar2011


11/78


Example. For every S-bimodule P, S(P) is a saturated S-bimodule.

Notice that S(I)(m, n) = 0 if m = n and S(n, n) = k[Sn].

Proposition 1.8. The category (sat-S-biMod, , S(I)) is a monoidal category.

Proof. Since Q P = S(Q P), the associativity relation comes from Proposi-tion 1.2. If P is a saturated S-bimodule, one has P S(I) = S(P) = P.

We sum up these results in the diagram

(V ect, k, k) (S-Mod, , I) (S-biMod,c, I)S

(sat-S-biMod,, S(I)),

where the first arrows are faithful functors of monoidal categories.

Remark. Since the concatenation product is symmetric and bilinear, its homolog-ical properties are well known. In the rest of the text, we will mainly study the

homological properties of the composition products (cf. Sections 3 and 5). Sincethe composition product is obtained by concatenation S from the connected com-position product c, we will only state the theorems for the connected compositionproduct c in the following of the text. There exist analogue theorems for thecomposition product , which are left to the reader.

1.4. Representation of the elements ofQcP. The product QcP(and QP)is isomorphic to a quotient under the equivalence relation that permutates theorder of the elements of Q and P. For instance, to study the homological propertiesof this product, we need to find natural representatives of the classes of equivalencefor the relation .In this section, all the S-bimodules are reduced (cf. 1.1).

Definition (SB

i, ). We denote bySB

i, the set of block permutations of type(i, . . . , i) times

ofSi . (cf. Conventions).

Remark. The set SBi, is a subgroup ofSi .

Let be a partition of the integer n. We consider the n-tuple defined by ik :=|{j / ij = k}|. Denote by SB the subgroup image of

nk=1 S

Bk, i

kin Sn.

Proposition 1.9. The product Qc P(m, n) is isomorphic to the S-bimodule de-fined by the following formula

k[SmS

Bl

] Sl Q(l, k) Sk k[Sck,

] S P(, ) S k[SB

Sn],

where the sum runs over partition of the integer n, l partition of the integer m,N inN and a-tuple, k b-tuple such that |k| = || = N.

Proof. After choosing the order of the elements of P given by the partition ofthe integer n, the remaining relations that permutate the elements of P involveonly elements ofSB .

We can do the same kind of work with the definition ofc using non-planar graphs.The goal here is to choose a planar representative of indexed graphs.

Definition (Ordered partitions of [n]). An ordered partition of [n] is a sequence(1, . . . , k) such that {1, . . . , k} is a partition of [n] and such that min(1) 0 is null.

The properad Bi coding infinitesimal Hopf algebras. This properad isgenerated by the S-bimodule

V =

V(1, 2) = m.k k[S2],V(2, 1) = .k[S2] k,V(m, n) = 0 otherwise.

=cc

cc

,

And its relations are given by

R =

m(m, 1) m(1, m) (, 1) (1, ) m (m, 1)(1, ) (1, m)(, 1).

=

cccc

cc cccc

cc

cc cc

cc

cc

cc

cc c

c c

cc

c.

20


30Mar2011


21/78


These Bi-gebras are associative and non-commutative analogs of Liebialgebras. They correspond to the infinitesimal Hopf algebras defined byM. Aguiar (cf. [Ag1], [Ag2] and [Ag3]).

The properad 12

Bi, degenerated analogue of the properad of bialgebras.The generating S-bimodule V is the same as in the case of Bi, composedby a product and a coproduct. The relations R are the following ones

R =

m(m, 1) m(1, m) (, 1) (1, ) m ().

=

cccc

cc cccc

cccc cc cc cc

cc

cc ().

This example has been introduced by M. Markl in [Ma2] to describe aminimal model for the prop of bialgebras.

Counterexamples.

The previous properad is a quadratic version of the properad Bi codingbialgebras. This last one is defined by generators and relations but is notquadratic. The definition of Bi is the same than 12 Bi where the relation ()is replaced by the following one

() : m (m, m) (1324) (, )

=cc

cc w

wqqqqww

ww qq

qq ww

Since the relation () includes a composition involving 4 generating oper-ations, the properad Bi is neither quadratic nor weight graded.

Consider the prop coding unitary infinitesimal Hopf algebras defined byJ.-L. Loday in [L4]. Its definition is the same than the prop Bi where thethird relation is

m (m, 1)(1, ) (1, m)(, 1) (1, 1) =cc

cc

cc

c

c

cc .

Since this relation is neither connected nor homogenous of weight 2, onehas that this prop is not quadratic and it does not come from a properad.

3. Differential graded framework

In this section, we generalize the previous notions in the differential graded frame-work. For instance, we define the compositions products of differential gradedS-bimodule. We give the definitions of quasi-free P-modules and quasi-free proper-ads.

21


30Mar2011


22/78

BRUNO VALLETTE

3.1. The category of differential graded S-bimodules. We now work in thecategory dg-Mod of differential graded k-modules. Recall that this category can beendowed with a tensor product V k W defined by the following formula

(V k W)d :=

i+j=d

Vi k Wj ,

where the boundary map on V k W is given by

(v w) := (v) w + (1)|v|v (w),

for v w an elementary tensor and where |v| denotes the homological degree of v.The symmetry isomorphism involve Koszul-Quillen sign rules

v, w : v w (1)|v||w|w v.

More generally, the commutation of two elements (morphisms, differentials, objects,

etc ...) of degree d and e induces a sign (1)de.

Definition (dg-S-bimodule). A dg-S-bimodule P is a collection (P(m, n))m, nNof differential graded modules with an action of the symmetric group Sm on the leftand an action ofSn on the right. These groups acts by morphisms of dg-modules.We denote by Pd(m, n) the sub-(Sm, Sn)-module composed by elements of degreed of the chain complex P(m, n).

3.2. Composition products of dg-S-bimodules. The composition product cand of S-bimodules can be generalized to the differential graded framework.The definition remains the same except that the equivalence relation is basedon symmetry isomorphisms and therefore involves Koszul-Quillen sign rules. Forinstance, one has

q1 qi qi+1 qb p1 pa

(1)|qi||qi+1| q1 qi+1 qi qb p1 pa ,

where = (1 . . . i + 1 i . . . a)k and = (1 . . . i + 1 i . . . a)1

l.

Remark. To lighten the notations, we will often omit the induced representations and in the following of the text.

We define the image of an element q1 qb p1 pa of Q(l, k) Skk[Sc

k, ] Sl P(, ) under the boundary map by the following formula

(q1 qb p1 pa) =b

=1(1)|q1|++|q1|q1 (q) qb p1 pa +

a=1

(1)|q1|++|qb|+|p1|++|p1|q1 qb p1 (p) pa.

Lemma 3.1. The differential is constant on equivalence classes under the relation.

Proof. The proof is straightforward and left to the reader.

One can also generalize the concatenation product to the category of dg-S-bimodules.

22


30Mar2011


23/78


Definition (Saturated differential graded S-bimodule). A saturated differentialgraded S-bimodule is a dg-S-bimodule P such that S(P) = P. We denote thiscategory sat-dg-S-biMod.

One can extend Proposition 1.9 and Proposition 1.10 in the differential gradedframework.

Proposition 3.2. Let Q and P be two differential graded S-bimodules. Theirconnected composition product Qc P(m, n) is isomorphic to the dg-S-bimodule

k[SmS

Bl


] S P(, ) S k[SB

Sn]

and to the dg-S-bimodule

(1)Q(|1|, k1) k k (b)Q(|b|, kb)Sk k[Sck,

] S(P(j1, |1|)(1) k k P(ja, |a|)(a)

,

Once again, we have the same propositions for the composition product . Weare going to study the homological properties of the composition product of twodg-S-bimodules.

Remark that the chain complex (Qc P, ) corresponds to the total complex of abicomplex. We define the bidegree of an element

(q1, . . . , qb) (p1, . . . , pa) of k[SmS

Bl

]SlQ(l, k)Skk[Sck,

]SP(, )Sk[SB

Sn]

by (|q1|+ +|qb|, |p1|+ +|pa|). The horizontal differential h : Qc P Qc Pis induced by the one of Q and the vertical differential v : Q c P Q c P is

induced by the one of P. One has explicitly,

h

(q1, . . . , qb) (p1, . . . , pa)

=

b=1

(1)|q1|++|q1|(q1, . . . , (q), . . . , qb) (p1, . . . , pa)

and v

(q1, . . . , qb) (p1, . . . , pa)

=a

=1

(1)|q1|++|qb|+|p1|++|p1|(q1, . . . , qb) (p1, . . . , (p), . . . , pa).

One can see that = h + v and that h v = v h.

Like every bicomplex, this one gives rise to two spectral sequences Ir

(Q

P) etIIr(Q P) that converge to the total homology H(Q P, ). Recall that thesetwo spectral sequences verify

I1(Qc P) = H(Qc P, ), I2(Qc P) = H(H(Qc P, v), h)

and I I 1(Qc P) = H(Qc P, h), II2(Qc P) = H(H(Qc P, h), v).

Proposition 3.2 gives that the composition product Qc P can be decomposed asthe direct sum of chain complexes

k[SmS

Bl


] S P(, ) S k[SB

Sn].

23


30Mar2011


24/78

BRUNO VALLETTE

This decomposition is compatible with the structure of bicomplex. Therefore, thespectral sequences can be decomposed in the same way

Ir(Qc P) =

Ir

k[SmS

Bl


] S P(, ) S k[SB

Sn]

and IIr(Qc P) =

IIr

k[SmS

Bl


] S P(, ) S k[SB

Sn]

From these two formulas, one has the following proposition.

Proposition 3.3. When k is a field of characteristic 0, one has

I1

k[SmS

Bl


] S P(, ) S k[SB

Sn]

= k[SmSBl ] Sl Q(l, k) Sk k[Sck, ] S H (P(, )) S k[SB Sn] andII1

k[Sm

S

Bl


] S P(, ) S k[SB

Sn]

= k[SmS

Bl

] Sl H

Q(l, k)

Sk k[Sck,

] S P(, ) S k[SB

Sn].

Proof. By Mashkes Theorem, we have the rings of the form k[S] and k[S] aresemi-simple. Therefore, the modules k[Sm

SB

l] Sl Q(l, k) Sk k[S

ck,

] are k[S]

projective modules (on the right) and the modules k[SBSn] are k[S] projective

modules (on the left) .


I2

k[Sm

S

Bl


] S P(, ) S k[SB

Sn]

= II2 k[SmSBl ] Sl Q(l, k) Sk k[Sck, ] S P(, ) S k[SB Sn]

= k[SmS

Bl

] Sl HQ(l, k) Sk k[Sck,

] S HP(, ) S k[SB

Sn].

Proof. We use the same arguments to show that

I2

k[SmS

Bl


] S P(, ) S k[SB

Sn]

= k[SmS

Bl

] Sl H

Q(l, k)

Sk k[Sck,

] S H (P(, )) S k[SB

Sn].

We conclude by Kunneths formula

H

Q(l, k)

= H (Q(l1, k1) Q(la, ka)) =

HQ(l1, k1) HQ(la, ka) = HQ(l, k).

Remark. Let : M M and : N N be two morphisms of dg-S-bimodules. The morphism c : Mc N M c N is a morphism ofbicomplexes. It induces morphisms of spectral sequences Ir( c ) : I

r(McN) Ir(M c N) and IIr(c ) : IIr(Mc N) IIr(M c N).

More generally, we have the following proposition.


H(QcP)(m, n) =

k[SmS

Bl

]SlHQ(l, k)Skk[Sck,

]SHP(, )Sk[SB

Sn].

Thats-to-say, H(Qc P) = (HQ)c (HP).

24


30Mar2011


25/78


Proof. It is a direct corollary of Mashkes Theorem (for coinvariants) and Kun-neths Theorem (for tensor products).

Once again, the similar results apply to the connected composition products .

Remark. These propositions generalize the results of B. Fresse for the compositionproduct of operads to the composition products c and of properads and props(cf. [Fr] 2.3.).

3.3. Differential properads and differential props. We give here a differentialvariation of the notions of properads and props.

Definition (Differential properad). A differential properad (P, , ) is a monoidin the monoidal category (dg-S-biMod, c, I) of differential graded S-bimodules.

The composition morphism of a differential properad is a morphism of chain

complexes of degree 0 compatible with the action of the symmetric groups Sm andSn. One has the derivation type relation :

(((p1, . . . , pb) (p1, . . . , p

a))) =

b=1

(1)|p1|++|p1|((p1, . . . , (p), . . . , pb) (p1, . . . , p

a)) +

a=1

(1)|p1|++|pb|+|p1|++|p

1|((p1, . . . , pb) (p

1, . . . , (p

), . . . , p

a)).

Proposition 3.5 gives the following property.

Proposition 3.6. When k is a field of characteristic 0, the homology groups

(H(P), H(), H()) of a differential properad (P, , ) form a graded properad.Definition (Differential weight graded S-bimodule). A differential weight gradedS-bimodule M is a direct sum over N of differential S-bimodules M =

N M

().The morphisms of differential weight graded S-bimodules are the morphisms ofdifferential S-bimodules that preserve this decomposition. This forms a categorydenoted gr-dg-S-biMod.

Remark that the weight is a graduation separate from the homological degree thatgives extra information.

Definition (Differential weight graded properad). A differential weight gradedproperad is a monoid in the monoidal category of differential weight graded S-bimodules.

A differential weight graded properad P such that P(0) = I is called connected.

Remark. Dually, we can define the notion of differential coproperad and weightgraded differential coproperad. A differential coproperad is a coproperad where thecoproduct is a morphism of dg-S-bimodules verifying a coderivation type relation.

Similarly one has the notions of differential weight graded prop and coprop.

3.4. Free and quasi-free P-modules. We define a variation of the notion of freeP-module, namely the notion ofquasi-free P-module, and we prove some of its basicproperties.

25


30Mar2011


26/78

BRUNO VALLETTE

Let P be a (weight graded) dg-properad. Recall from 2.5, that the free differentialP-module on the right over a dg-S-bimodule M is given by L = Mc P, wherethe differential comes from the one of M and the one of P as explained before.When (P, , , ) is an augmented dg-properad, one has that the indecomposablequotient of the free dg-P-module L = McP is isomorphic to M, as dg-S-bimodule.

In the following part of the text, we will encounter differential P-modules which arenot strictly free. They are built on the dg-S-bimodule Mc P but the boundarymap is the sum of the canonical differential with an homogenous morphism ofdegree 1.

Definition (Quasi-free P-module). Let P be differential properad. A (right)quasi-free P-module L is a dg-S-bimodule of the form Mc P, where the dif-

ferential is the sum + d of the canonical differential on the product M

c Pwith an homogenous morphism d : Mc P Mc P of degree 1 such that

d ((m1, . . . , mb) (p1, . . . , pa)) =b

=1

(1)|m1|++|m1|(m1, . . . , d(m), . . . , mb) (p1, . . . , pa).

In the weight graded framework, we ask d to be an homogenous map for the weightsuch that

d : M()

M

(1)

().

In this case, the boundary map preserves the global weight of L.

Therefore, the boundary map : Mc P Mc P verifies the following relation

((m1, . . . , mb) (p1, . . . , pa)) =b

=1

(1)|m1|++|m1|(m1, . . . , (m), . . . , mb) (p1, . . . , pa) +

a=1

(1)|m1|++|mb|+|p1|++|p1|(m1, . . . , mb) (p1, . . . , (p), . . . , pa).

Since 2 = 0, the identity 2 = 0 is equivalent to d + d + d

2 = 0.

If one writes d(m) = (m1, . . . , mb)

(p1, . . . , pa) Mc P, then one has

(m1, . . . , d(m), . . . , mb) (p1, . . . , pa)

(m1, . . . , m1, . . . , m

b , . . . , mb)

((p1, . . . , pa)

(p1, . . . , pa)) Mc P.

Notice that the morphism d is determined by its restriction on MMcGG Mc P .

Proposition 3.7. The morphism MMcGG Mc P is a morphism of dg-S-bimodules

if and only if d = 0.

When P is a augmented dg-properad, the condition for the morphism Mc tobe a morphism of differential graded modules is weaker.

26


30Mar2011


27/78


Proposition 3.8. Let (P, , , ) be an augmented dg-properad. The morphism

Mc PMcGG M is a morphism of dg-S-bimodules is and only if the following

composition is null

Mc Pd GG Mc P

McGG M = 0.

Corollary 3.9. In this case, the indecomposable quotient of L = Mc P isisomorphic to M as a dg-S-bimodule.

Definition (Morphism of weight graded quasi-free P-module). Let L = Mc Pand L = M c P be two weight graded quasi-free P-modules. A morphism : L L of weight graded differential P-modules is a morphism ofweight gradedquasi-free P-modules if preserves the graduation on the left

: M()

cP M

()

cP.

In the following of the text, the examples of weight graded quasi-free P-modulesthat we will have to treat will have a particular form.

Definition (Analytic quasi-free P-module). Let P be a weight graded differentialproperad. An analytic quasi-free P-module is quasi-free P-module L = Mc P suchthat the weight graded differential S-bimodule M is given by the image M = (V)of a weight graded differential S-bimodule V under a split analytic functor (cf.2.7). Denote =

nN (n), where (n) is an homogenous polynomial functor

of degree n and such that (0) = I. Moreover, V must verify V(0) = 0 so that

(V)(0)

= I. As we have seen in Proposition 2.5, (V) is bigraded. The morphismd is supposed to verify

d : (n)(V)() (V)

n1

c P1

,

where the graduation () is the global graduation coming from the one of V and(n) is the degree of the analytic functor .

Examples. The examples of analytic quasi-free P-modules fall into two parts.

When V = P, we have the example of the (differential) bar construction(V) = B(P) = Fc(P) (cf. section 4). The augmented bar constructionB(P)c P is an analytic quasi-free P-modules.

When P is quadratic properad generated by a dg-S

-bimodule V, the Koszuldual coproperad P is a split analytic functor in V (cf. section 7). Therefore,the Koszul complex P c P is an analytic quasi-free P-module.

We have immediately the following proposition.

Proposition 3.10. Let P be an augmented weight graded dg-properad and L =(V)c P be an analytic quasi-free P-module. We have

Mc Pd GG Mc P

McGG M = 0.

Therefore, we can identify the indecomposable quotient of L to (V).

27


30Mar2011


28/78

BRUNO VALLETTE

Remark. Dually, we have the notions of differential cofree C-comodule on a dg-S-bimodule M, given by Mc C, quasi-cofree C-comodule and analytic quasi-cofreeC-comodule. The augmented cobar construction Bc(C)cC is an example of analyticquasi-free C-comodule.

3.5. Free and quasi-free properads. We define here the notion of free differen-tial properad. As in the case of P-modules, we will have to deal with differentialproperads that are not strictly free. Therefore, we introduce the notion of quasi-freeproperad which is a free properad where the differential is given by the sum of thecanonical one with a derivation.Let (V, ) be a dg-S-bimodule. We have seen in 2.7 that the free properad on V isgiven by the weight graded S-bimodule F(V) which is the direct sum on connectedgraphs (without level) where the vertices are indexed by elements of V,

F(V) =

gGc

N

V(|Out()|, |In()|)

.This construction corresponds to the quotient of the simplicial S-bimodule FS(V) =

nN(V+)cn under the relation generated by Ic V Vc I. In the differential

graded framework, one has to take the signs into account. In this case, we quotientFS(V) by the relation

(1, , 2) (1, 1,

2) (1)

|2|+|1|(1, 1, 1, 2)

(1, , 2)

, for instance, where belongs to V(2, 1).

Proposition 3.11. The canonical differential defined on FS(V) by the one of Vis constant on the equivalence classes for the relation .

Proof. The verification of signs is straightforward and is the same as in Lemma 3.1.

Theorem 3.12. The differential properad (F(V), ) is free on the dg-S-bimodule(V, ).

Remark. Since the analytic decomposition of the functor F(V) given in Propo-sition 2.4 is stable under the differential , we have that F(V) is a weight gradeddifferential properad.The projection F(V) I is a morphism of dg-properads.

The functor F shares interesting homological properties.

Proposition 3.13. Over a fieldk of characteristic 0, the functorF : dg-S-biModdg-properad is exact. We have H(F(V)) = F(H(V)).

Proof. The proof of this proposition is the same as the proof of Theorem 4 ofthe article of M. Markl and A. A. Voronov [MV] and lies, once again, on Mashkestheorem.

Definition (Derivation). Let P be a dg-properad. An homogenous morphism ofS-bimodules d : P P of (homological) degree |d| is called a derivation if it

28


30Mar2011


29/78


satisfies the following identity

(((p1, . . . , pb) (p1, . . . , p

a))) =

b=1

(1)(|p1|++|p1|)|d|((p1, . . . , (p), . . . , pb) (p1, . . . , p

a)) +

a=1

(1)(|p1|++|pb|+|p1|++|p

1|)|d|((p1, . . . , pb) (p

1, . . . , (p

), . . . , p

a)).

For example, the differential of a differential properad is a derivation of degree 1such that 2 = 0. Notice that the sum + d of the differential with a derivationd of degree 1 is a derivation of the same degree. In this case, since 2 = 0 theequation (+d)2 is equivalent to d + d + d2 = 0.

Remark. Dually, one can define the notion of coderivation on a differential copr-operad C.

The next proposition gives the form of the derivations on the free differential prop-erad F(V).

Proposition 3.14. LetF(V) be the free dg-properad on the dg-S-bimodule V. Forany homogenous morphism : V F(V), there exists a unique derivation d :F(V) F(V) with the same degree such that its restriction to V is equal to .This correspondence is one-to-one.Moreover, if : V F(r)(V) F(V), we have d

F(s)(V)

F(r+s1)(V).

Proof. Let g be a graph with n vertices. We define d on a representative of anequivalence class for the relation (vertical move of a vertex up to sign) associated

to the graph g. This corresponds to a choice of an order for writing the elementsof V that index the vertices of g. Write this representative

ni=1 i, with i V.

Hence, we define d by the following formula

d(n

i=1

i) =n

i=1

(1)|1|++|i1|

i1j=1

j

(i) n

j=i+1

j

.One can see that d is well defined on the equivalence classes for the relation .Applying d is given by applying to every element of V indexing a vertex of g.(Since the application is an application of S-bimodules, the morphism d is welldefined on the equivalence classes for the relation ).The product of the free properad F(V) is surjective. Hence, it shows that everyderivation is of this form.

If the image under of the elements of V is a sum of elements of F(V) that can bewritten by graphs with r vertices ( creates r 1 vertices), the form of d showsthat d

F(s)(V)

F(r+s1)(V).

Dually, we have the same proposition for the coderivations of the cofree connectedcoproperad.

Proposition 3.15. Let Fc(V) be the cofree connected dg-coproperad on the dg-S-bimodule V. For any homogenous morphism : Fc(V) V, there exists a uniquecoderivation d : Fc(V) Fc(V) such that the composition

Fc(V)d GG Fc(V)

proj GG V

29


30Mar2011


30/78

BRUNO VALLETTE

is equal to . This correspondence is one-to-one.Moreover, if : Fc(V) V is null on the components Fc(r)(V) F

c(V), for

s = r, we have d

F(s+r1)(V)

F(s)(V), for all s > 0.

Graphically, applying the coderivation d on an element of Fc(V) represented by agraph g corresponds to apply on every admissible sub-graph of g.

We can now state the definition of a quasi-free properad.

Definition. (Quasi-free properad) A quasi-free properad (P, ) is a dg-properadof the form P = F(V), with (V, ) a dg-S-bimodule and where the differential : F(V) F(V) is the sum of the canonical differential with a derivation dof degree 1, = + d.

The previous proposition shows that every derivation d on F(V) is determinedby its restriction on V. Therefore, the inclusion V F(V) is a morphism ofdg-S-bimodules if and only if is null. On the contrary, the projection F(V) Vis a morphism of dg-S-bimodules under a weaker condition.

Proposition 3.16. The projection F(V) V is a morphism of dg-S-bimodules ifand only if (V)

r2 F(r)(V).

In this case, we say that the differential is decomposable.Dually, we have the notion of quasi-cofree coproperad.

Definition (Quasi-cofree coproperad). A quasi-cofree coproperad (C, ) is a dg-coproperad of the form C = Fc(V), with (V, ) a dg-S-bimodule and where thedifferential : Fc(V) Fc(V) is the sum of the canonical differential with a

coderivation d of degree 1, = + d.

The previous proposition on coderivations of Fc(V) shows that the projectionFc(V) V is a morphism of dg-S-bimodules if and only if is null and thatthe inclusion V Fc(V) is a morphism of dg-S-bimodules if and only if is nullon Fc(1)(V) = V.

Examples. The fundamental examples of quasi-free properad and quasi-cofreecoproperad are given by the cobar construction F

1C

and the bar construction

Fc

P

. These two constructions are studied in the next section.

4. Bar and cobar construction

We generalize the bar and cobar constructions of associative algebras (cf. Seminaire

H. Cartan [C]) and for operads (cf. E. Getzler, J. Jones [GJ]) to properads. First,we study the properties of the partial composition products and coproducts. Wedefine the reduced bar and cobar constructions and then the bar and cobar con-structions with coefficients. These constructions share interesting homological prop-erties. For instance, they induce resolutions for differential properads and props(cf. Theorem 7.8 and Section 7). To show these resolutions, we prove that theaugmented bar and cobar constructions are acyclic, at the end of this section.

4.1. Partial composition product and coproduct. We define the notions ofsuspension and desuspension of a dg-S-bimodules. Then, we give the definitionsand first properties of the partial composition products and coproducts.

30


30Mar2011


31/78


4.1.1. Suspension and desuspension of a dg-S-bimodule. Let be the gradedS-bimodule defined by

(0, 0) = k. where is an element of degree +1,(m, n) = 0 otherewise.

Definition (Suspension V). The suspension of the dg-S-bimodule V is the dg-S-bimodule V := V.

Taking the suspension of a dg-S-bimodule V corresponds to taking the tensorproduct of every element of V with . To an element v Vd1, one associatess v (V)d, which is often denoted v. Therefore, (V)d is naturally isomor-phic to Vd1. Following the same rules as in the previous section, the differentialon V is given by the formula (v) = (v), for every v in V. The suspensionV corresponds to the introduction of an extra element of degree +1. This involvesKoszul-Quillen sign rules when permutating objects.

In the same way, we define the dg-S-bimodule 1 by1(0, 0) = k.1 where 1 is an element of degree 1,1(m, n) = 0 otherwise.

Definition (Desuspension 1V). The desuspension of the dg-S-bimodule V isthe dg-S-bimodule 1V := 1 V.

4.1.2. Partial composition product. Let (P, , , ) be an augmented prop-erad. Since 2-level connected graphs are endowed with a bigraduation given by thenumber of vertices on each level, we can decompose the composition product on

connected graphs in the following way = r, sN (r, s), where(r, s) : (I P

r

)c (I Ps

) P.

The product (r, s) gives the composition of r non-trivial operations with s non-trivial operations.

Definition (Partial composition product). We call partial composition product therestriction (1, 1) of the composition product to (I P

1

)c (I P1

).

Notice that the partial composition product is the composition product of two non-trivial elements ofP where at least one output of the first one is linked to one inputof the second one.

Remark. When P is an operad, this partial composition product is exactly thepartial product denoted i by V. Ginzburg and M.M. Kapranov in [GK]. In theframework of 12 -PROPs introduced by M. Markl and A.A. Voronov in [MV], theseauthors only consider partial composition products where the upper element hasonly one output or where the lower element has only one input. The partial compo-sition product defined by W.L. Gan in [G] on dioperads only involves graphs wherethe two elements of P are linked by a unique edge. In these cases, the relatedmonoidal product is generated by the partial composition product (every compo-sition can be written with a finite number of partial products). One can see thatthe composition product of a properad has the same property.

31


30Mar2011


32/78

BRUNO VALLETTE

Lemma 4.1. Let(P, , , ) be a augmented dg-properad. The partial compositionproduct (1, 1) induces an homogenous morphism of degree 1

: Fc(2)(P) = (I P1

)c (I P1

) P.

Recall that when P is a dg-S-bimodule, the free properad F(P) on P is bigraded.The first graduation comes from the number of operations of P used to representan element of F(P). We denote this graduation F(n)(P). And the secondgraduation , the homological degree, is given by the sum of the homological degreesof the operations of P.

Proof. Let (1, . . . , 1, q, 1, . . . , 1)(1, . . . , 1, p, 1, . . . , 1) represent a homoge-nous element of homological degree |q| + |p| + 2 of Fc(2)(P). We define by thefollowing formula

((1, . . . , 1, q, 1, . . . , 1)(1, . . . , 1, p, 1, . . . , 1)) :=

(1)|q|(1, 1)(1, . . . , 1, q, 1, . . . , 1)(1, . . . , 1, p, 1, . . . , 1).

Since P is a differential properad, this last element has degree |q| + |p| + 1.

With Proposition 3.15, we can associate a coderivation d : Fc(P) Fc(P) tothe homogenous morphism any morphism .

Proposition 4.2. The coderivation d has the following properties :

(1) The equation d + d = 0 is true if and only if the partial compositionproduct (1, 1) is a morphism of dg-S-bimodules.

(2) We have d2 = 0.

Proof.

(1) Following Proposition 3.15, applying d to an element of Fc(P) repre-sented by a graph g corresponds to apply to every possible sub-graph ofg. Here, we apply to every couple of vertices linked by at least on edgeand such that there exists no vertex between. For a couple of such verticesindexed by q and p, we have

d ((1, . . . , 1, q, 1, . . . , 1)(1, . . . , 1, p, 1, . . . , 1)) =

(1)|q|

(1, 1)((1, . . . , 1, q, 1, . . . , 1)(1, . . . , 1, p, 1, . . . , 1))

=

(1)|q|+1

(1, 1)((1, . . . , 1, q, 1, . . . , 1)(1, . . . , 1, p, 1, . . . , 1))

and

d ((1, . . . , 1, q, 1, . . . , 1)(1, . . . , 1, p, 1, . . . , 1)) =

d (1, . . . , 1, (q), 1, . . . , 1)(1, . . . , 1, p, 1, . . . , 1) +(1)|q|(1, . . . , 1, q, 1, . . . , 1)(1, . . . , 1, (p), 1, . . . , 1)

=

(1)|q|(1, 1)((1, . . . , 1, (q), 1, . . . , 1)(1, . . . , 1, p, 1, . . . , 1)) +

(1, 1)((1, . . . , 1, q, 1, . . . , 1)(1, . . . , 1, (p), 1, . . . , 1)).

Therefore, d + d = 0 implies that

(1, 1)((1, . . . , 1, q, 1, . . . , 1)(1, . . . , 1, p, 1, . . . , 1))

=

(1, 1)((1, . . . , 1, (q), 1, . . . , 1)(1, . . . , 1, p, 1, . . . , 1)) +

(1)|q|(1, 1)((1, . . . , 1, q, 1, . . . , 1)(1, . . . , 1, (p), 1, . . . , 1)),

32


30Mar2011


33/78


which means that the partial composition product (1, 1) is a morphism ofdg-S-bimodule. In the other way, applying d + d corresponds to do asum of expressions of the previous type.

(2) To show that d2 = 0, we have to consider the pairs of couples of verticesof a graph g. There are two possible cases : the pairs of couples of verticessuch that the four vertices are different (a), the pairs of couples of verticeswith one common vertex (b).

(a) To an element of the form

X q1 p1 Y q2 p2 Z,

we apply twice beginning with one different couple each time. It

gives(1)|X|+|q1|(1)|X|+|q1|+|p1|+1+|Y|+|q2| + (1)|X|+|q1|+|p1|+|Y|+|q2|(1)|X|+|q1|

X (1, 1)(q1 p1) Y (1, 1)(q2 p2) Z = 0.

(b) In this case, there are two possible configurations The common vertex is between the two other vertices (following

the flow of the graph) (cf. Figure 4 ).

p

q

r

Figure 4

Therefore, to an element of the form

X r q p Y,

if we apply twice beginning by a different couple each time.We have

(1)|X|+|r|(1)|X|+|r|+|q|X (1, 1)((1, 1)(r q) p) Y +

(1)|X|+|r|+1+|q|(1)|X|+|r|X (1, 1)(r (1, 1)(q p)) Y = 0,

by associativity of the partial product (1, 1) (which comes fromthe associativity of ).

33


30Mar2011


34/78

BRUNO VALLETTE

p

r

||||||||q

Figure 5

Otherwise, the common vertex is above (cf. Figure 5) or underthe two others.We same kind of calculus gives this time

(1)|X|+|r|+1+|q|(1)|X|+|r|X (1, 1)(r (1, 1)(q p)) Y +

(1)(|r|)+1(|q|+1)+|X|+|q|+1+|r|(1)|X|+|q|X (1, 1)(q (1, 1)(r p)) Y = 0.

And the associativity of (1, 1) gives

(1, 1)(q (1, 1)(r p)) = (1)|r||q|(1, 1)(r (1, 1)(q p)).

We conclude by remarking that the image under the morphism d2 is a sumof expressions of such type.

Remark. This proposition lies on the natural Koszul-Quillen sign rules. In thearticle of [GK], these signs appear via the operad Det.

Definition (Pair of adjacent vertices). Any pair of vertices of a graph linked by atleast on edge and without a vertex between is called a pair of adjacent vertices. Itcorresponds to sub-graphs of the form F(2)(V) = F

c(2)(V) and are the composable

pairs under the coderivation d.

Remark. The image under the coderivation d is given by the composition of allpairs of adjacent vertices. In order to define the homology of graphs, M. Kontsevichhas introduced in [Ko] the notion of edge contraction. This notion corresponds tothe composition of a pair of vertices linked by only one edge. In the case of operads

(trees), 12 -PROPs and dioperads (graphs of genus 0), adjacent vertices are linkedby only one edge. Therefore, this notion applies there. But in the framework ofproperads, this notion is too restrictive and its natural generalization is given bythe coderivation d.

4.1.3. Partial coproduct. We dualize here the results of the previous section.

Definition (Partial coproduct). Let (C, , , ) be a coaugmented coproperad.The following composition is called partial coproduct

C

C c C (I C1

)c (I C1

).

34


30Mar2011


35/78


It corresponds to the part of the image of represented by graphs with two vertices.We denote it (1, 1).

Lemma 4.3. Let(C, , , ) be a differential coaugmented coproperad. The partialcoproduct induces an homogenous morphism of degree 1

: 1C F(2)(1C) = (I C

1

)c (I C1

).

Proof. Let c be an element of C. Using the same kind of notations as Sweedlerfor Hopf algebras, we have

(1, 1)(c) =

(c, c)

(1, . . . , 1, c, 1, . . . , 1)(1, . . . , 1, c, 1, . . . , 1).

Hence, we define (1c) by the formula

(1c) :=

(c, c)

(1)|c|(1, . . . , 1, 1 c, 1, . . . , 1)(1, . . . , 1, 1 c, 1, . . . , 1).

Remark. The sign appearing in the definition of is not essential here. Hisrole will be obvious in the proof of the bar-cobar resolution (cf. Theorem 5.8).

With Proposition 3.14, we can associate a derivation d : F(1C) F(1C) ofdegree 1 to the homogenous morphism which verifies the following properties.

Proposition 4.4.

(1) The equation d + d = 0 is true if and only if the partial coproduct

(1, 1) is a morphism of dg-S

-bimodules.(2) We have d2 = 0.

Proof. The proof of Proposition 3.14 shows that the image of an element ofF(1C) represented by a graph g under the derivation d is a sum where theapplication is applied on each vertex of g. Therefore, the arguments to show thisproposition are the same as for the previous proposition. For instance, the secondpoint comes from the coassociativity of (1, 1), which is induced by the one of ,and from the rules of signs.

Remark. The derivation d corresponds to take the partial coproduct of everyelement indexing a vertex of a graph. Therefore, this derivation is the naturalgeneralization of the notion of vertex expansion introduced by M. Kontsevich in[Ko] in the framework of graph cohomology. In the case of operads, the derivation

d is the same morphism as the one defined by the vertex expansion of trees.

4.2. Definitions of the bar and cobar constructions. With the propositionsgiven above, we can now define the bar and cobar constructions for properads.

4.2.1. Reduced bar and cobar constructions.

Definition (Reduced bar construction). To any differential augmented properad(P, , , ), we can associate the quasi-cofree coproperad B(P) := Fc(P) withthe differential defined by = + d, where is the morphism induced by thepartial product of P (cf. Lemma 4.1). This quasi-cofree coproperad is called thereduced bar construction of P.

35


30Mar2011


36/78

BRUNO VALLETTE

We denote B(s)(P) := Fc

(s)

(P). Therefore, d defines the following chain complex

d GG B(s)(P)

dGG B(s1)(P)d GG

d GG B(1)(P)d GG B(0)(P).

Proposition 4.2 gives that d + d = 0 and that d2 = 0. Hence, we have

2 = 0and we see that the reduced bar construction is the total complex of a bicomplex.Dually, we have the following definition :

Definition (Reduced cobar construction). To any differential coaugmented copr-operad (C, , , ) we can associate the quasi-free properad Bc(C) := F(1C)with the differential defined by = + d , where is the morphism induced bythe partial coproduct of C (cf. Lemma 4.3). This quasi-free properad is called thereduced cobar construction of C.

Once again, the derivation d

defines a cochain complexBc(0)(C)

d GG Bc(1)(C)d GG

dGG Bc(s)(C)d GG Bc(s+1)(C)

d GG .

The reduced cobar construction is the total complex of a bicomplex.

Remark. When P (or C) is an algebra or an operad (a coalgebra or a cooperad),we find the classical definitions of bar and cobar construction ( cf. Seminaire H.Cartan [C] and [GK]).

4.2.2. Bar construction with coefficients. Let (P, , , ) be an augmenteddifferential properad. On B(P) we can define two homogenous morphisms r :B(P) B(P)c P and l : B(P) Pc B(P) of degree 1 by

r : B(P) = Fc(P)

Fc(P)c F

c(P) Fc(P)c (I P1 ),and by

l : B(P) = Fc(P)

Fc(P)c F

c(P) (I P1

)c Fc(P).

Remark that these two morphisms correspond to extract one non-trivial operationfrom the top or the bottom of graphs that represent elements of Fc(P).

Lemma 4.5. The morphism r induces an homogenous morphism dr of degree 1

dr : B(P)c P B(P)c P.

And the dg-S-bimodule B(P)c P endowed with the differential defined by the sumof the canonical differential with the coderivation d of B(P) and the morphism

dr is an analytic quasi-free P-module (on the right).Proof. Since ( + d)2 = 0, the equation ( + d + dr )

2 = 0 is equivalent to( + d)dr + dr( + d) = 0 anddr

2 = 0,

These two equations are proved in the same arguments as in the proof of Propo-sition 4.2. The functor B(P) is analytic in P, therefore the quasi-free P-moduleB(P)c P is analytic.

The morphism dr corresponds to take one by one operations indexing top verticesof the graph of an element of B(P) and compose them with operations of P thatare on the first level of B(P)c P.

36


30Mar2011


37/78


38/78

BRUNO VALLETTE

Corollary 4.8.

The reduced bar construction is isomorphic to the bar construction withscalar coefficients L = R = I. We have B(P) = B(I, P, I).

The chain complex B(L, P, P) = L c B(P) c P (resp. B(P, P, R) =Pc B(P)c R) is an analytic quasi-free P-module on the right (resp. onthe left).

In the same way as Proposition 3.13 and Theorem 5.9, we have that the bar con-struction is an exact functor.

Proposition 4.9. The bar constructionB(L, P, R) preserves quasi-isomorphisms.

Proof. Let : P P be a quasi-isomorphism of augmented dg-properads.Let : L L be a quasi-isomorphism of right dg-P,P-modules and let :

R R be a quasi-isomorphism of left dg-P,P-modules. Denote by the inducedmorphism B(L, P, R) B(L, P, R). To show that is a quasi-isomorphism, weintroduce the following filtration Fr := srB(s)(L, P, R), where s is the numberof vertices of the underlying graph indexed by elements of P. One can see that thisfiltration is stable under the differential d of the bar construction. The canonicaldifferentials , L and R go from B(s)(L, P, R) to B(s)(L, P, R) and the maps d,dR and dR go from B(s)(L, P, R) to B(s1)(L, P, R). Therefore, this filtrationinduces a spectral sequence Es, t such that E

0s, t = B(s)(L, P, R)s+t and such that

d0 = + L + R. By the same ideas as in the proof of Proposition 3.13, wehave that E0s, t() is a quasi-isomorphism. Since the filtration Fr is bounded belowand exhaustive, the result is given by the classical convergence theorem of spectralsequences (cf. J.A. Weibel [W] 5.5.1).

4.2.3. Cobar construction with coefficients. We dualize the previous argu-ments to define the cobar construction with coefficients.

Let (C, , , ) be a differential coaugmented coproperad and let (L, l) and (R, r)be two differential C-comodules on the right and on the left.The two C-comodules induce the following morphisms :

R : Rr

C c R (I C1

)c R,

L : Lr

Lc C Lc (I C1).

Lemma 4.10. The applications R and L both induce an homogenous morphism of

degree 1 on Lc Bc(C)c R of the form

dR

: Lc Bc(C)c RRGG Lc Bc(C)c (I C

1

)c R GG

Lc Bc(C)c (I 1C1

)c RLcBc(C)cR GG Lc Bc(C)c R ,

38


30Mar2011


39/78


L : dL : Lc Bc(C)c R

L GG Lc (I C

1

)c Bc(C)c R GG

Lc (I 1C1

)c Bc(C)c RLcBc(C)cR GG Lc Bc(C)c R .

Proposition 4.11. The differential d on the S-bimodule L c Bc(C) c R is thesum of the following morphisms :

the canonical differential induced by the one of C on Bc(C), the canonical differential L induced by the one of L, the canonical differential R induced by the one of R, the derivation d defined on Bc(C), the homogenous morphism d

Rof degree 1,

the homogenous morphism dL

of degree 1.

Definition (Cobar construction with coefficients). The dg-S-bimodule LcBc(C)cR endowed with the differential d is called the cobar construction with coefficientsin L and R and is denoted Bc(L, C, R).

Corollary 4.12.

The reduced cobar construction Bc(C) is isomorphic to the cobar construc-tion with coefficients in the scalar C-comodule I.

The chain complex Bc(L, C, C) = L c Bc(C) c C (resp. B(C, C, R) =C c Bc(C)c R) is an analytic quasi-cofree C-comodule on the right (resp.on the left)

4.3. Acyclicity of the augmented bar and cobar constructions. When P isan algebra A, thats-to-say when the dg-S-bimodule P is concentrated in P(1, 1) =A, one has that the augmented bar construction is acyclic. To prove it, one in-troduces directly a contracting homotopy (cf. Seminaire H. Cartan [C]). In thecase of operads, B. Fresse has proved that the augmented bar construction on leftP B(P) is acyclic using a contracting homotopy. This homotopy lies on the factthe monoidal product is linear on the left. The acyclicity of the augmented barconstruction on the right comes from this result and the use of comparison lemmasfor quasi-free modules (cf. [Fr] section 4.6).In the framework of properads, since the composition product c is not linearon the left nor on the right, we can not use the same method. We begin bydefining a filtration on the chain complex

Pc B(P), d

. This filtration induces a

convergent spectral sequence. Then, we introduce a contracting homotopy on thechain complexes

E0p, , d

0

, for p > 0, to show that they are acyclic.

4.3.1. Acyclicity of the augmented bar construction. Let (P, , , ) be aaugmented dg-properad. The dg-S-bimodule Pc B(P) is the image of the functor

P (I P)c Fc(P).

Since this functor is a sum a polynomial functors, it is an split analytic functor(cf. 2.7). For an element of (P, , , ), the graduation is given by the number of

39


30Mar2011


40/78

BRUNO VALLETTE

vertices indexed by elements of P. Once again, we denote this (total) graduationby

Pc B(P) =sN

Pc B(P)

(s)

.

We consider the filtration defined by

Fi = Fi

P B(P)

:=si

Pc B(P)

(s)

.

Therefore, Fi is composed by elements P B(P) represented by graphs indexed atmost i elements of P.

Lemma 4.13. The filtration Fi of the dg-S-bimodule Pc B(P) is stable under thedifferential d of the augmented bar construction.

Proof. Lemma 4.5 give the form of the differential d of the augmented bar con-struction. It is the sum of the three following terms :

(1) The differential coming from the one of P. This one does not change thenumber of elements of P. Hence, one has (Fi) Fi.

(2) The coderivation d of the reduced bar construction B(P). This morphismcorresponds to the composition of pairs of adjacent vertices. Therefore, itverifies d(Fi) Fi1.

(3) The morphism dl . Applying this morphism corresponds to taking an op-eration of P of B(P) from the bottom and compose it with elements of Pon the last level of the graph,

Pc B(P) Pc (I P)c B(P) Pc B(P).

The total number of elements of P is decreasing and one has dl(Fi) Fi.

This lemma shows that the filtration Fi induces a spectral sequence, denote Ep, q.The first term is equal to

E0p, q = Fp

(Pc B(P))p+q

/Fp1

(Pc B(P))p+q

,

where p+q is the homological degree. Hence, the module E0p, q is given by the graphs

with exactly p vertices indexed by elements ofP. We have E0p, q =

Pc B(P)

(p)

p+q

and the differential d0 is a sum of the morphisms d0 = + dl . The morphism is induced by the one of P and the morphism dl is equal to the morphism dlwhen it does not change the total number of vertices indexed by P. Therefore, themorphism dl takes one operation of P in B(P) from the bottom and inserts it on

the line of elements of P (in the last level) without composing it with operationsofP. Otherwise, the morphism dl implies a non-trivial composition with elementsof P, in this case dl is null. (cf. figure 6).

The spectral sequence Ep, q is concentrated in the half-plane p 0. We compute

the homology of the chain complexes

E0p, , d0

to show that the spectral sequence

collapses at rank E1p, q.

Lemma 4.14. At rank E1p, q, we have

E1p, q =

I if p = q = 0,0 otherwise.

40


30Mar2011


41/78


ffff

ffff

||||

||||

p4

PPPP

PPPP

PPPP

PPP

wwwwwwwwwp5

zzzz

zzzz

p3

zzzz

zzzz

hhhh

hhhh

p1

dl

~~

Q1

p2

p1

||||

||||

p2

wwww

wwww

w

ffff

ffff

Figure 6. Form of the morphism dl

Proof. When p = 0, we have

E00, q =

I if q = 0,0 otherwise.

and the differential d0 is null on the modules E00, q. Hence, the homology of these

modules is equal to

E10, q =

I if q = 0,0 otherwise.

When p > 0, we introduce a contracting homotopy h for each chain complexesE0p,, d

0

.

Thanks to Proposition 3.2, we can represent an element of PB(P) by (p1, . . . , pr) (b1, . . . , bs). When p1 I, we define

h ((p1, . . . , pr)(b1, . . . , bs)) = 0,

otherwise h is given by the formula

h ((p1, . . . , pr)(b1, . . . , bs)) =(1)(|p1|+1)(|p2|++|pr|)(1, . . . , 1, p2, . . . , pr)

(b, bi+1, . . . , bs),

where b = F(P) (p1 (b1, . . . , bi)) with b1, . . . , bi the elements of B(P) linkedvia at least one edge to the vertex indexed by p1 in the graphic representative of(p1, . . . , pr)(b1, . . . , bs). This morphism h is homogenous of degree +1 (P P)and does not change the number of elements of P. Therefore, we have h : E0p, q E0p, q+1. Intuitively, the application h takes the first non-trivial operation between

the ones of P on the last level, suspends it, and lifts it between the ones of B(P).Let verify that hd0 + d0h = id.

(1) The rules of signs and suspensions show that h + h = 0.

41


30Mar2011


42/78

BRUNO VALLETTE

(2) We have hd

l+ d

lh = id. We first compute d

l:

dl ((p1, . . . , pr)(b1, . . . , bs)) =(1)|p

|(|pj+1|++|pr|+|b1|++|bk|)

(p1, . . . , p, pj+1 . . . , pr)(b1, . . . , bk, bk+1, . . . , bs),

where bk+1 = F(P)(p bk+1). Then, we have

hdl ((p1, . . . , pr)(b1, . . . , bs)) =(1)|p

|(|pj+1|++|pr |+|b1|++|bk|)+(|p1|+1)(|p2|++|p|++|pr|)

(1, . . . , 1, p2, . . . , p, pj+1 . . . , pr)

(b, bi+1, . . . , bk, b

k+1, . . . , bs).

The same calculus for dlh gives

dlh ((p1, . . . , pr)(b1, . . . , bs)) =

(p1, . . . , pr)(b1, . . . , bs) (1)|p

|(|pj+1|++|pr |+|b1|++|bk|)+(|p1|+1)(|p2|++|p|++|pr|)

(1, . . . , 1, p2, . . . , p, pj+1 . . . , pr)(b, bi+1, . . . , bk, bk+1, . . . , bs),

the sign 1 comes from the commutation of p with p1.

Since h is an homotopy, the chain complexes

E0p,, d0

are acyclic for p > 0, whichmeans that we have

E1p, q = 0 pour tout q si p > 0.

We can now conclude the proof of the acyclicity of the augmented bar construction.

Theorem 4.15. The homology of the chain complex

Pc B(P), d

is given by :H0

Pc B(P)

= I,Hn

Pc B(P)

= 0 otherwise.

Proof. Since the filtration Fi is exhaustive (Pc B(P) =

i Fi) and boundedbelow (F1

Pc B(P)

= 0), we can apply the classical theorem of convergence of

spectral sequences (cf. [W] 5.5.1). Hence, we get that the spectral sequence Ep, qconverges to the homology of Pc B(P)

E1p, q = Hp+q

Pc B(P), d

.

And we conclude using Lemma 4.14.

Corollary 4.16. The augmentation morphism

Pc B(P)PcFc(P)GG Ic I = I

is a quasi-isomorphism.

This last result can be generalized in the following way :We define the augmentation morphism (P, P, P) by the composition

B(P, P, P) = Pc B(P)c PPcFc(P)cP

Pc Ic P = Pc P PPP = P.

42


30Mar2011


43/78


Theorem 4.17. LetR be a left differential P-module. The augmentation morphism(P, P, R)

B(P, P, R) = PP B(P, P, P)P RPP(P,P,P)PRGG PP PP R = R

is a quasi-isomorphism.

Proof. We define the same kind of filtration based on the number of elements ofP indexing the graphs.

Remark. Using the same arguments, we have the same results for the augmentedbar construction on the right B(P)c P.

4.3.2. Acyclicity of the augmented cobar construction. Dually, we can provethat the augmented cobar construction Bc(C)c C is acyclic. But in order to make

the spectral sequence converge, we need to work with a weight graded coproperadC.

Theorem 4.18. For every weight graded coaugmented coproperad C, the homologyof the augmented cobar construction

Bc(C)c C, d

is given by

H0

Bc(C)c C

= I,Hn

Bc(C)c C

= 0 otherwise.

Proof. We define a filtration Fi on

Bc(C)c C, d

by

Fi =

si

Bc(C)c C

(s)

.

This filtration is stable under the differential d :

(1) Since the differential does not change the number of elements of C, wehave (Fi) Fi.

(2) The derivation d of the reduced cobar construction Bc(C) splits into twoparts each element indexing a vertex. Therefore, the global number ofvertices indexed by elements of C is raised by 1. The derivation d verifiesd(Fi) Fi1.

(3) The morphism dr decomposes elements of C into two parts and includeson part in Bc(C). The global number of elements of C indexing vertices isincreasing. We have dr(Fi) Fi.

This filtration induces a spectral sequence Ep, q. Its first term is given by

E0p, q = Fp

(Bc(C)c C)p+q

/Fp1

(Bc(C)c C)p+q

=

(Bc(C)c C)(p)

p+q

.

The differential d0

is the sum of two terms d0

= + dr , where d

r corresponds to

dr when it does not strictly increase the number of elements of C. The morphismdr takes elements of C in the first level, desuspends them and include them in

Bc(C).We show then that

E1p, q =

I if p = q = 0,0 otherwise.

We define the image under the homotopy h of an element (b1, . . . , br) (c1, . . . , cs)of Bc(C) C by first decomposing b1 B

c(C)(n) :

b11 GG b1 1c ,

43


30Mar2011


44/78

BRUNO VALLETTE

where b1 Bc(C)(n1) and c C. If the element

1c is linked above to elementsof I, then we suspend it and include it in the line of elements of C. Using the samearguments, one can show that hd0 + d0h = id.

Since the coproperad C is weight graded, one can decompose the chain complexBc(C)c C into a direct sum indexed by the total weight :

Bc(C)c C =N

(Bc(C) C)().

This decomposition is compatible with the filtration Fi and with the homotopy h.Therefore, we have

E1p, q

(Bc(C)c C)

()

= 0 for every p, q when > 0 and

E1p, q(Bc(C)c C)(0) = I if p = q = 0,0 otherwise.Since the filtration Fi is bounded on the sub-complex (Bc(C)c C)()

F0

(Bc(C)c C)()

= (Bc(C)c C)() et F1

(Bc(C)c C)

()

= 0,

we can apply the classical theorem of convergence of spectral sequences ( cf. [W]5.5.1). This gives that the spectral sequence Ep, q

(Bc(C)c C)()

converges to the

homology of (Bc(C)c C)(). We conclude by taking the direct sum on .

Remark. Using the same methods, we can show that the augmented bar construc-tion on the left is acyclic.

5. Comparison lemmas

We prove here comparison lemmas for quasi-free P-modules and quasi-free proper-ads. These lemmas are generalizations to the composition product c of classicallemmas for the tensor product k. Notice that B. Fresse has given in [Fr] suchlemmas for the composition product of operads. These technical lemmas allowus to show that the bar-cobar construction of a weight graded properad P is aresolution of P.

5.1. Comparison lemma for quasi-free modules.

5.1.1. Filtration and spectral sequence of an analytic quasi-free module.Let P be a weight graded differential properad P and let L = Mc P be an analytic

quasi-free P-module on the right, where M = (V). Denote M()d be the sub-S-bimodule ofM composed by elements of homological degree d and global weight

(cf. 3.4).We define the following filtration on L :

Fs(L) :=

|d|+||s

M()

d(l, k) Sk k[S

ck,

] S Pe(, ),

where the direct sum () is over the integers m, n and N and the tuples l, k, , .

Lemma 5.1. The filtration Fs is stable under the differential d of L.

Proof. The differential d is the sum of three terms.

The differential M, induced by the one of M, decreases the homologicaldegree |d| by 1. Therefore, we have M(Fs) Fs1.

44


30Mar2011


45/78


The differential P , induced by the one of P, decreases the homologicaldegree |e| by 1. We have P(Fs) Fs.

By definition of the analytic quasi-free P-modules, one has that the mor-phism d decreases the total graduation || by at least 1 and the homologicaldegree |d| by 1. We have d(Fs) Fs2.

This filtration gives a spectral sequence Es, t. The first term of this spectral se-

quence E0s, t is isomorphic to the following dg-S-bimodule :

E0s, t =

sr=0

|d|=sr, |e|=t+r

||=r

M()

d(l, k) Sk k[S

ck,

] S Pe(, ).

Using the same notations as in section 3.2, we have

E0s, t =s

r=0

I0sr, t+r

||=r

M() c P

.On can decompose the weight graded differential S-bimodule L with its weightL =

L

(). Since this decomposition is stable under the filtration Fs, the spectralsequence Es,,t is isomorphic to

Es, t(L) =N

Es, t(L()).

By definition of the analytic quasi-free P-modules, the weight of the elements ofVis at least 1. Hence, we have

E0s, t(L()) =

min(s, )r=0

I0sr, t+r

||=r

M() c PL() .Proposition 5.2. LetL be an analytic quasi-free P-module. The spectral sequenceEs, t converges to the homology of L

Es, t = Hs+t(L, d).

The differential d0 on E0s, t is given by P and the differential d1 on E1s, t is given

by M. Therefore, we have

E2s, t =s

r=0

I2sr, t+r

||=r

M() c P

,

which can be written

E2s, t =s

r=0

||=r

|d|=sr|e|=tr

(H(M))()

dc (H(P))e.

Proof. Since the filtration is exhaustive

s Fs = L and bounded below F1 = 0,the classical theorem of convergence of spectral sequences (cf. [W] 5.5.1) showsthat Es, t converges to the homology of L.

The form of the differentials d0 and d1 comes from the study of the action ofd = M + P + d on the filtration Fs. And the formula of E

2s, t comes from

Proposition 3.4.

45


30Mar2011


46/78

BRUNO VALLETTE

5.1.2. Comparison lemma for analytic quasi-free modules. To show thecomparison lemma, we need to prove the following lemma first.

Lemma 5.3. Let : P P be a morphism of weight graded connected dg-properads. Let(L, ) be an analytic quasi-free P-module and(L, ) be an analyticquasi-free P-module. Denote L = M and L = M the indecomposable quotients.We consider the action of P on L induced by the morphism . Let : L L bea morphism of weight graded differential P-modules.

(1) The morphism preserves the filtration Fs and leads to a morphism ofspectral sequences

E() : E(L) E(L).

(2) Let : M M be the morphism of dg-S-bimodules induces by . The

morphism E0() : Mc P M c P is equal to E0 = c .

Proof.

(1) Let (m1, . . . , mb) (p1, . . . , pa) be an element of Fs(Mc P). It meansthat |d| + || s. Since, is a morphism of P-modules, we have

(m1, . . . , mb) (p1, . . . , pa)

=

(m1, . . . , mb) (p1, . . . , pa)

=

((m1), . . . , (mb)) ((p1), . . . , (pa))

.

Denote

(mi) = (mi1, . . . , m

ibi

) i (pi1, . . . , piai

).

The application is a morphism of dg-S-bimodules. Therefore, we havedi = |di| + |ei| which implies |di| di and i |di| |d|. Since is amorphism of weight graded P-modules, we have i |i| ||. We concludethat

(m1, . . . , mb) (p1, . . . , pa)

Fs(L).

(2) The morphism E0s, t() is given by the following morphism on the quotients

E0s, t : Fs(Ls+t)/Fs1(Ls+t) Fs(Ls+t)/Fs1(L

s+t).

Let (m1, . . . , mb) (p1, . . . , pa) be an element of E0s, t, where || = r s,

|d| = s r and |e| = t + r. We have

E0s, t()

(m1, . . . , mb) (p1, . . . , pa)

= 0

if and only if (m1, . . . , mb) (p1, . . . , pa) Fs1(L). We have seen

that

(m1, . . . , mb) (p1, . . . , pa)

=

((m1), . . . , (mb)) ((p1), . . . , (pa))

.

Therefore

(m1, . . . , mb) (p1, . . . , pa)

belongs to Fs(Ls+t)\Fs1(Ls+t)

if and only if (mi) = mi = (mi). Consider (mi) = (mi1, . . . , m

ibi

)

i (pi1, . . . , piai

) where at least one pij does not belong to I. By definition

of a weight graded properad, the weight of these pij is at least 1. Since

preserves the total weight , we have |i| < i. Finally, the morphismE0() is equal to c .

46


30Mar2011


47/78


48/78

BRUNO VALLETTE

By the preceding lemma, we have

E2s, t(()) =

min(s, )r=0

I2sr, t+r

||=r

() c

.With the induction hypothesis, we get that E2s, t(

()) : E2s, t(L())

E2s, t(L()) is an isomorphism for s < d + . Let show that

E2d+,(()) : E2d+,(L

()) E2d+,(L())

is still an isomorphism. To do that, we consider the mapping cone of () :cone(()) = 1L()L(). On this cone, we define the following filtration

Fs(cone(())) := Fs1(

1L()) Fs(L()).

This filtration induces a spectral sequence which verifies

E1, t(cone(())) = cone(E1, t(

())).

The mapping cone of E1, t(()) verifies the long exact sequence

Hs+1

cone(E1, t(()))

Hs

E1, t(L

()) HsE1, t(())

Hs

E1, t(L())

Hs

cone(E1, t(

()))

Hs1

E1, t(L())

which finally gives the following long exact sequence () :

GG E2s+1, t(cone(())) GG E2s, t(L

())E2s, t(

())GG E2s, t(L

())

GG E2s, t(cone(())) GG E2s1, t(L()) GG

We have seen that for all t < ,

E2s, t(L()) = E2s, t(L

()) = 0.

Therefore, the long exact sequence () shows that E2s, t(cone(())) = 0 for

all s, when t < .In the same way, we have seen that E2s, t(

()) is an isomorphism for s 0,we have F(r)(M)() = F(r)(M)() = 0 when r > and

Fs(F(M)()) =

sr

F(r)(M)().

One can see that the filtration Fs is stable under the differential = M + d. (Wehave M(Fs) Fs and d(Fs) Fs1). Therefore, this filtration induces a spectralsequence Es, t. The first term of this spectral sequence is given by the formula

E0s, t = Fs(F(M)()s+t)/Fs1(F(M)

()s+t) = Fs(M)

()s+t,

and the differential d0 is equal to the canonical differential M. From this descrip-tion and the properties of , we get that E0s, t() = F(s)()s+t.

The filtration Fs of F(M)() is bounded below and exhaustive (F1 = 0 and

F0 = F(M)()). Hence, the classical convergence theorem of spectral sequences(cf. [W] 5.5.1) shows that the spectral sequence Es, t converges to the homology of

F(M)().

We can now prove the equivalence :

() If : M M is a quasi-isomorphism, since the functor F is exact (cf.Proposition 3.13), we have that

F(s)()s+t = E0s, t(

()) : E0s, t(F(M)()) E0s, t(F(M

)())

is a quasi-isomorphism. Hence, E1(()) is an isomorphism. The convergence of thespectral sequence shows that () : F(M)() F(M)() is a quasi-isomorphism.

() In the other way, if is a quasi-isomorphism, then each () is a quasi-isomorphism. We will show by induction on that () : M() M() is aquasi-isomorphism.For = 0, we have M(0) = M(0) = 0 and the morphism (0) is a quasi-isomorphism.Suppose that the result is true up to and show it for + 1. In this case, since

E0s, t(F(M)(+1)) = F(s)(M)

(+1)s+t , one can see that E

1s, t(

(+1)) is an isomorphismfor every t, so long as s < 1.Once again, we consider the mapping cone of the application (+1). We define thesame filtration on the mapping cone cone((+1)) as on F(M)(+1) :

Fs(cone((+1))) := Fs(

1F(M)(+1)) Fs(F(M)(+1)).

This filtration induces a spectral sequence such thatE0s,(cone(

(+1))) = cone(E0s, (+1)).

As in the proof of Theorem 5.4, we have the following long exact sequence

GG E1s, t+1(F(M)(+1)) GG E1s, t+1(F(M

)(+1)) GG E1s, t(cone((+1)))

GG E1s, t(F(M)(+1)) GG E1s, t(F(M

)(+1)) GG E1s, t1(cone((+1))) GG

Since E1s, t((+1)) is an isomorphism, for s < 1, the long exact sequence shows

that E1s, t(cone((+1)) = 0 for every t so long as s < 1. The form of the filtration

gives that E1s, t(cone((+1))) = 0 for every t when s 0. The spectral sequence

53


30Mar2011


54/78

BRUNO VALLETTE

Es, t(cone((+1))) collapses at rank E1 (only the row s = 1 is not null). It implies

thatE1, t(cone(

(+1))) = E11, t(cone((+1))).

Applying the classical convergence theorem of spectral sequences, we get that the

Bruno Vallette- A Koszul Duality for props

Documents

Transcript of Bruno Vallette- A Koszul Duality for props