Introduction to cyclic homology Christian Voigtcvoigt/papers/cyclic.pdf · The basic object of...

Introduction to cyclic homology

Christian Voigt

CHAPTER 1

Preliminaries

Throughout the text we will work over the field C of complex numbers. Inparticular, vector spaces, linear maps and algebras will always be defined over thecomplex numbers. This is convenient for the purposes of noncommutative geometry,however, there are interesting applications of Hochschild and cyclic homology in thesetting of more general commutative ground rings. Actually, most of the material wediscuss in chapter 3 may be developped in the same way over arbitrary commutativerings.We point out that in our terminology an algebra will not be required to possess aunit. Again, this is convenient for noncommutative geometry, but it is importantto note that this terminology is not commonly accepted.

1. Algebras and modules

The basic object of study in cyclic homology are algebras. We shall thus beginwith the definition of an algebra.

Definition 1.1. An algebra is a vector space A together with a bilinear mapµ : A×A→ A written as µ(a, b) = ab and called multiplication such that

(ab)c = a(bc)

for all a, b, c ∈ A. A unital algebra is an algebra with an element 1 ∈ A such that1a = a1 = a for all a ∈ A.An algebra homomorphism f : A → B between algebras is a linear map such thatf(ab) = f(a)f(b) for all a, b ∈ A. A unital homomorphism f : A → B betweenunital algebras is a homomorphism such that f(1) = 1.

The easiest example of an algebra is the zero vector space A = 0. More gen-erally, one may equip any vector space with the zero multiplication to obtain analgebra. We will discuss more interesting examples of algebras below.There are a few standard construction with algebras. Let us have a look at two ofthem. Firstly, given two algebras A and B their direct sum A ⊕ B is the algebradefined by the multiplication (a1, b1)(a2, b2) = (a1a2, b1b2).Secondly, there is an easy way to adjoin a unit element to an algebra A. One definesA+ = A⊕ C as a vector space but with the multiplication

(a, α)(b, β) = (ab+ αb+ βa, αβ).

It is easy to check that A+ becomes a unital algebra with unit element (0, 1) in thisway. The algebra A+ is called the unitarization of A. We have a natural algebrahomomorphism ι : A→ A+. Remark that the unit element of A+ is different fromthe unit of A if the algebra A itself is unital.If A happens to be unital, the algebra A+ can be described as follows.

Excercise 1.2. Let A be a unital algebra with unit element 1A. Then the mapφ : A+ → A ⊕ C given by φ(a, α) = (a + α · 1A, α) is an isomorphism of unitalalgebras.

The unitarization of an algebra is characterized by the following property.

3

4 1. PRELIMINARIES

Excercise 1.3. Let A be an algebra and let B be a unital algebra. For every al-gebra homomorphism f : A→ B there exists a unique unital algebra homomorphismF : A+ → B such that the diagram

A A+-ι

f@@@@RB?

F

is commutative.

Let us discuss some examples of algebras.

1.1. Matrix algebras. We denote by Mn(C) the vector space of n × n-matrices with entries in C with the usual addition and multiplication. It is easy tocheck that Mn(C) is a unital algebra. More generally, if A is an arbitrary algebrawe obtain the algebra Mn(A) = of n × n-matrices with entries in A. This algebrais unital iff A is unital.

1.2. Smooth functions on manifolds. Let M be a smooth manifold andlet C∞(M) be the linear space of complex-valued smooth functions on M . ThenC∞(M) becomes a unital algebra with pointwise multiplication of functions. Onemay also consider the algebra C∞c (M) of smooth functions with compact support.Clearly, C∞c (M) is unital iff M is compact.

1.3. Group rings. Let Γ be a discrete group and let CΓ be the vector spacewith basis Γ. Elements in CΓ can be written as finite sums

n∑j=1

αjtj

with αj ∈ C and tj ∈ Γ. One defines a multiplication on CΓ by extending the groupmultiplication Γ×Γ→ Γ to a bilinear map CΓ×CΓ→ CΓ. It is easy to check thatCΓ becomes a unital algebra in this way. Associativity of the multiplication followsfrom associativity of the group law and the unit element is given by e = 1e ∈ CΓwhere e ∈ Γ is the unit element.

Returning to the general theory, we come now to the definition of modules overan algebra.

Definition 1.4. Let A be an algebra. A (left) module over A is a vector spaceM together with a bilinear map A×M →M such that

(ab)m = a(bm)

for all a, b ∈ A and m ∈ M . A unitary (left) module over a unital algebra A is anA-module M such that 1m = m for every m ∈ M . An A-module homomorphismf : M → N between (unitary) A-modules is a linear map which satisfies f(am) =af(m) for all a ∈ A and m ∈M .In a similar way one defines (unitary) right A-modules and their homomorphisms.

If M and N are left A-modules we write HomA(M,N) for the vector space ofA-module homomorphisms between M and N . We will frequently write AM of MA

to indicate that M is a left or right A-module, respectively. Every algebra A canbe viewed as a left or right module over itself using the multiplication map.Let End(M) denote the unital algebra of linear endomorphisms of the vector spaceM . An A-module structure on M may be described as a homomorphism φ : A →End(M) such that φ(a)(m) = am. Having this in mind, the following statement isa consequence of excercise 1.3.

2. PROJECTIVE AND INDUCTIVE LIMITS 5

Excercise 1.5. Let AM be a module over A. Then M becomes a unitarymodule over A+ by declaring 1m = m for all m ∈ M . Conversely, every unitaryA+-module can be viewed as an A-module by restricting the action to A.

Let us discuss another operation with algebras. The opposite algebra Aop ofan algebra A has the same underlying vector space as A and is equipped with theopposite multiplication

a • b = b · awhere b · a denotes the multiplication in A.An algebra A is called commutative if ab = ba for all a, b ∈ A. In this case theopposite algebra Aop is equal to A.Next we shall see that it is in principle sufficient to consider only left modules.

Excercise 1.6. Let M be a left module over A. Then M is a right Aop-moduleby setting

ma = am

for all m ∈M and a ∈ Aop.

However, usually modules over an algebra A appear naturally as left or rightmodules and it is convenient not to work with the algebra Aop.We conclude this section with the definition of a bimodule.

Definition 1.7. Let A and B be algebras. An A-B-bimodule is a vector spaceM which is both a left A-module and a right B-module such that

(am)b = a(mb)

for all a ∈ A,m ∈M and b ∈ B. If A and B are unital, a unitary A-B-bimodule isan A-B-bimodule M such that 1m = m = m1 for every m ∈M .A bimodule homomorphism f : M → N between (unitary) A-B-bimodules is a linearmap which is both a (unitary) A-module homomorphism and a (unitary) B-modulehomomorphism.

A basic example of an A-A-bimodule is the algebra A itself with the left andright action by multiplication.A submodule N of an A-module M is a linear subspace N ⊂M such that an ∈ Nfor all n ∈ N , that is, if it is an A-module with the restricted action. The quotientM/N of an A-module by a submodule is the ordinary quotient space with theA-module structure induced by M . Similar definitions are made for bimodules.

2. Projective and inductive limits

In this section we discuss projective and inductive limits of modules over analgebra.We begin with direct products. Let A be an algebra and let (Mj)j∈J be a familyof A-modules. The direct product of this family is the vector space∏

j∈JMj

with componentwise action of A. For every i ∈ J the canonical projection πi :∏j∈JMj →Mi is an A-module map. The direct product is a unitary A-module iff

all modules Mj are unitary. Moreover we have the following universal property.

Excercise 1.8. Let (Mj)j∈J be a family of A-modules. For every A-module Nand every family (fj)j∈J of A-module homomorphisms fj : N →Mj there exists a

6 1. PRELIMINARIES

unique A-module homomorphism f : N →∏j∈JMj such that the diagrams

N∏j∈JMj-f

fi@@@@RMi

?

πi

are commutative for all i ∈ J .

As a generalization of direct products one defines projective limits. First recallthe definition of a partially ordered set.

Definition 1.9. A set J is partially ordered if there is a relation ≤ defined onJ such that

a) j ≤ j for all j ∈ J (reflexivity).b) j ≤ i and i ≤ j implies i = j (symmetry).c) j ≤ i and k ≤ j implies k ≤ i (transitivity).

A partially ordered set is called directed if for all i, j ∈ J there exists k ∈ J suchthat i ≤ k and j ≤ k.

Every set is partially ordered using the trivial relation stipulating only j ≤ jfor all j ∈ J . Note that this partial ordering is directed only if J consists of a singleelement. An easy example of a directed set is the set N of natural numbers withits natural ordering. Actually, for our purposes this will be the most importantexample of a directed set.An inverse system of A-modules over a partially ordered set J is a family (Mj)j∈Jof A-modules together with A-module maps πji : Mi → Mj for all j ≤ i such thatπii = id for all i and πkjπji = πki whenever k ≤ j ≤ i. The projective limit of aninverse system is the A-submodule

lim←−j∈J

Mj ⊂∏j∈J

Mj

consisting of all families (mj)j∈J such that mj = πji(mi) whenever j ≤ i. Again,for every i ∈ J the canonical projection πi : lim←−j∈JMj →Mi is an A-module map.

The inverse limit is a unitary A-module if all modules Mj are unitary and we havethe following universal property.

Excercise 1.10. Let (Mj)j∈J be an inverse system of A-modules over thedirected set J . For every A-module N and every family (fj)j∈J of A-module ho-momorphisms fj : N →Mj satisfying fj = πjifi for all j ≤ i there exists a uniqueA-module homomorphism f : N → lim←−j∈JMj such that the diagrams

N lim←−j∈JMj-f

fi@@@@RMi

?

πi


In the special case where J is partially ordered with the trivial partial orderrelation discussed above we reobtain the definition and characterization of directproducts.Dual to the notion of a direct product one defines direct sums. Let again (Mj)j∈J

2. PROJECTIVE AND INDUCTIVE LIMITS 7

be a family of A-modules over an algebra A. The direct sum of this family is thevector space⊕

j∈JMj = (xj)j∈J ∈

∏j∈J

Mj |xj = 0 for all but finitely many j ∈ J

with addition and module action inherited from∏j∈JMj . For every i ∈ J there is

a canonical A-module map ιi : Mi →⊕

j∈JMj .

Excercise 1.11. Let (Mj)j∈J be a family of A-modules. For every A-moduleN and every family (fj)j∈J of A-module homomorphisms fj : Mj → N there existsa unique A-module homomorphism f :

⊕j∈JMj → N such that the diagrams

Mi

⊕j∈JMj-ιi

fi@@@@RN?

f


An important special case arises if all modules Mj are equal to A+.

Definition 1.12. Let J be a set and A be an algebra. The free A-module overJ is the direct sum

AJ =⊕j∈J

A+

of copies of A+.

The next excercise describes the universal property of free modules.

Excercise 1.13. Let AJ be the free A-module over the set J and let M beany A-module. For every map f : J → M there exists a unique A-module mapF : AJ →M such that the diagram

J AJ-ι

f@@@@RM?

F

is commutative.

As a generalization of direct sums one defines inductive limits. Essentiallythis consists of reversing the order of arrows in all statements in the definitionof projective limits. Let J be a partially ordered set. An inductive system ofA-modules is a family (Mj)j∈J of A-modules together with A-module maps πji :Mi → Mj for all i ≤ j such that πkjπji = πki whenever i ≤ j ≤ k. The inductivelimit of an inductive system is the quotient A-submodule⊕

j∈JMj → lim−→

j∈JMj

obtained by dividing out the subspace generated by all elements of the form mj −πji(mi) whenever i ≤ j. For every i ∈ J the canonical map ιi : Mi → lim−→j∈JMj

is an A-module map. The inductive limit is a unitary A-module if all modules Mj

are unitary and we have the following universal property.

8 1. PRELIMINARIES

Excercise 1.14. Let (Mj)j∈J be an inductive system of A-modules over thedirected set J . For every A-module N and every family (fj)j∈J of A-module ho-momorphisms fj : Mj → N satisfying fjπji = fi for all i ≤ j there exists a uniqueA-module homomorphism f : lim−→j∈JMj → N such that the diagrams

Mi lim−→j∈JMj-ιi

fi@@@@RN?

f


As above, in the special case where J is partially ordered with the trivial partialorder relation we reobtain the definition and characterization of direct sums.We point out that in the context of projective and inductive limits the terminologyis not unique in the literature. Sometimes projective limits are called inverse limitsand inductive limits are called direct limits. An inductive system of modules is alsocalled a directed system.Finally we remark that in the special case A = 0 we (re-)obtain the definitions ofdirect products, sums as well as projective and inductive limits of vector spaces.

3. Tensor products

In this section we define and study tensor products of modules over algebras.We begin with the tensor product of modules. Let MA and AN be modules overan algebra A and let V be a vector space. A bilinear map f : M ×N → V is calledA-bilinear if f(ma, n) = f(m, an) for all m ∈M,n ∈ N, a ∈ A.

Definition 1.15. Let MA and AN be A-modules. A vector space M ⊗A Ntogether with an A-bilinear map ⊗ : M ×N 3 (m,n) 7→ n⊗ n ∈M ⊗A N is calledtensor product of M and N over A if for every vector space V and every A-bilinearmap f : M ×N → V there exists a unique linear map F : M ⊗A N → V such thatthe diagram

M ×N M ⊗A N-⊗

f@@@@RV?

F

is commutative.

Lemma 1.16. The tensor product M ⊗A N is uniquely determined up to iso-morphism by MA and AN .

Proof. Let M ⊗A N and M A N be tensor products of M and N and let⊗ : M ×N →M ⊗A N and : M ×N →M A N be the corresponding bilinearmaps. By the universal property, there exist linear maps h : M ⊗A N → M A Nand k : M A N → M ⊗A N such that = h⊗ and ⊗ = k. Hence = hkand ⊗ = kh⊗. By the uniqueness assertion we deduce hk = id and kh = id. HenceM ⊗A N and M A N are isomorphic. Even without knowing existence of the tensor product one may prove the followingproperties directly from the definition.

Excercise 1.17. Let M ⊗A N be a tensor product. Then

a) M⊗AN is generated as a vector space by elementary tensors m⊗n with m ∈M ,n ∈ N .

3. TENSOR PRODUCTS 9

b) (w + x)⊗ y = w ⊗ y + x⊗ y for w, x ∈M and y ∈ N .c) x⊗ (y + z) = x⊗ y + x⊗ z for x ∈M and y, z ∈ N .d) xa⊗ y = x⊗ ay for x ∈M , y ∈ N and a ∈ A.

Excercise 1.18. Let M1 and M2 be right A-modules and let N1 and N2 be leftA-modules. If f1 : M1 → M2 and f2 : N1 → N2 are A-module maps there exists aunique linear map f1 ⊗ f2 : M1 ⊗A N1 → M2 ⊗ N2 such that (f1 ⊗ f2)(m ⊗ n) =f1(m)⊗ f2(n).

We shall now show that tensor products always exist.

Proposition 1.19. Let MA and AN be modules. Then there exists a tensorproduct M ⊗A N .

Proof. We let M⊗AN be the quotient of the vector space P with basis M×Nby the relations

(w + x, z) = (w, z) + (x, z), (x, y + z) = (x, y) + (x, z), (xa, y) = (x, ay)

for all w, x ∈M , y, z ∈ N and a ∈ A+. The map ⊗ : M ×N →M ⊗AN is inducedby the canonical map ι : M ×N → P . Now let f : M ×N → V be an A-bilinearmap. Then there exists a unique linear map F : P → V such that Fι = f . Since fis assumed to be A-bilinear we see that F induces a linear map F : M ⊗A N → Vwhich satisfies F⊗ = f . Now assume that G : M ⊗A N → V is another linearmap such that G⊗ = f . It follows that the resulting map Gπ : P → V is equalto Fπ : P → V where π : P → M ⊗A N is the canonical projection. Since π issurjective this implies F = G. We may view modules IfMA and AN as unitary modulesMA+ and A+N in a naturalway. It is straightforward to check that the natural map M ⊗A N → M ⊗A+ N isan isomorphism. Hence we do not have to care wether we consider modules over Aor unitary modules over A+.Next we show that in some cases the tensor product of A-modules can be describedin a more concrete way.

Proposition 1.20. Let MA be an arbitrary A-module and let N = AJ be thefree left A-module over J . Then

M ⊗A N ∼=⊕j∈J

M.

Proof. An A-bilinear map f : M × AJ → V is uniquely determined by thelinear maps fj : M → V given by fj(m) = f(m, ej) where ej denotes the element of⊕

j∈JM determined by the unit element of A+ in the jth position. It follows that⊕j∈JM satisfies the universal property of a tensor product. Hence the assertion

is a consequence of lemma 1.16. A similar assertion holds if MA is a free right A-module and N is arbitrary. As aconsequence we obtain the following result.

Corollary 1.21. Let MA = IA and AN = AJ be free modules. Then

M ⊗A N ∼=⊕

(i×j)∈I×J

A+.

Proof. This follows from proposition 1.20 and the natural isomorphism⊕j∈J

(⊕i∈I

A+

)=

⊕(i×j)∈I×J

A+

which in turn is a consequence of the universal property of direct sums. In the special case A = 0 we simply write M ⊗N for the tensor product over thezero algebra. Recall that a module over the zero algebra is simply a vector space.

10 1. PRELIMINARIES

Corollary 1.22. Let V and W be vector spaces with bases (ui)i∈I and (vj)j∈J ,respectively. Then M ⊗N is a vector space with basis (ui ⊗ vj)(i,j)∈I×J .

Let us come back to the general situation. If M and N happen to be bimodulesthen the same holds true for their tensor product. More precicsely, let AMB and

BNC be bimodules. We want to define an A-C-bimodule structure on M ⊗B Nusing the formulas

a(m⊗ n) = am⊗ n, (m⊗ n)c = m⊗ nc.However, to be precise we define for each a ∈ A a map la : M ×N →M ⊗B N by

la(m,n) = am⊗ nand verify that la is B-bilinear. Let La : M ⊗BN →M ⊗BN be the correspondinglinear map. Then we define the left module structure A× (M ⊗B N) → M ⊗B Nby (a, x) 7→ La(x). Similarly one has to proceed for the right action of C.

Excercise 1.23. Verify that AM⊗BNC becomes an A-C-bimodule in this way.

Proposition 1.24. Let MA,ANB and BP be modules. Then there exists anatural isomorphism

(M ⊗A N)⊗B P ∼= M ⊗A (N ⊗B P ).

Proof. Both spaces are universal for trilinear maps f : M×N×P → V whichsatisfy f(ma, n, p) = f(m, an, p) and f(m,nb, p) = f(m,n, bp) for all elements inM,N,P and A,B, respectively. The assertion follows easily from this.

Excercise 1.25. If AMB and ANC are bimodules the vector space HomA(M,N)of A-module homomorphisms between M and N becomes an B-C-bimodule usingthe formula (bfc)(m) = f(mb)c.

We shall now formulate an important property of tensor products.

Proposition 1.26. Let AMB ,B NC and APD be bimodules. Then there existsa natural isomorphism

C HomA(M ⊗B N,P )D =C HomB(N,HomA(M,P ))D

of C-D-bimodules.

Proof. One defines a map φ : HomA(M ⊗BN,P )→ HomB(N,HomA(M,P ))by φ(f)(m)(n) = f(m⊗n). Conversely, one defines ψ : HomB(N,HomA(M,P ))→HomA(M⊗BN,P ) by ψ(f)(m⊗n) = f(n)(m). We leave it as an excercise to checkthat these maps are well-defined inverse isomorphisms. We conclude this section with a discussion of the tensor product of algebras. LetA and B be algebras and consider the tensor product A⊗B.

Excercise 1.27. There is a multiplication on A⊗B given by (a1⊗b1)(a2⊗b2) =a1a2 ⊗ b1b2 which turns A⊗B into an algebra.

Using excercise 1.5 and excercise 1.6 we see that an A-B-bimodule is the samething as a unitary left A+⊗(Bop)+-module. One should however be careful with theunitarizations at this point. The following example shows that not every A⊗Bop-module is the restriction of an A-B-bimodule.Consider the commutative algebra

A = B = f ∈ C∞[0, 1]|f(0) = 0with pointwise multiplication and let M be the linear space of all functions f ∈C∞[0, 1] such that f ′(0) = 0. We claim that M is an A⊗B-module using the action

(f ⊗ g)h = fgh

4. PROJECTIVE MODULES 11

given by pointwise multiplication of functions. The essential point here is to checkthat this action is well-defined. However, using the Leibniz rule we get

∂

∂x(fgh)(0) =

∂

∂x(f)(0)g(0)h(0) + f(0)

∂

∂x(g)(0)h(0) + f(0)g(0)

∂

∂x(h)(0) = 0

provided f ∈ A, g ∈ B and h ∈M . This shows that (f ⊗ g)h is an element of M asrequired. Assume that M is the restriction of an A-B-bimodule and consider theelement

m = f · χ · 1

in M where χ ∈M is the constant function with value 1 and 1 denotes the unit ofB+. According to the definitions, we have

(mg)(x) = f(x)g(x)

for all g ∈ B and x ∈ [0, 1]. Choosing functions gε ∈ B such that gε(x) = 1 forx > ε for all ε > 0 we deduce m(x) = f(x) for all x > 0. Since m is continuous thisimplies m = f ∈ M which is a contradiction since the derivative of f at zero doesnot necessarily vanish.

4. Projective modules

Projective modules play an important role in homological algebra. In thissection we discuss their basic properties.

Definition 1.28. Let A be an algebra. A module AP is called projective if forevery epimorphism π : M → N of A-modules and every A-module map f : P → Nthere exists an A-module map F : P →M such that the diagram

M N-π

F

P

?

f

is commutative.

Excercise 1.29. For every algebra A the A-module A+ is projective. Moreover,for every set J the free module AJ is projective. More generally, a direct sum ofprojective modules is projective.

An A-submodule M of an A-module P is called a direct summand if there existsan A-submodule N in P such that the natural map M⊕N → P is an isomorphism.Equivalently, there exists an A-module map π : P → M such that πι = id whereι : M → P is the natural inclusion.

Excercise 1.30. If M is isomorphic to a direct summand in a projective mod-ule, then M is itself projective.

An epimorphism π : M → N of A-modules is called split if there exists anA-module homomorphism σ : N →M such that πσ = id.

Proposition 1.31. Let AP be a module. The following are equivalent:

a) P is projective.b) Every epimorphism π : M → P splits.c) P is isomorphic to a direct summand in a free module.

12 1. PRELIMINARIES

Proof. a) ⇒ b) Let π : M → P be an epimorphism. According to theprojectivity of P there exists an A-module map σ : P →M such that the diagram

M P-π

σ

P

?

id

is commutative. This shows that π splitsb) ⇒ c) Consider the free A-module over the set P . There exists a canonical A-module map π :

⊕p∈P A

+ → P characterized by πιp(1) = p where 1 denotes the

unit in A+. Since π is clearly surjective, there exists a splitting σ for π. This showsthat P is a direct summand in the free module

⊕p∈P A

+.

c⇒ a) This implication is contained in excercise 4.14 and excercise 4.15. If A is unital we see using excercise 1.2 that there is a natural A-module isomomor-phism A+ ∼= A⊕Cτ where Cτ is the zero module. It follows that A is a projectiveA-module in this case.

Proposition 1.32. Let A be a unital algebra. Then a unitary A-module P isprojective iff it is a direct summand in

⊕j∈J A for some index set J .

Proof. According to the previous considerations and excercise 4.15, directsummands in the A-module

⊕j∈J A are projective for every index set J . Note

that such direct summands are automatically unitary. Conversely, assume that Pis projective and unitary. Then the natural A-module homomorphism

f :⊕p∈P

A→ P

determined by fιp(a) = ap is surjective. Using that P is projective there exists asplitting σ for π and it follows that P is a direct summand in

⊕p∈P A.

We next prove the dual basis lemma. For this we need some more terminology. A(unitary) module M over a (unital) algebra A is called finitely generated if thereexist elements m1, . . . ,mn ∈M for some n ∈ N such that the smallest A-submodulecontaining m1, . . . ,mn is equal to M .For every module MA denote by M∗ the right A-module HomA(M,A). Then thereis a natural map

db : M ⊗AM∗ → EndA(M)

defined by db(m⊗f)(x) = mf(x). This map is called the dual-basis homomorphism.

Proposition 1.33 (Dual basis lemma). Let PA be a unitary module over theunital algebra A. The following are equivalent:

a) P is finitely generated and projective.b) There exist f1, . . . , fn ∈ HomA(P,A) and p1, . . . , pn ∈ P such that

p =

n∑j=1

pjfj(p).

c) The dual basis homomorphism db : P ⊗A P ∗ → EndA(P, P ) is an isomorphism.

Proof. a)⇒ b) According to proposition 1.32 there are A-module homomor-phisms π : An =

⊕nj=1A→ P and σ : P → An such that πσ = id. Define fj = πjσ

where πj : An → A is the projection onto the j-th component and pj = π(ιj(1)).These elements satisfy the desired relation.b)⇒ a) Since p =

∑pjfj(p) for all P the module P is finitely generated. Moreover

the homomorphism π :⊕n

j=1A → P determined by pi = πιi(1) is surjective. Let

4. PROJECTIVE MODULES 13

σ : P →⊕n

j=1A be defined by σ(p) =∑pjfj(p). Then πσ(p) = p for all p ∈ P .

Hence P is a direct summand in∑nj=1A and thus projective according to proposi-

tion 1.32.b)⇒ c) Consider the map φ : EndA(P )→ P ⊗A P ∗ given by φ(f) =

∑f(pj)⊗ fj .

We havedbφ(f)(x) =

∑f(pj)fj(x) = f(x).

Hence dbφ = id. Conversely,

φdb(p⊗ f) =∑

pf(pj)⊗ fj =∑

p⊗ f(pj)fj = p⊗ f.

This shows φdb = id. Hence db is an isomorphism.c)⇒ b) If db is an isomorphism choose elements pj and fj such that db(

∑pj⊗fj) =

id. Then pj and fj satisfy the required relation. Let us have a look at some examples.

Proposition 1.34. Let Cτ be the vector space C with the zero multiplication.Then Cτ is not a projective Cτ -module.

Proof. Assume that Cτ is projective. Then there exists a Cτ -linear splittingσ for the multiplication map C+

τ ⊗ Cτ → Cτ . Consider

σ(1) =

n∑j=1

(aj , αj)⊗ βj

in C+τ ⊗ Cτ . Then

∑nj=1 αjβj = 1 since πσ(1) = 1. By Cτ -linearity of σ we have

n∑j=1

γ(aj , αj)⊗ βj =

n∑j=1

(γαj , 0)⊗ βj = 0

for all γ ∈ Cτ . This is a contradiction. The following unitary example is more interesting.

Proposition 1.35. Consider the algebra A = C∞[0, 1] of smooth functions onthe interval [0, 1] and let C be the unitary A-module defined by f ·α = f(0)α. ThenC is not a projective A-module.

Proof. Assume that the A-module C is projective. Then there exists a sectionσ : C → A for the natural A-linear projection π : A → C given by π(f) = f(0).Consider the function σ(1) ∈ A. Since f(x)σ(1)(x) = f(0)σ(1)(x) for all f ∈ Aby A-linearity we see that σ(1)(x) = 0 for all x > 0. Since σ(1) is continuous thisimplies σ(1) = 0 which is a contradiction to πσ = id. Finally, let us have a look at an example of a projective module which is not free.Consider a discrete group Γ and the unital homomorphism ε : CΓ→ C defined byε(t) = 1 for all t ∈ Γ. The map ε is called the augmentation homomorphism of CΓ.

Proposition 1.36. Let Γ be a finite group different from the trivial group.Then the unitary CΓ-module C defined by the augmentation homomorphism is pro-jective but not free.

Proof. It is clear that C is not free by dimension reasons. There exists aCΓ-linear splitting σ for the surjection ε : CΓ→ C given by

σ(1) =∑t∈Γ

1

nt

where n is the number of elements in Γ. Hence C is projective according to propo-sition 1.32. In fact, the CΓ-module C defined by the augmentation homomorphism is projectiveiff the group Γ is finite.

14 1. PRELIMINARIES

5. Morita theory

In this section we describe the notion of Morita equivalence of unital algebras.Morita equivalence is an important concept for noncommutative geometry sincemany natural examples of algebras modelling noncommutative spaces are only de-termined up to Morita equivalence.

Definition 1.37. Let A and B be unital algebras. A Morita context for A andB consists of two unitary bimodules APB and BQA and bimodule maps

〈−,−〉A : P ⊗B Q→ A, 〈−,−〉B : Q⊗A P → B

such that

〈p1, q〉A p2 = p1〈q, p2〉B〈q1, p〉B q2 = q1〈p, q2〉A

for all pi, p ∈ P and qi, q ∈ Q. A Morita context is called strict if the maps〈−,−〉A : P ⊗B Q→ A, 〈−,−〉B : Q⊗A P → B are surjective.

Definition 1.38. Two unital algebras A and B are called Morita equivalent ifthere exists a strict Morita context for A and B.

Clearly every unital algebra A is Morita equivalent to itself and the relationof Morita equivalence is symmetric. The next excercise shows that this relation istransitive.

Excercise 1.39. Let A,B and C be unital algebras. If A is Morita equivalentto B and B is Morita equivalent to C, then A is Morita equivalent to C.

Hence Morita equivalence satisfies the axioms of an equivalence relation.The modules P and Q in a strict Morita context are often called equivalence bi-modules.We will now prove a basic result on Morita equivalent algebras.

Theorem 1.40. Let A and B be Morita equivalent unital algebras and let Pand Q be equivalence bimodules. Then

a) The maps 〈−,−〉A and 〈−,−〉B are isomorphisms.b) P is finitely generated projective as left A-module and as right B-module.

Q is finitely generated projective as left B-module and as right A-module.c) There are isomorphisms

A ∼= EndB(Q,Q) ∼= EndB(P, P ) B ∼= EndA(P, P ) ∼= EndA(Q,Q)

as algebras.

Proof. a) Since 〈−,−〉A is surjective there exists∑xi ⊗ yi ∈ P ⊗B Q such

that∑〈xi, yi〉A = 1. Now assume that

∑〈vj , wj〉A = 0. Then we have∑

vj ⊗ wj =∑

vj ⊗ wj〈xi, yi〉A =∑

vj〈wj , xi〉B yi =∑〈vj , wj〉A xi ⊗ yi = 0

and hence 〈−,−〉A is injective. By symmetry, the same holds true for 〈−,−〉B .b) The isomorphism in proposition 1.26 shows that 〈−,−〉A ∈ HomA−A(P ⊗BQ,A)corresponds to a map φ : Q→ HomA(P,A) = P ∗. Since 〈−,−〉B is surjective thereexist elements qi ∈ Q and pi ∈ P such that

∑〈qj , pj〉B = 1. Then

p = p∑〈qj , pj〉B =

∑〈p, qj〉Apj =

∑φ(qi)(p)pj

for all p ∈ P . According to the dual basis lemma 1.33 (or rather its version for leftmodules) this shows that AP is finitely generated and projective. The assertionsfor PB ,B Q and QA are proved in a similar way.c) The right A-module structure of Q induces a unital algebra homomorphism

5. MORITA THEORY 15

φ : A→ EndB(Q) given by φ(a)(q) = qa. Let us show that f is an isomorphism. Ifφ(a)(q) = qa = 0 for all q ∈ Q we have

a =∑〈xi, yi〉Aa =

∑〈xi, yia〉A = 0.

Hence f is injective. For f ∈ EndB(Q) we have

f(q) = f(q1) =∑

f(q〈xi, yi〉A) =∑

f(〈q, xi〉Byi)

=∑〈q, xi〉Bf(yi) =

∑q〈xi, f(yi)〉A

and hence f = φ(∑〈xi, f(yi)〉A) is in the image of φ. Again, the remaining asser-

tions follow by symmetry. The most important example of a Morita equivalence is given as follows. Consider aunital algebra A and the algebra Mn(A) of matrices over A. Moreover let APMn(A)

be the space An of all row vectors and Mn(A)QA be the space An of all columnvectors of length n with entries in A. The module actions are given by matrixmultiplication.

Excercise 1.41. Show that P and Q define equivalence bimodules for A andMn(A).

Assume that A is a unital algebra such that every finitely generated projectiveunitary A-module is isomorphic to a finite direct sum of copies of A. Then it followsfrom theorem 1.40 that every Morita equivalence between A and another algebra isof the form described before. This applies in particular to the algebra C of complexnumbers.

CHAPTER 2

Homological algebra

Homological algebra is a set of tools to study the homology of chain complexes.We will need some of these tools for our study of cyclic homology. There are severalgood textbooks on homological algebra, we follow closely the treatment in the bookof Weibel [14].

1. Chain complexes

Definition 2.1. Let A be an algebra. A chain complex of A-modules is asequence C = (Cn)n∈Z of A-modules Cn together with module homomorphismsdn : Cn → Cn−1 such that dndn+1 = 0 for all n ∈ Z. A chain map f : C → Dbetween chain complexes is a family fn : Cn → Dn of A-module homomorphismssuch that the diagrams

Cnd //

fn

Cn−1

fn−1

Dn

d // Dn−1

are commutative for all n ∈ N.

A chain complex of A-modules for A = 0 is simply called a chain complex. Alsoin the general case we will occasionally omit the algebra A in our terminology.A chain complex is called bounded below if there exists N ∈ Z such that Cn = 0for all n < N . Similarly, it is called bounded above if there exists N ∈ Z such thatCn = 0 for all n > N . A chain complex is called bounded if it is bounded belowand above. In the sequel we will meet mainly bounded below chain complexes.It is common to write d : Cn → Cn−1 instead of dn. We will also do this, in this waythe important algebraic property of the differential is d2 = dd = 0. The elementsx ∈ Cn of a chain complex C are called n-chains or simply chains. Elements of theform d(x) for some x ∈ Cn+1 are called n-boundaries. The space of all n-boundariesis denoted by Bn(C). Similarly, elements x ∈ Cn satisfying d(x) = 0 are called n-cycles. The space of all cycles is denoted by Zn(C). The relation d2 = 0 impliesBn ⊂ Zn for all n.

Definition 2.2. The n-th homology group of a chain complex C is the spaceHn(C) = Zn/Bn.

Note that the homology Hn(C) of a chain complex (of A-modules) is in fact avector space (even an A-module). It is easy to check that a chain map f : C → Dinduces a map Hn(f) : Hn(C) → Hn(D) on homology for all n. An importantsituation is when these induced maps are isomorphisms.

Definition 2.3. A chain map f : C → D is called a quasiisomorphism if theinduced maps Hn(f) : Hn(C)→ Hn(D) are isomorphisms for all n.

A chain complex is called acyclic if Hn(C) = 0 for all n. Clearly, a chaincomplex C is acyclic iff the trivial chain map 0 → C is a quasiisomorphism. Iff : C → D is a chain map, then ker(f) and im(f) are again chain complexes.

17

18 2. HOMOLOGICAL ALGEBRA

Moreover C/ ker(f) and D/ im(f) become chain complexes in a natural way. Wewill see below that the induced map Hn(f) : Hn(C) → Hn(D) of an injective(surjective) chain map is not injective (surjective) in general.

Definition 2.4. Two chain maps f, g : C → D are called homotopic if thereexists a map h : C → D of degree 1, that is, a family of homomorphisms hn : Cn →Dn+1 such that dhn + hn−1d = fn − gn for all n, or simply

dh+ hd = f − g.

A chain map f : C → D is called a homotopy equivalence if there exists a chainmap g : D → C such that fg is homotopic to the identity map on D and gf ishomotopic to the identity map on D. Two chain complexes C and D are calledhomotopy equivalent if there exists a homotopy equivalence between C and D.A chain complex C is called contractible if it is chain homotopy equivalent to thetrivial chain complex 0.

We write f ∼ g if the chain maps f and g are homotopic. If f : C → D isa homotopy equivalence then a map g : D → C satisfying fg ∼ id and gf ∼ idis called a homotopy inverse of f . Note that a homotopy inverse is in general notuniquely determined.The following excercise shows that homotopy is an equivalence relation.

Excercise 2.5. Let f, g, h : C → D be chain maps. We have the followingimplications.

a) f ∼ f .a) f ∼ g implies g ∼ f .b) f ∼ g and g ∼ h implies f ∼ h.

Lemma 2.6. Let f, g : C → D be homotopic. Then we have Hn(f) = Hn(g)for all n. In particular, every homotopy equivalence is a quasiisomorphism.

Proof. Let x ∈ Cn be a cycle. Then f(x)− g(x) = dh(x) + hd(x) = dh(x) =0 ∈ Hn(D). Hence Hn(f) = Hn(g). If f : C → D is a homotopy equivalencewith homotopy inverse g : D → C the previous assertion implies that Hn(f) is anisomorphism with inverse Hn(g) for all n. Apart from ordinary complexes we will also need bicomplexes for the definition ofcyclic homology.

Definition 2.7. A bicomplex is a family C = (Cmn)(m,n)∈Z×Z of modules Cmntogether with horizontal differentials dh : Cmn → Cm−1n and vertical differentialsdv : Cmn → Cmn−1 such that

dhdh = 0, dvdv = 0, dhdv + dvdh = 0.

A chain map f : C → D between bicomplexes is a family fmn : Cmn → Dmn ofmaps which commute with both differentials dh and dv.

It is convenient to visualize bicomplexes in the plane. To do this one insertsthe module Cmn in the point (m,n) ∈ R2 and connects adjacent points by arrowsrepresenting the differentials. Motivated by such a picture, one says that C is afirst quadrant bicomplex if Cmn = 0 if m < 0 or n < 0. Hence a first quadrant

1. CHAIN COMPLEXES 19

double complex looks like this:

......

? ?C02 C12d

h

?

dv

?

dv

C00 C10dh

C01 C11d

h

?

dv

?

dv

C21

C22d

h

? ?

dv

dh

dh

? ?

dv

......

? ?C32

dh

? ?

dv

C20dh

dh

? ?

dv

...

?C42

dh

?

dv

C30 C40dh

C31 C41d

h

?

dv

· · ·

· · ·

· · ·

Note that the squares occuring here are not commutative. In fact, they are anti-commutative in the sense that we have dhdv = −dvdh according to the definitionof a bicomplex.Apart from considering particular rows and columns in a bicomplex C there areessentially two canonical ways to associate an ordinary complex to C. The directproduct total complex of C is defined by

Tot(C)n =∏

p+q=n

Cpq

with differential d = dh + dv. The direct sum total complex of C is defined by

tot(C)n =⊕p+q=n

Cpq

and equipped with the same differential. Clearly there is a natural chain maptot(C)→ Tot(C). This map is an isomorphism, for instance, if C is a first quadrantbicomplex. In general however, it is not even a quasiisomorphism. The homologiesof Tot(C) and tot(C) may differ drastically.In connection with Hochschild cohomology and cyclic cohomology we will also usethe concept of a cochain complex.

Definition 2.8. A cochain complex is a sequence C = (Cn)n∈Z of modules Cntogether with homomorphisms dn : Cn → Cn+1 such that dn+1dn = 0 for all n ∈ Z.A chain map f : C → D between cochain complexes is a family fn : Cn → Dn suchthat the diagrams

Cnd //

fn

Cn+1

fn+1

Dn

d // Dn+1


A cochain complex is called bounded below if there exists N ∈ Z such thatCn = 0 for all n < N . It is called bounded above if there exists N ∈ Z such thatCn = 0 for all n > N .The elements x ∈ Cn of a cochain complex C are called n-cochains or simplycochains. Elements of the form d(x) for some x ∈ Cn−1 are called n-coboundaries.The space of all n-coboundaries is denoted by Bn(C). Similarly, elements x ∈ Cn


satisfying d(x) = 0 are called n-cocycles. The space of all cocycles is denoted byZn(C). The relation d2 = 0 implies Bn ⊂ Zn for all n.

Definition 2.9. The n-th cohomology group of a cochain complex C is thespace Hn(C) = Zn/Bn.

Similarly, one may define bicomplexes in the cohomological framework. Weshall not write down explicitly the corresponding definitions.Every cochain complex C can be transformed into a chain complex and vice versaby setting Cn = C−n with the corresponding differential. Hence the concepts of achain complex and a cochain complexes are essentially equivalent. However, mostof the time certain constructions are most naturally viewed as chain complexes orcochain complexes. We leave it to the reader to adapt the notions and results onchain complexes presented in this chapter to the case of cochain complexes.

2. Exact sequences

In this section we discuss the notion of an exact sequence and some fundamentalresults of homological algebra.Let A be an algebra. A sequence

· · · // Cn+1d // Cn

d // Cn−1// · · ·

of A-modules and homomorphisms is called exact if im(d) = ker(d) ⊂ Cn for alln ∈ Z. One also speaks about a long exact sequence in this case. Note that a longexact sequence may be viewed as an acyclic complex.If Cn = 0 exept for three consecutive numbers, such a sequence is called a shortexact sequence and written as

K // i // Ep // // Q.

Explicitly, a short exact sequence consists of A-modules K,E and Q and A-modulehomomorphisms i, p such that i is a monomorphism, p is an epimorphism andim(i) = ker(p).A chain map f : C → D is called a monomorphism (epimorphism) if all mapsfn : Cn → Dn are monomorphisms (epimorphisms). A short exact sequence ofcomplexes is a diagram

K // i // Ep // // Q

of chain complexes and chain maps such that i is a monomorphism, p is an epimor-phism and im(i) = ker(p). Equivalently, in each degree the associated short exactsequence of modules is exact.The following result is of fundamental importance in homological algebra.

Proposition 2.10. Let K // i // Ep // // Q be a short exact sequence of

chain complexes. Then there exists natural connecting homomorphisms ∂ : Hn(Q)→Hn−1(K) for all n such that the sequence

· · · // Hn(K)Hn(i) // Hn(E)

Hn(p) // Hn(Q)∂ // Hn−1(K) // · · ·

is exact.

The connecting homomorphism ∂ : Hn(Q) → Hn−1(K) is constructed as fol-lows. If x ∈ Qn is a cycle with homology class [x] we lift x to a chain y ∈ En andapply d. The resulting element z = dy ∈ En−1 satisfies pdy = 0 and hence lies infact in Kn−1. Moreover we clearly have dz = 0 and thus z defines a homology class[z] ∈ Hn−1(K).

2. EXACT SEQUENCES 21

Excercise 2.11. The homology class [z] ∈ Hn−1(K) depends only on the ho-mology class of x. That is, it is independent of the representative for [x] and of thechoice of y.

Hence we may define ∂ : Hn(Q)→ Hn−1(K) by ∂([x]) = [z].For the proof of proposition 2.10 we shall use the snake lemma.

Lemma 2.12 (Snake lemma). Let A be an algebra and consider a diagram ofA-modules

K1//

f

E1//

g

Q1//

h

0

0 // K2// E2

// Q2

with exact rows. Then there is an exact sequence

ker(f) // ker(g) // ker(h)∂ // coker(f) // coker(g) // coker(h).

Here the connecting map ∂ : ker(h) → coker(f) is defined in the same way asabove. The proof of the snake lemma is done by checking case by case and left tothe reader.Let us now show that the sequence in proposition 2.10 is exact. Consider thediagram

Kn/dKn+1//

d

En/dEn+1//

d

Qn/dQn+1//

d

0

0 // Zn−1(K) // Zn−1(E) // Zn−1(Q)

where Zn−1(K) denotes the space of (n−1)-cycles in the complex K, and similarlyfor E and Q. It is easy to check directly that the rows in this diagram are exact.Applying the snake lemma 2.12 yields an exact sequence

Hn(K) // Hn(E) // Hn(Q)∂ // Hn−1(K) // Hn−1(E) // Hn−1(Q)

where the maps are given as described in proposition 2.10. Pasting together theexact sequences thus obtained yields the assertion.Another result which will be used in many situations is the five lemma.

Lemma 2.13 (Five lemma). Consider a diagram of A-modules of the form

M1//

∼=

M2//

∼=

M3//

f

M4//

∼=

M5

∼=

N1// N2

// N3// N4

// N5

and assume that both rows are exact. Then f is an isomorphism.

Proof. This is a diagram chase best done visually. Let us only show that f isinjective. Assume that f(x) = 0. Then the image of x in M4 is zero. Hence thereexists x2 in M2 which maps to x. The image of x2 in N2 comes from an elementy1 of N1. Hence there exists x1 ∈ M1 such that the image of x1 in M2 is equal tox2. It follows that the image of x2 in M3 is zero. Hence x is zero.


3. Projective resolutions and derived functors

In this section study projective resolutions of modules and the derived functorof the tensor product functor.

Definition 2.14. Let A be an algebra and let M be an A-module. A projectiveresolution P of M consists of a long exact sequence

M oo εP0oo P1

oo P2oo P3

oo · · ·

of A-modules such that all Pj are projective.

Our first aim is to show that every module has a projective resolution.

Lemma 2.15. Let M be an A-module. Then there exists a projective resolutionfor M .

Proof. Let P0 = AM be the free module over the set M and let ε : P0 → Mbe the natural A-module map. By construction, ε is surjective and P0 is free,hence projective. Now let P1 = A ker(ε) be the free module over the set ker(ε)and let d1 : P1 → P0 be the natural map. Then im(d1) = ker(ε) and P1 is againprojective. We may next consider ker(d1) and continue in this way to obtain aprojective resolution of M by free A-modules. In many cases one may construct smaller projective resolutions of a module. If Pis already a projective module then the most evident projective resolution of P isgiven by P0 = 0, Pj = 0 for j > 0 and ε = id.For the general theory it is important that projective resolutions may be compared.

Proposition 2.16. Let M and N be A-modules and let P and Q be projectiveresolutions of M and N , respectively. If f : M → N is an A-module homomorphismthere exist A-module homomorphisms fj : Pj → Qj for all j such that the diagram

M oo ε

f

P0oo

f0

P1oo

f1

P2oo

f2

P3oo

f3

· · ·

N oo εQ0oo Q1

oo Q2oo Q3

oo · · ·

is commutative. Moreover, if (gj)j≥0 is another family of such homomorphisms,then the chain maps f and g thus defined are homotopic.

Proof. The maps fj are constructed inductively. Since ε : Q0 → N is surjec-tive and P0 is projective, there exists an A-module map f0 : P0 → Q0 such thatfε = εf0. Now assume that fj has been constructed. Consider the diagram

· · · oo Pj−1oo dj

fj−1

Pj oodj+1

fj

Pj+1oo · · ·

· · · oo Qj−1oo dj Qj oo

dj+1

Qj+1oo · · ·

The image of the map fjdj+1 is contained in ker(dj) = im(dj+1). Again, sincedj+1 : Qj+1 → im(dj+1) is surjective and Pj+1 is projective, there exists fj+1 suchthat dj+1fj+1 = fjdj+1.To prove the second assertion, it suffices to consider the case f = 0. We have to showthat any family of maps fj as above is homotopic to zero. Again, the contractinghomotopy h will be constructed inductively. We define h−1 = 0 : M → Q0.Since f is a chain map, the image of f0 is contained in ker(ε) = im(d1). Byprojectivity of P0, there exists h0 : P0 → Q1 such that d1h0 = f0. Now assume

3. PROJECTIVE RESOLUTIONS AND DERIVED FUNCTORS 23

that hj : Pj → Qj+1 has been constructed such that dj+1hj + hj−1dj = fj . Thenfj+1 − hjdj+1 maps into ker(dj+1) = im(dj+2) since

dj+1fj+1 − dj+1hjdj+1 = dj+1fj+1 + hj−1djdj+1 − fjdj+1 = 0.

By projectivity of Pj+1, there exists hj+1 : Pj+1 → Qj+2 such that dj+2hj+1 =fj+1 − hjdj+1. This yields the claim. As a consequence, one has the following result.

Excercise 2.17. Two projective resolutions of a module M are homotopy equiv-alent.

We will now define the derived functor of the tensor product.

Definition 2.18. Let MA and AN be modules over an algebra A and choosea projective resolution P of AN . Then

TorAn (M,N) = Hn(M ⊗A P ).

Using excercise 4.26 we see that, up to natural isomorphism, the definition ofTor(M,N) is independent of the resolution P .

Excercise 2.19. Let AN be a module and let

0 // K // E // Q // 0

be a short exact sequence of left A-modules. Then the induced sequence of vectorspaces

0 // HomA(N,K) // HomA(N,E) // HomA(N,Q)

is exact.If NA is projective then in addition the map HomA(N,E) → HomA(N,Q) is sur-jective.

Proposition 2.20. Let AN be a module and let

0 // K // E // Q // 0

be a short exact sequence of right A-modules. Then the induced sequence

K ⊗A N // E ⊗A N // Q⊗A N // 0

is exact.

Proof. According to proposition 1.26 we have

Hom(M ⊗A N,V ) ∼= HomA(N,Hom(M,V ))

for every module MA and every vector space V . According to excercise 2.19 thisimplies that the sequence

0 // Hom(Q⊗A N,V ) // Hom(E ⊗A N,V ) // Hom(K ⊗A N,V )

is exact for every vector space V . Since we are now only dealing with vector spacesit follows that already the sequence

K ⊗A N // E ⊗A N // Q⊗A N // 0

is exact. This yields the claim. As a consequence of proposition 2.20 one obtains in particular the following state-ment.

Excercise 2.21. Let MA and AN be modules over an algebra A. Then

TorA0 (M,N) ∼= M ⊗A N


All constructions in this section can as well be carried out for unitary modulesover unital algebras. This is in fact the standard way to define derived functors ofmodules. We could thus give the following definition. Let MA and AN be unitarymodules over a unital algebra A and choose a projective resolution P of AN ofunitary A-modules. Then the unitary derived functor of the tensor product is

torAn (M,N) = Hn(M ⊗A P ).

In the same way as above one may prove that this does not depend on the choiceof P up to isomorphism. However, using proposition 1.32 one obtains even thefollowing statement.

Excercise 2.22. Let A be a unital algebra and let M and N be unitary A-modules. Then there is a natural isomorphism

TorAn (M,N) ∼= torAn (M,N)

for all n.

Hence it makes essentially no difference if we work with unital algebras andunitary modules or with arbitrary algebras and arbitrary modules.

4. Inductive and projective limits of chain complexes

In this section we study the homology of inductive and projective limits of chaincomplexes.Let J be a partially ordered set and let (Cj)j∈J be an inductive system of chaincomplexes. That is, we are given chain complexes Cj and a compatible family ofchain maps fji : Ci → Cj for all i ≤ j. Then we may form the inductive limit

C = lim−→j∈J

Cj

by letting Cn = lim−→j∈J Cjn be the inductive limits in each degree. It is straightfoward

to check that the inductive limit C is again a chain complex. As a special case onemay consider direct sums.

Excercise 2.23. Let (Cj)j∈J be a family of chain complexes. Then the naturalmap ⊕

j∈JH∗(Cj)→ H∗

(⊕j∈J

Cj)

is an isomorphism.

Recall that a partially ordered set J is directed if for every i, j ∈ J there existsk ∈ J such that i ≤ k and j ≤ k.

Lemma 2.24. Let (Cj)j∈J be an inductive system of chain complexes over adirected set J . Then the natural map

lim−→j∈J

H∗(Cj)→ H∗

(lim−→j∈J

Cj)

is an isomorphism.

Proof. Let C be the inductive limit of the complexes Cj . There is a compat-ible family of chain maps ιj : Cj → C and hence an induced map ι : lim−→H∗(C

j)→H∗(C). Let us show that this map is injective and surjective. If c ∈ C is acycle there exists j ∈ J such that c = ιj(cj) for cj ∈ Cj . Moreover dc = 0implies ιkj(dcj) = 0 ∈ Ck for some k ≥ j. Hence ιkj(cj) ∈ Ck is a cycle andι([ιkj(cj)]) = [c]. This show that ι is surjective. If ι([cj ]) = 0 in H∗(C) there existsb ∈ Ck and l ∈ J such that ιlj(cj) = ιlk(db) = dιlkb in Cl. Here we use the fact

4. INDUCTIVE AND PROJECTIVE LIMITS OF CHAIN COMPLEXES 25

that J is directed. It follows that ιlj([cj ]) = 0 in H∗(Cl) and hence the class [cj ] in

lim−→H(Cj) is zero. This shows that ι is injective. For projective limits one start with the dual definitions. Let J be a partially or-dered set and let (Cj)j∈J be a projective system of chain complexes. That is, weare given chain complexes Cj and a compatible family of chain maps fji : Ci → Cj

for all j ≤ i. We form the projective limit

C = lim←−j∈J

Cj

componentwise and obtain a chain complex C. Let us consider the special case ofdirect products.

Excercise 2.25. Let (Cj)j∈J be a family of chain complexes. Then the naturalmap ∏

j∈JH∗(Cj)→ H∗

( ∏j∈J

Cj)

is an isomorphism.

The case of general projective limits is more complicated. We consider only thespecial case where the index set J is the set of natural numbers with the canonicalordering.First we have to explain what lim←−

1

j∈NMj for a projective system (M j)j∈N of modules

or chain complexes over N is. Consider the map σ :∏j∈NM

j →∏j∈NM

j given by

σ((xj)j∈N) = (πj,j+1(xj+1))j∈N.

The kernel of id−σ can be identified with lim←−j∈NMj . By definition lim←−

1

j∈NMj is

the cokernel of id−σ. Hence we have a short exact sequence

lim←−j∈NMj // // ∏

j∈NMj id−σ // ∏

j∈NMj // // lim←−

1

j∈NMj .

In favorable circumstances, the term lim←−1

j∈NMj vanishes.

Lemma 2.26. Let (M j)j∈N be an inverse system with surjective structure maps.

Then lim←−1

j∈NMj = 0.

Proof. If all structure maps are surjective, the map id−σ is surjective as well.Hence lim←−

1

j∈NMj = 0 by the definition of lim←−

1

j∈N.

In the hypothesis of the previous lemma the condition of having surjective structuremaps can be relaxed. One says that the projective system (Mj)j∈N satisfies theMittag-Leffler condition if for all j ∈ N there exists k such that the images of themaps πjl : M l →M j are equal for all l ≥ k. We remark that the assertion of lemma2.26 remains true for projective systems satisfying the Mittag-Leffler condition.

Proposition 2.27. Let (Cn)n∈N be a projective system of chain complexeswith surjective structure maps. If we denote by C the projective limit of the system(Cn)n∈N, there is a short exact sequence

lim←−1

j∈NHn+1(Cj) // // Hn(C) // // lim←−j∈NHn(Cj)

for each n.

Proof. We have an exact sequence

lim←−j∈N Cj // // ∏

j∈N Cj id−σ // ∏

j∈N Cj // // lim←−

1

j∈N Cj


of chain complexes. According to the assumption, the term lim←−1

j∈N Cj is zero and

we get a short exact sequence of chain complexes

lim←−j∈N Cj // // ∏

j∈N Cj id−σ// // ∏

j∈N Cj

which induces a long exact sequence

· · · // Hn(C) // ∏j∈NHn(Cj)

id−σ // ∏j∈NHn(Cj) // · · ·

in homology. Here we use excercise 2.25 to describe the homology of the directproduct complexes. By definition, lim←−

1

j∈NHn+1(Cj) is the cokernel of the map

id−σ at the (n + 1)th stage in this sequence. It maps injectively into Hn(C) bythe boundary map. Moreover, the kernel of id−σ at the nth stage is equal tolim←−j∈NHn(Cj). By exactness, this yields the assertion.

5. Presimplicial modules

For the description of Hochschild homology it is convenient to use the follow-ing concept. Historically, it originates from the study of simplicial homology andsingular homology.

Definition 2.28. A presimplicial module C is a sequence of vector spaces Cnfor n ≥ 0 together with maps

dj : Cn → Cn−1 for j = 0, . . . , n

called face maps such that

didj = dj−1di for 0 ≤ i < j ≤ nand all n. A presimplicial map f : C → D between presimplicial modules is a familyof linear maps Cn → Dn such that dif = fdi for all face maps di.

Of course, one might as well consider A-modules over an algebra together withmodule homomorphisms satisfying the above relations. We will not need this,though.Historically, the following observation was one of the starting points of homologicalalgebra.

Excercise 2.29. Let C be a presimplicial module. Then C becomes a complexwith boundary operators d : Cn → Cn−1 given by

d =

n∑j=0

(−1)jdj .

Observe that every map f : C → D of presimplicial modules induces a chainmap between the associated complexes.

Definition 2.30. Let f, g : C → D be presimplicial maps. A presimplicialhomotopy between f and g is a family of linear maps hj : Cn → Dn+1 for j =0, . . . , n such that d0h0 = f and dn+1hn = g while

dihj =

hj−1di, 0 ≤ i < j ≤ ndihi−1, 0 < i = j ≤ nhjdi−1, 1 ≤ j + 1 < i ≤ n+ 1

for all n.

We write f ∼ g if two maps f and g are connected by a presimplicial homo-topy. The following excercise shows in particular that presimplicial homotopy is anequivalence relation.

5. PRESIMPLICIAL MODULES 27

Excercise 2.31. Consider presimplicial maps from C to D. Then

a) f ∼ f .b) f ∼ g implies g ∼ f and −f ∼ −g.c) f ∼ g and g ∼ h implies f ∼ h.c) f1 ∼ g1 and f2 ∼ g2 implies (f1 + f2) ∼ (g1 + g2).

Excercise 2.32. Let f, g : C → D be maps of presimplicial modules whichare connected by a presimplicial homotopy. Then the associated chain maps arehomotopic.

Let us remark that there exists also the notion of a simplicial module. It isobtained from the definition of a presimplicial module by requiring in addition theexistence of certain degeneracy maps satisfying some conditions. For the applica-tions we have in mind, such degeneracy maps exist only if we work with unitalalgebras. Since we do not want to restrict attention to unital algebras it is morenatural to consider presimplicial modules.

CHAPTER 3

Hochschild homology and cyclic homology

In this chapter we define Hochschild homology and cyclic homology and studytheir basic properties. There are different approaches to these theories, and eachof these approaches has its particular virtues. We will begin with the definitionusing the mixed complex of noncommutative differential forms and then deduce thedescription of Hochschild homology for unital algebras as a derived functor. Wealso define cyclic homology based on mixed complexes. After having treated theSBI-sequence we will introduce Hochschild cohomology and cyclic cohomology. Wediscuss periodic cyclic homology and cohomology and the relation of these theoriesto ordinary cyclic homology and cohomology.

1. Noncommutative differential forms

In this section we define and study a noncommutative replacement of the alge-bra of differential forms A(M) on a smooth manifold M .

Definition 3.1. Let A be an algebra. For n > 0 we let Ωn(A) = A+ ⊗A⊗n bethe space of noncommutative n-forms over A. In addition we set Ω0(A) = A.

Here we have used the notation

A⊗n = A⊗A⊗ · · · ⊗Ato denote the tensor product of n copies of A. Elements in Ωn(A) are written inthe suggestive form a0da1 · · · dan for a0 ∈ A+ and a1, . . . , an in A. We also writeda1 · · · dan if a0 = 1 ∈ A+.Let us first consider the case n = 1. We define a left A-module structure on Ω1(A)by setting

a(a0da1) = aa0da1.

A right A-module structure on Ω1(A) is defined according to the Leibniz ruled(ab) = dab+ adb by

(a0da1)a = a0d(a1a)− a0a1da.

Excercise 3.2. Verify that Ω1(A) becomes an A-A-bimodule in this way.

The next statement shows that higher differential forms may be constructedout of the bimodule Ω1(A). We will use the notation

Ω1(A)⊗An = Ω1(A)⊗A Ω1(A)⊗A · · · ⊗A Ω1(A)

for the tensor product of n copies of Ω1(A) over A.

Excercise 3.3. There is a natural isomorphism

Ωn(A) ∼= Ω1(A)⊗A Ω1(A)⊗A · · · ⊗A Ω1(A) = Ω1(A)⊗An

for every n ≥ 1.

As a consequence, the spaces Ωn(A) are equipped with an A-A-bimodule struc-ture in a natural way. Explicitly, the left A-module structure on Ωn(A) is givenby

a(a0da1 · · · dan) = aa0da1 · · · dan

29

30 3. HOCHSCHILD HOMOLOGY AND CYCLIC HOMOLOGY

and the right A-module structure may be written as

(a0da1 · · · dan)a = a0da1 · · · dan−1d(ana)

+

n−1∑j=1

(−1)n−ja0da1 · · · d(ajaj+1) · · · danda+ (−1)na0a1da2 · · · danda.

Moreover we view Ω0(A) = A as an A-bimodule in the obvious way using themultiplication in A.According to excercise 3.3 one may define a map Ωn(A) ⊗ Ωm(A) → Ωn+m(A) byconsidering the natural projection

Ω1(A)⊗An ⊗ Ω1(A)⊗Am → Ω1(A)⊗An ⊗A Ω1(A)⊗Am = Ω1(A)⊗A(m+n).

Let us denote by Ω(A) the direct sum of the spaces Ωn(A) for n ≥ 0. Then themaps Ωn(A)⊗ Ωm(A)→ Ωn+m(A) assemble to a map Ω(A)⊗ Ω(A)→ Ω(A).

Excercise 3.4. In this way the space Ω(A) becomes an algebra. Actually,Ω(A) is a graded algebra if one considers the natural grading given by the degree ofa differential form.

Let us now define a linear operator d : Ωn(A)→ Ωn+1(A) by

d(a0da1 · · · dan) = da0 · · · dan, d(da1 · · · dan) = 0

for a0, . . . , an ∈ A. It follows immediately from the definition that d2 = 0.A differential form in Ω(A) is called homogenous of degree n if it is contained inthe subspace Ωn(A).

Excercise 3.5. The graded Leibniz rule

d(ωη) = dωη + (−1)|ω|ωdη

holds on Ω(A) for homogenous forms ω and η.

Hence the operator d has similar properties like the exterior differential onordinary differential forms. Actually one might think of this operator as an analogueof the exterior differential.At this point it would be tempting to define the de Rham homology of an algebraA to be the homology of Ω(A) with respect to the differential d. However, it iseasy to check that Ω(A) is contractible with respect to this boundary operator. Acontracting homotopy h : Ωn(A)→ Ωn−1(A) is given by

h(a0da1 · · · dan) = 0, h(da1 · · · dan) = a1da2 · · · dan

for a0, a1 . . . , an ∈ A. Hence we do not obtain any interesting information in thisway. Instead we have to consider more interesting boundary operators.We define a linear operator b : Ωn(A)→ Ωn−1(A) by

b(a0da1 · · · dan) = (−1)n−1(a0da1 · · · dan−1an − ana0da1 · · · dan−1)

= (−1)n−1[a0da1 · · · dan−1, an]

for a0 ∈ A+ and a1 · · · an ∈ A. Here [x, y] = xy − yx denotes the ordinary com-mutator for elements in the algebra Ω(A). The operator b is called the Hochschildoperator. Using the explicit formula for the right A-module structure of Ω(A) weobtain

b(a0da1 · · · dan) = a0a1da2 · · · dan

+

n−1∑j=1

(−1)ja0da1 · · · d(ajaj+1) · · · dan + (−1)nana0da1 · · · dan−1.

1. NONCOMMUTATIVE DIFFERENTIAL FORMS 31

which is closer to the traditional form of the definition of the Hochschild boundarymap.

Lemma 3.6. The Hochschild operator b satisfies b2 = 0.

Proof. We compute for ω = a0da1 · · · dan and x, y ∈ Ab2(ωdxdy) = b((−1)n+1(ωdxy − yωdx)) = b((−1)n+1(ωd(xy)− ωxdy − yωdx))

= (−1)n(−1)n+1(ωxy − xyω − (ωxy − yωx)− (yωx− xyω)) = 0

which yields the claim. We proceed to construct more operators as follows. The Karoubi operator κ :Ωn(A)→ Ωn(A) is given by

κ = id−(bd+ db)

and the Connes operator B : Ωn(A)→ Ωn+1(A) is defined by

B =

n∑j=0

κjd.

Using d2 = 0 we obtain κd = dκ. Moreover this implies immediately B2 = 0. Letus record explicit formulas for the operators κ and B on Ωn(A). Clearly one hasκ(a) = a for a ∈ Ω0(A) = A.

Excercise 3.7. For all n > 0 one has

κ(a0da1 · · · dan) = (−1)n−1dana0da1 · · · dan−1

on Ωn(A).

For the Connes operator we compute

B(a0da1 · · · dan) =

n∑i=0

(−1)nidan+1−i · · · danda0 · · · dan−i

using excercise 3.7. We need the following lemma concerning relations between theoperators constructed above.

Lemma 3.8. On Ωn(A) the following relations hold:

a) κn+1d = db) κn = id +bκndc) bκn = bd) κn+1 = id−dbe) (κn+1 − id)(κn − id) = 0f) Bb+ bB = 0

Proof. a) follows directly from the explicit formula for κ obtained in excercise3.7. b) Using again the formula for κ we compute

κn(a0da1 · · · dan) = da1 · · · dana0

= a0da1 · · · dan + (−1)nb(da1 · · · danda0)

= a0da1 · · · dan + bκnd(a0da1 · · · dan).

c) follows by applying the Hochschild boundary b to both sides of b). d) Apply κto b) and use a). e) is a consequence of b) and d). f) We compute

Bb+bB =

n−1∑j=0

κjdb+

n∑j=0

bκjd =

n−1∑j=0

κj(db+ bd) + κnbd

= id−κn + κnbd = id−κn(id−bd) = id−κn(κ+ db) = 0

where we use d) and b). We can rephrase parts of this discussion using the following definition.


Definition 3.9. A mixed complex M is a sequence of vector spaces Mn togetherwith differentials b of degree −1 and B of degree +1 satisfying b2 = 0, B2 = 0 and

[b, B] = bB +Bb = 0.

on Mn for all n.

Proposition 3.10. Let A be an algebra. The space Ω(A) of noncommutativedifferential forms together with the operators b and B is a mixed complex.

2. Hochschild homology

In this section we define and study the Hochschild homology of an algebra.

Definition 3.11. Let A be an algebra. The Hochschild homology of A is thehomology of Ω(A) with respect to the Hochschild boundary b. We denote by HHn(A)the n-th Hochschild homology group of A.

Let us identify the homology group HH0(A).

Lemma 3.12. Let A be an algebra. Then HH0(A) is the quotient A/[A,A] ofA by the linear span of all commutators.

Proof. The image of the Hochschild boundary b : Ω1(A) → A is equal to[A,A] since b(a0da1) = a0a1 − a1a0. As a consequence we obtain immediately

Corollary 3.13. Let A be a commutative algebra. Then HH0(A) = A.

Let A be an arbitrary algebra and consider the direct sum decomposition

Ωn(A) = A+ ⊗A⊗n = (A⊕ C)⊗A⊗n = A⊗n+1 ⊕A⊗n.for n > 0.

Excercise 3.14. Using this decomposition the Hochschild complex Ω(A) of Acan be identified with the total complex of the bicomplex

......

? ?A⊗3 A⊗31− t

?

b

?

−b′

A A1− t

A⊗2 A⊗21− t

?

b

?

−b′

where the vertical operators are defined by

b(a0 ⊗ a1 ⊗ · · ·⊗an) =

n−1∑j=0

(−1)ja0 ⊗ · · · ⊗ ajaj+1 ⊗ · · · ⊗ an

+ (−1)nana0 ⊗ a1 ⊗ · · · ⊗ an−1

and

b′(a0 ⊗ a1 ⊗ · · · ⊗ an) =

n−1∑j=0

(−1)ja0 ⊗ · · · ⊗ ajaj+1 ⊗ · · · ⊗ an.

2. HOCHSCHILD HOMOLOGY 33

The horizontal operator is constructed using the map t given by

t(a0 ⊗ a1 ⊗ · · · ⊗ an) = (−1)nan ⊗ a0 ⊗ · · · ⊗ an−1.

We denote by C(A) the complex given by the first column of this bicomplex.That is, Cn(A) = A⊗n+1 with boundary operator b. This complex will be calledthe unital Hochschild complex of A. Remark that we use the letter b both for theboundary operator in Ω(A) and the boundary operator in C(A). However, thisshould not lead to confusion - the operator b on C(A) is just the restriction of theoperator b on Ω(A) to the first column.The second column of the above bicomplex is denoted by Bar(A) and called theBar-complex of A. We have Barn(A) = A⊗n with boundary operator −b′. Byconstruction, we have a short exact sequence

C(A) // // Ω(A) // // Bar(A)

of complexes. Using this exact sequence we shall now obtain a different descriptionof Hochschild homology in the case that the algebra A is unital.

Excercise 3.15. If A is a unital algebra the Bar-complex Bar(A) is contractibleusing the contracting homotopy s : Barn(A)→ Barn+1(A) given by

s(a0 ⊗ a1 ⊗ · · · ⊗ an+1) = 1⊗ a0 ⊗ a1 ⊗ · · · ⊗ an+1.

Proposition 3.16. Let A be a unital algebra. Then the inclusion of the unitalHochschild complex C(A) into Ω(A) is a homotopy equivalence.

Proof. Define a map ρ : Ω(A)→ C(A) by

ρ(a0da1 · · · dan) = a0 ⊗ a1 ⊗ · · · ⊗ an, ρ(da1 · · · dan) = −(1− t)s(a1 ⊗ · · · ⊗ an)

in degree n. Then one has

bρ(a0da1 · · · dan) = ρb(a0da1 · · · dan)

for a0, a1, . . . , an ∈ A and and

bρ(da1 · · · dan) = −b(1− t)s(a1 ⊗ · · · ⊗ an) = (1− t)b′s(a1 ⊗ · · · ⊗ an)

= (1− t)(1− sb′)(a1 ⊗ · · · ⊗ an) = ρb(da1 · · · dan)

which shows that ρ is a chain map. If ι : C(A) → Ω(A) denotes the canonicalinclusion then ρι = id. Moreover ιρ is homotopic to the identity using the homotopygiven by s on Bar(A) and by 0 on C(A). Hence we obtain

Proposition 3.17. Let A be a unital algebra. Then the Hochschild homology ofA is naturally isomorphic to the homology of the unital Hochschild complex C(A).

Recall that the opposite algebra Aop of A has the same underlying vector spaceand the opposite multiplication a • b = ba. Let us form the tensor product algebraAe = A ⊗ Aop. If A is unital then Ae is again unital and called the extendedalgebra of A. Note that there is a bijective correspondence between unitary (left)Ae-modules and unitary A-A-bimodules.Let us define an A-bimodule structure on Barn(A) by the formula

a(a0 ⊗ a1 ⊗ · · · ⊗ an+1)b = aa0 ⊗ a1 ⊗ · · · ⊗ an+1b.

If A is unital the complex Bar(A) consists of projective unitary Ae-modules anddue to excercise 3.15 we obtain the following statement.

Lemma 3.18. Let A be a unital algebra. Then Bar(A) is a projective resolutionof the A-bimodule A.


Excercise 3.19. Let A be unital. The map φ : C(A) → A ⊗Ae Bar(A) givenby

φ(a0 ⊗ a1 ⊗ · · · ⊗ an) = a0 ⊗ 1⊗ a1 ⊗ · · · ⊗ an ⊗ 1

is an isomorphism of chain complexes.

According to the definition of Tor in section 3 we now deduce the followingresult.

Proposition 3.20. Let A be a unital algebra. Then there is a natural isomor-phism

HHn(A) ∼= TorAe

n (A,A)

for all n.

This result is important since it allows to compute the Hochschild homology ofa unital algebra using arbitrary projective resolutions of the A-bimodule A. Oftenthe computation of Hochschild homology groups relies on finding particularly nicesuch resolutions. As a very simple example we will illustrate this in calculating theHochschild homology of the complex numbers.

Lemma 3.21. The Hochschild homology of C is given by

HH0(C) = C

and HHn(C) = 0 for n > 0.

Proof. Observe that we have an algebra isomorphism Ce ∼= C. Hence theCe-module C is projective. It follows that there exists a projective resolution oflength 0 for C. As a consequence HHn(C) = 0 for n > 0. We have HH0(C) = Csince C is commutative. We will see below a more interesting example of a computation based on a specificprojective resolution. Before we proceed we remark that the definition in the non-unital case is made in such a way that the following result holds.

Lemma 3.22. Let A be an algebra. Then there is a split short exact sequence

HHn(A) // // HHn(A+) // // HHn(C)

for all n.

Proof. Consider the normalized bar complex of A+ defined by

barn(A) = A+ ⊗A⊗n ⊗A+

for n ≥ 0. Then, as above, bar(A) is a projective resolution of A+ by unitary A+-bimodules. SinceA+ is unital, the Hochschild homologyHH(A+) may be computedby A+⊗(A+)e bar(A). A straightforward calculation shows A+⊗(A+)e bar0(A) = A+

and

A+ ⊗(A+)e barn(A) ∼= Ω(A)

for n > 0. Moreover the differential is precisely the Hochschild boundary of Ω(A)under this identification. It follows that the natural projection A+ → C inducesisomorphisms

HHn(A) ∼= HHn(A+)

for n > 0 and HH0(A+) = HH0(A)⊕ C = HH0(A)⊕HH0(C). In the remaining part of this section we shall consider tensor algebras. Let V be avector space. The tensor algebra TV over V is defined by

TV =

∞⊕j=1

V ⊗j

2. HOCHSCHILD HOMOLOGY 35

with multiplication given by concatenation of tensors. Here one uses the canonicalisomorphisms V ⊗n ⊗ V ⊗m ∼= V ⊗n+m for all n,m ∈ N. There is an obvious mapι : V → TV given by inclusion of tensors of length one. Remark that, according toour definition, the tensor algebra TV does not possess a unit element. The tensoralgebra satisfies the following universal property.

Excercise 3.23. Let TV be the tensor algebra over a vector space V . For everyalgebra A and any linear map f : V → A there exists a unique homomorphismF : TV → A such that the diagram

V TV-ι

f@@@@R

A?

F

is commutative.

We shall now calculate the Hochschild homology of the unitarized tensor algebra(TV )+. Define P0 = (TV )+ ⊗ (TV )+ and P1 = (TV )+ ⊗ V ⊗ (TV )+ and considerthe complex

(TV )+ oo µP0oo d

P1oo 0

where µ denotes the multiplication map and d is defined by

d(x⊗ v ⊗ y) = (x⊗ v)⊗ y − x⊗ (v ⊗ y).

In addition we set P−1 = (TV )+. Evidently, the maps µ and d are (TV )+-(TV )+-bimodule homomorphisms. Let us show that this complex is exact. We define amap s−1 : (TV )+ → P0 by s−1(x) = x ⊗ 1. Moreover we define s0 : P0 → P1 bys0(x⊗ 1) = 0 and

s0(x⊗ v1 ⊗ · · · ⊗ vn) = −(x⊗ v1 ⊗ · · · ⊗ vn−1)⊗ vn ⊗ 1

−n−1∑j=2

(x⊗ v1 ⊗ · · · ⊗ vj−1)⊗ vj ⊗ (vj+1 · · · ⊗ vn)− x⊗ v1 ⊗ (v2 ⊗ · · · ⊗ vn)

for v1 ⊗ · · · ⊗ vn ∈ V ⊗n ⊂ TV for n > 0. Clearly one has µs−1 = id.

Excercise 3.24. Verify the relations s−1µ+ ds0 = id and s0d = id.

It follows that P defines a projective resolution of length 1 of the bimodule(TV )+. Algebras A allowing for resolutions of length ≤ 1 of A+ by projectiveunitary A+-A+-bimodules are called quasifree. Hence TV is a quasifree algebra.According to the general theory, the complexes Bar((TV )+) and P are homotopyequivalent. Let us explicitly write down a homotopy equivalence f : Bar((TV )+)→P . We let f0 : Bar0((TV )+) = (TV )+⊗(TV )+ → P0 be the identity map. In degreeone we define

f1(x⊗ (v1 ⊗ · · · ⊗ vn)⊗ y) =

n∑j=1

(x⊗ v1 ⊗ · · · ⊗ vj−1)⊗ vj ⊗ (vj+1 ⊗ · · · ⊗ y).

Let us also define g : P → Bar((TV )+) by g0 = id and g1 : (TV )+ ⊗ V ⊗ (TV )+ →(TV )+ ⊗ (TV )+ ⊗ (TV )+ by

g1(x⊗ v ⊗ y) = x⊗ v ⊗ y.

Clearly one has fg = id.

Excercise 3.25. The maps f and g are chain maps.


Since these maps cover the identity in degree −1 it follows already by thegeneral theory that f and g are inverse homotopy equivalences.

Excercise 3.26. Construct explicitly a homotopy between gf and the identitymap on Bar((TV )+).

According to proposition 3.20 we may compute the Hochschild homology of(TV )+ using the resolution P . Tensoring this resolution over the extended algebra((TV )+)e with (TV )+ it follows that the Hochschild homology of (TV )+ is thehomology of the complex

0 oo (TV )+ oo (TV )+ ⊗ V oo 0

where the boundary maps an element x⊗ v to x⊗ v − v ⊗ x.Let us denote by τ : TV → TV the linear map given by τ(v1 ⊗ · · · ⊗ vn) =vn ⊗ v1 ⊗ · · · ⊗ vn−1. We denote by (TV )τ the space of elements fixed by τ and let(TV )τ be the quotient of TV by all elements x − τ(x) with x ∈ TV . With thesedefinitions we obtain immediately the following result.

Proposition 3.27. The Hochschild homology of (TV )+ is given by

HH0((TV )+) = C⊕ (TV )τ

HH1((TV )+) = (TV )τ

HHn((TV )+) = 0 for n > 1.

Under this identification the copy of C in degree zero corresponds to multiplesof the unit element of (TV )+. Using lemma 3.22 we obtain the following result.

Proposition 3.28. The Hochschild homology of the tensor algebra TV is givenby

HH0(TV ) = (TV )τ

HH1(TV ) = (TV )τ

HHn(TV ) = 0 for n > 1.

3. Cyclic homology

In this section we define cyclic homology and study some of its basic properties.Let A be an algebra. According to proposition 3.10 we can form the bicomplex

......

? ?Ω3(A) Ω2(A)B

?

b

?

b

Ω2(A) Ω1(A)B

?

b

?

b

Ω0(A)

Ω1(A) Ω0(A)B

?

b

Ω0(A)B

?

Ω1(A)B

? ?

b

Ω0(A)B

......

? ?

3. CYCLIC HOMOLOGY 37

which is by definition the (B, b)-bicomplex of A.

Definition 3.29. Let A be an algebra. The cyclic homology of A is the homol-ogy of the total complex of the (B, b)-bicomplex of A. We denote by HCn(A) then-th cyclic homology group of A.

Remark that we do not have to specify wether we use direct products or directsums to define the total complex since the (B, b)-bicomplex is located in the firstquadrant.It is easy to describe the cyclic homology group HC0(A).

Lemma 3.30. Let A be an algebra. Then HC0(A) = HH0(A) is equal toA/[A,A].

Proof. This follows immediately by an inspection of the (B, b)-bicomplex. Observe that the first column of the (B, b)-bicomplex is precisely the Hochschildcomplex of A. Moreover, the quotient of the (B, b)-bicomplex by the first column isnaturally isomorphic to another copy of the (B, b)-bicomplex. Taking into accountthe corresponding degree shifts on the total complexes, proposition 2.10 immedi-ately implies the following result.

Proposition 3.31. For every algebra A there is a natural long exact sequence

· · · // HHn(A)I // HCn(A)

S // HCn−2(Q)B // HHn−1(A) // · · ·

This long exact sequence is called the SBI-sequence. The SBI-sequence is animportant tool to compute cyclic homology groups.It is often useful to have some information on the boundary map in the SBI-sequence. Actually, the boundary map B : HCn(A)→ HHn+1(A) is closely relatedto the operator B on differential forms.

Lemma 3.32. Let A be an algebra. The map B : HCn(A) → HHn+1(A) isinduced by the map B : Ωn(A)→ Ωn+1(A).

Proof. Consider a cycle z = (zn−2j)j≥0 of degree n in the total complex ofthe cyclic bicomplex where zk ∈ Ωk(A). We may lift z to a cycle of dimension n+2by adding 0 in Ωn+2(A). Then, by definition of the boundary map B we obtain thecycle B(zn) ∈ Ωn+1(A) representing a Hochschild homology class of degree n + 1.This proves the claim. It is clear that the definition of Hochschild homology and cyclic homology for al-gebras can be extended to arbitrary mixed complexes. We write HHn(M) andHCn(M) for the Hochschild and cyclic homology of a mixed complex M . Asabove, these theories are related by an SBI-sequence. One says that a map ofmixed complexes f : M → N induces an isomorphism in Hochschild homology ifHHn(f) : HHn(M)→ HHn(N) is an isomorphism for all n. The same terminologyis used for cyclic homology.

Lemma 3.33. Let f : M → N be a map of mixed complexes. Then f inducesan isomorphism in Hochschild homology iff it induces an isomorphism in cyclichomology.

Proof. This is a consequence of the five lemma 2.13. Remark that in theSBI-sequence there are much more entries containing cyclic homology groups thenentries with Hochschild homology. However, consider the last part of the SBI-sequence

// HH1(M) //

HC1(M) //

0 // HH0(M) //

HC0(M) //

0

// HH1(N) // HC1(N) // 0 // HH0(N) // HC0(N) // 0


for M and N . If f induces an isomorphism in Hochschild homology it follows that falso induces an isomorphism on HC0 and HC1. Inductively, we see that f inducesan isomorphism on HCn for all n. In particular we obtain the following result.

Corollary 3.34. Let f : A → B be an algebra homomorphism. Then finduces an isomorphism in Hochschild homology iff it induces an isomorphism incyclic homology.

Let us now apply the SBI-sequence to determine the cyclic homology of thecomplex numbers.

Proposition 3.35. The cyclic homology of the complex numbers is given by

HC2n(C) = C, HC2n+1(C) = 0

for all n.

Proof. Clearly HC0(C) = HH0(C) = C and exactness of the SBI-sequenceshows HC1(C) = 0. Moreover HHn(C) = 0 for n > 0 implies that S : HCn+2(C)→HCn(C) is an isomorphism for all n. This proves the claim. Note that the isomorphism C ∼= HC2n+2(C) → HC2n(C) ∼= C implemented by Sis the identity map under the above identifications.Let us also consider tensor algebras.

Proposition 3.36. Let V be a vector space. The cyclic homology of the tensoralgebra TV is given by

HC0(TV ) = (TV )τ , HCn(TV ) = 0

for all n > 0.

Proof. Clearly HC0(TV ) = HH0(TV ) = (TV )τ according to proposition3.28. Let us show that the boundary map B : HC0(TV ) → HH1(TV ) is anisomorphism. Due to lemma 3.32 this map is induced by d : TV → Ω1(TV ).Using the map ρ in proposition 3.16 the element d(v1 ⊗ · · · ⊗ vn) is mapped tov1 ⊗ · · · ⊗ vn ⊗ 1− 1⊗ v1 ⊗ · · · ⊗ vn in C1((TV )+). This, in turn, is mapped to

−n∑j=1

(vj+1 ⊗ · · · ⊗ vn ⊗ v1 ⊗ · · · ⊗ vj−1)⊗ vj

in (TV )+ ⊗ V under chain map C1((TV )+) → (TV )+ ⊗ V induced by the map fdefinded in the previous section. Hence, on homology the map B : (TV )τ → (TV )τ

is given by

B(v1 ⊗ · · · ⊗ vn) = −n∑j=1

vj+1 ⊗ · · · ⊗ vn ⊗ v1 ⊗ · · · ⊗ vj−1 ⊗ vj .

It is easy to see that this map is surjective. To check injectivity observe that

B(x) + nx = (τ − id)

n−1∑j=0

(j + 1)τ j(x)

for x ∈ TV homogenous of degree n.Now exactness of the SBI-sequence shows HC1(TV ) = 0 and HC2(TV ) = 0.Since HHn(TV ) = 0 for n > 1 it follows that S : HCn+2(TV ) → HCn(TV ) is anisomorphism for all n > 0. This proves the claim. We record the following statement concerning the behaviour of cyclic homologywith respect to unitarizations.

3. CYCLIC HOMOLOGY 39

Lemma 3.37. Let A be an algebra. Then there is a natural split short exactsequence

HCn(A) // // HCn(A+) // // HCn(C)

for all n. That is, there are natural isomorphisms HCn(A+) ∼= HCn(A) if n is oddand HCn(A+) ∼= HCn(A)⊕ C if n is even.

Proof. According to lemma 3.22 the natural homomorphisms A → A+ andC → A+ determine a map of mixed complexes Ω(A) ⊕ Ω(C) → Ω(A+) which in-duces an isomorphism in Hochschild homology. Hence this map of mixed complexesdetermines an isomorphism in cyclic homology as well according to lemma 3.33. In particular we obtain according to proposition 3.38 the following description ofthe cyclic homology for unitarized tensor algebras.

Proposition 3.38. Let V be a vector space. The cyclic homology of the uni-tarized tensor algebra (TV )+ is given by

HC0((TV )+) = C⊕ (TV )τ , HC2n((TV )+) = C, HC2n−1((TV )+) = 0

for all n > 0.

It is frequently useful to describe cyclic homology by other complexes. We shallbe interested in particular in the cyclic bicomplex. The cyclic bicomplex CC(A) ofan algebra A is given by

......

? ?A⊗3 A⊗31− t

?

b

?

−b′

A A1− t

A⊗2 A⊗21− t

?

b

?

−b′

A⊗3 A⊗3N

? ?

b

A AN

A⊗2 A⊗2N

? ?

b

......

? ?A⊗3 A⊗31− t

? ?

−b′

A A1− t

A⊗2 A⊗21− t

? ?

−b′

...

?A⊗3N

?

b

A AN

A⊗2 A⊗2N

?

b

· · ·

· · ·

· · ·

Here the operators b, b′ and t already have been defined in section 2. The operatorN : A⊗n+1 → A⊗n+1 is given by

N(a0 ⊗ a1 ⊗ · · · ⊗ an) =

n∑j=0

(−1)njan+1−j ⊗ · · · ⊗ an ⊗ a0 ⊗ · · · ⊗ an−j .

It is easy to check that N(id−t) = 0 and (id−t)N = 0.

Excercise 3.39. For all n the relation Nb = b′N holds on A⊗n+1.

Excercise 3.39 shows together with excercise 3.14 that the cyclic bicomplex isindeed a first quadrant bicomplex. Moreover, we have already seen in excercise3.14 that the total complex of the first two columns of this bicomplex is naturallyisomorphic to the Hochschild complex Ω(A) of A. We may use this observation toidentify the total complexes of the (B, b)-bicomplex and the cyclic bicomplex of A.

Excercise 3.40. Under this identification of the total complex of CC(A) withthe total complex of the (B, b)-bicomplex of A, the operator N is corresponds to theboundary operator B.


Hence we obtain the following statement.

Proposition 3.41. For every algebra A the cyclic homology HC∗(A) is equalto the homology of the total complex associated to CC(A).

Finally remark that the homology of the rows in the cyclic bicomplex vanishesexcept in degree zero. More precisely, we have im(id−t) = ker(N) and ker(id−t) =im(N). This is seen using the maps h0 : A⊗n+1 → A⊗n+1 and h1 : A⊗n+1 → A⊗n+1

given by

h0(x) =1

n+ 1x, h1(x) = − 1

n+ 1

n∑j=0

(j + 1)tj(x)

for n ≥ 0. We have Nh0(x) = x for x ∈ ker(id−t) and (id−t)h1(x) = x forx ∈ ker(N). In a slightly different form the latter relation already appeared in theproof of proposition 3.38.

4. Hochschild cohomology and cyclic cohomology

In algebraic topology one considers the (singular) homology of a topologicalspace as well as its cohomology. Singular cohomology is obtained by dualizing thechain complex defining singular homology. In a similar way there are dual theoriesto Hochschild homology and cyclic homology. These theories will be discussed inthis section.If V is a vector space we denote by V ′ = Hom(V,C) its dual space. If f : V → Wis a linear map then it induces a linear map W ′ → V ′ which will be denoted by f ′.Applying the dual space functor to the Hochschild complex Ω(A) of an algebra Awe obtain by definition the Hochschild cochain complex Ω(A)′.

Definition 3.42. Let A be an algebra. The Hochschild cohomology of A is thecohomology of the Hochschild cochain complex Ω(A)′. We denote by HHn(A) then-th Hochschild cohomology group of A.

It is easy to identify the cohomology group HH0(A).

Lemma 3.43. Let A be an algebra. Then HH0(A) is the linear space of traceson A.

Proof. The kernel of the Hochschild coboundary b : A′ → Ω1(A)′ is the spaceof all linear maps τ : A → C such that bτ(a0da1) = τ(a0a1) − τ(a1a0) = 0. Thismeans precisely that τ is a trace. In particular, if A is a commutative algebra we have HH0(A) = A′.Since C is a field the dual space functor is exact, that is, if

K // i // Ep // // Q

is an exact sequence of vector spaces then the induced sequence

Q′ //p′ // E′

i′ // // K ′

is again exact. This implies the following result.

Proposition 3.44. Let A be an algebra. The Hochschild cohomology groupHHn(A) is canonically isomorphic to HHn(A)′.

Proof. The assertion follows from the observation that the exact sequence

im(b) // // ker(b) // // HHn(A)

induces an exact sequence

HHn(A)′ // // ker(b)′ // // im(b)′

4. HOCHSCHILD COHOMOLOGY AND CYCLIC COHOMOLOGY 41

and the fact that ker(b) ∼= Ωn(A)′/ im(b′) and im(b)′ = Ωn(A)′/ ker(b′) where nowb′ : Ω(A)′ → Ω(A)′ denotes the transposed of the Hochschild boundary. It is easyto check that the isomorphism φ : HHn(A)→ HHn(A)′ arising in this way is givenexplicitly by by φ(f)(z) = f(z). Hence one may easily obtain a description of the Hochschild cohomology of analgebra as soon as its Hochschild homology is known.Let us now come to cyclic cohomology. In the same way as above we construct thedual of the (B, b)-bicomplex of an algebra A. Explicitly, we have

......

6 6

Ω3(A)′ Ω2(A)′-B

6b

6b

Ω2(A)′ Ω1(A)′-B

6b

6b

Ω0(A)′

Ω1(A)′ Ω0(A)′-B

6b

Ω0(A)′-B

6

Ω1(A)′-B

6 6b

Ω0(A)′-B

......

6 6

and this is again a bicomplex.

Definition 3.45. Let A be an algebra. The cyclic cohomology of A is thecohomology of the total complex of the dual (B, b)-bicomplex of A. We denote byHCn(A) the n-th cyclic cohomology group of A.

As for homology we have the following result for the group HC0(A).

Lemma 3.46. Let A be an algebra. Then HC0(A) = HH0(A) is the space oftraces on A.

The SBI-sequence relates Hochschild cohomology and cyclic cohomology.

Proposition 3.47. There is a long exact sequence

· · · oo HHn(A) ooI

HCn(A) ooS

HCn−2(Q) ooB

HHn−1(A) oo · · ·

for every algebra A.

Corollary 3.48. Let f : A → B be an algebra homomorphism. Then finduces an isomorphism in Hochschild cohomology iff it induces an isomorphism incyclic cohomology.


We may also consider the dual of the cyclic bicomplex CC(A) which looks asfollows.

......

6 6

(A⊗3)′ (A⊗3)′-1− t

6b

6−b′

A′ A′-1− t

(A⊗2)′ (A⊗2)′-1− t

6b

6−b′

(A⊗3)′ (A⊗3)′-N

6 6b

-N

(A⊗2)′ (A⊗2)′-N

6 6b

......

6 6

(A⊗3)′ (A⊗3)′-1− t

6 6−b′

A′ A′-1− t

(A⊗2)′ (A⊗2)′-1− t

6 6−b′

...6

(A⊗3)′-N

6b

A′ A′-N

(A⊗2)′ (A⊗2)′-N

6b

- · · ·

-

-

· · ·

· · ·

This is a first quadrant bicomplex. Again, the cohomology of the rows of thisbicomplex is located in degre zero.

5. Periodic cyclic homology and cohomology

Since the cyclic bicomplex is periodic it may be continued to the left usingthis periodicity. More precisely, we consider the infinite product total complex ofthis periodic bicomplex and obtain by definition the periodic cyclic complex PC(A)given by ∏

j∈Z Ω2j(A)B+b // ∏

j∈Z Ω2j+1(A)B+boo

which is a Z2-graded complex. Equivalently, we may also view PC(A) as a periodiccomplex indexed by the integers.

Definition 3.49. Let A be an algebra. The periodic cyclic homology of A isthe homology of the periodic cyclic complex PC(A) of A.

It follows from the definitions that there are natural surjective chain mapsπ2n : PC(A)→ CC(A) which are compatible with the S-operator in the sense thatSπ2n+2 = π2n for all n. Actually one has

PC(A) ∼= lim←−j∈N

CC[2j](A)

where CC[2j](A) is the bicomplex CC(A) shifted by degree 2j, that is

CC[2j](A)pq = CC(A)(p+2j)q

and the limit is taken using the S-operator. The projective system (CC[2j](A))j∈Nclearly has surjective structure maps. The following statement then follows fromproposition 2.27.

Proposition 3.50. Let A be an algebra. Then there is a short exact sequence

lim←−1

j∈ZHC∗+2j+1(A) // // HP∗(A) // // lim←−j∈ZHC∗+2j(A)

Here projective limits over Z are taken in order to describe the (derived) pro-jective limit of the homologies Hn(CC[2j](A)) in terms of cyclic homology.As a consequence we may determine the periodic cyclic homology of the complexnumbers.

5. PERIODIC CYCLIC HOMOLOGY AND COHOMOLOGY 43

Lemma 3.51. The periodic cyclic homology of the complex numbers is given by

HP0(C) = C, HP1(C) = 0.

Proof. Due to proposition 3.35 we have HC2n(C) = C and HC2n+1(C) = 0for all n. Moreover, under this identification the operator S : HC2n+2(C) →HC2n(C) is the identity map. It follows that the structure maps in the inversesystem (HC∗+2j(C))j∈Z are all surjective and hence

HP∗(C) = lim←−j∈N

HC∗+2j(C).

This proves the claim.

Proposition 3.52. Let V be a vector space. The periodic cyclic homology ofthe tensor algebra TV is given by

HP0(TV ) = 0, HP1(TV ) = 0.

Proof. According to proposition 3.38 the S-operator is zero on HC∗(TV ).This implies lim←−HC∗(TV ) = 0 and lim←−

1HC∗(TV ) = 0 as well. Now the claimfollows from proposition 3.50. Actually, the projective system given by the cyclic homology of a tensor algebra isan easy example of a projective system satisfying the Mittag-Leffler condition.

Lemma 3.53. Let A be an algebra. Then there are natural split short exactsequences

HPj(A) // // HPj(A+) // // HPj(C)

for j = 0 and j = 1.

Proof. This follows from proposition 3.50 and lemma 3.37. As a consequence, proposition 3.54 implies the following statement.

Proposition 3.54. Let V be a vector space. The periodic cyclic homology ofthe unitarized tensor algebra (TV )+ is given by

HP0((TV )+) = C, HP1((TV )+) = 0.

As for Hochschild homology and cyclic homology, periodic cyclic homology maybe defined for arbitrary mixed complexes. Moreover, the analogue of proposition3.50 holds also in this more general situation. According to the five lemma andlemma 3.33, proposition 3.50 implies the following result.

Proposition 3.55. Let f : M → N be a map of mixed complexes which in-duces an isomorphism in Hochschild homology. Then the induced map HP∗(M)→HP∗(N) is an isomorphism as well.

We remark that the converse of proposition 3.55 is not true. A map of mixedcomplexes which induces an isomorphism in periodic cyclic homology is not neces-sarily a quasiisomorphism on the level of Hochschild homology or cyclic homology.According to proposition 3.54 an easy example is the map Ω(0)→ Ω(TV ) inducedby the homomorphism 0→ TV for some nonzero vector space V .For cohomology we have to take the dual PC(A)′ of the periodic cyclic complexPC(A) using direct sums. More precisely, PC(A)′ is defined by

⊕j∈Z Ω2j(A)′

B+b // ⊕j∈Z Ω2j+1(A)′

B+boo

which is again a Z2-graded complex.

Definition 3.56. Let A be an algebra. The periodic cyclic cohomology of A isthe homology of PC(A)′.


Lemma 3.57. Let A be an algebra. Then there is a natural isomorphism

lim−→j∈N

HC∗+2j(A) ∼= HP ∗(A).

We conclude this section by defining the pairing between periodic cyclic coho-mology and periodic cyclic homology. Let A be an algebra and consider the directproduct complex PC(A)′ × PC(A). It follows immediately from the definition ofthe differential in PC(A)′ that the obvious map

〈−,−〉 : PC(A)′ × PC(A)→ C[0], 〈φ, c〉 = φ(c)

is a chain map. Here C[0] denotes the trivial supercomplex with C in degree zeroand 0 and degree one. Hence we obtain an induced map

HP ∗(A)×HP∗(A)→ C[0]

in homology. This is the pairing between periodic cyclic homology and cohomology.

6. Morita invariance

In this section we shall show that Morita equivalent algebras have isomorphicHochschild homology and cyclic homology.

Proposition 3.58. Let A and B be Morita equivalent unital algebras. Thenthere is a natural isomorphism HH∗(A) ∼= HH∗(B).

Proof. Let APB and BQA be equivalence bimodules. Since 〈−,−〉A : P ⊗BQ→ A is an isomorphism there exist elements pi ∈ P and qi ∈ Q for i = 1, . . . ,mand some n ∈ N such that

m∑i=1

〈pi, qi〉A = 1

We construct a chain map φ : C(A) → C(B) as follows. On chains of degree k wedefine

φ(a0 ⊗ a1 ⊗ · · · ⊗ ak) =

m∑i0,...,ik=1

〈qi0 , a0pi1〉B ⊗ 〈qi1 , a1pi2〉B ⊗ · · · ⊗ 〈qik , akpi0〉B

and using the equation

〈qi, aipj〉B〈qj , ajpk〉B = 〈qi, aipj〈qj , ajpk〉B〉B = 〈qi, ai〈pj , qj〉Aajpk〉Bit is easy to check that φ is a chain map. In the same way we obtain elementsxj ∈ Q and yj in P for j = 1, . . . , n such that

m∑i=1

〈xi, yi〉B = 1

and a chain map ψ : C(B)→ C(A) by

ψ(b0 ⊗ b1 ⊗ · · · ⊗ bn) =

n∑j0,...,jk=1

〈xj0 , b0yj1〉A ⊗ 〈xj1 , b1xj2〉A ⊗ · · · ⊗ 〈xjk , bkyj0〉A.

The composition ψφ : C(A)→ C(A) is given by

ψφ(a0 ⊗ a1 ⊗ · · · ⊗ ak) =n∑

j0,...,jk=1

m∑i0,...,ik=1

〈xj0 , 〈qi0 , a0pi1〉Byj1〉A ⊗ 〈xj1 , 〈qi1 , a1pi2〉Byj2〉A ⊗ · · ·

· · · ⊗ 〈xjk , 〈qik , akpi0〉Byj0〉A.

6. MORITA INVARIANCE 45

We shall construct a presimplicial homotopy h on C(A) between id and ψφ asfollows. In degree k we define

hr(a0 ⊗ a1 ⊗ · · · ⊗ ak) =n∑

j0,...,jr=1

m∑i0,...,ir=1

a0〈pi0 , xj0〉A ⊗ 〈yj0 , qi0〉Aa1〈pi1 , xj1〉A ⊗ · · ·

· · · ⊗ 〈yjr−1, qir−1

〉Aar〈pr, xr〉A ⊗ 〈yjr , qir 〉A ⊗ ar+1 ⊗ · · · ⊗ akfor r = 0, . . . , k. Let us check that this is indeed a presimplicial homotopy in degreek = 1. We have to prove

d0h1 = h0d0, d1h1 = d1h0, d0h0 = id, d2h1 = ψφ.

For the first equation we calculate

d0h1(a0 ⊗ a1) =

n∑j0,j1=1

m∑i0,i1=1

a0〈pi0 , xj0〉A〈yj0 , qi0〉Aa1〈pi1 , xj1〉A ⊗ 〈yj1 , qi1〉A

=

n∑j1=1

m∑i1=1

a0a1〈pi1 , xj1〉A ⊗ 〈yj1 , qi1〉A = h0d0(a0 ⊗ a1)

The second equation follows in the same way. For the third equation we have

d0h0(a0 ⊗ a1) =

n∑j0=1

m∑i0=1

a0〈pi0 , xj0〉A〈yj0 , qi0〉A ⊗ a1 = a0 ⊗ a1

and the last equation is verified by calculating

d2h1(a0 ⊗ a1) =

n∑j0,j1=1

m∑i0,i1=1

〈yj1 , qi1〉Aa0〈pi0 , xj0〉A ⊗ 〈yj0 , qi0〉Aa1〈pi1 , xj1〉A

= ψφ(a0 ⊗ a1).

The fact that h satisfies the relations for a presimplicial homotopy in other degreesis proved in a similar way. We leave the verification to the reader.As a consequence we deduce that the complexes C(A) and C(B) are homotopyequivalent. This proves the claim.

Corollary 3.59. Let A and be B be Morita equivalent unital algebras. Thenthere are natural isomorphisms HC∗(A) ∼= HC∗(B) and HP∗(A) ∼= HP∗(B).

Proof. It is evident that the chain map φ : C(A) → C(B) constructed inproposition 3.58 is a map of cyclic modules. Hence it induces a map HC∗(A) →HC∗(B) which an isomorphism according to lemma 3.33. The assertion for theperiodic theory follows from proposition 3.55. It is instructive to consider explicitly the case of matrix algebras. Let A be a unitalalgebra and let ι : A→Mn(A) be the algebra homomorphism given by ι(a) = ae11.Here aeij for 1 ≤ i, j ≤ n denotes the matrix with the only nonzero entry a indegree (i, j). Remark that the homomorphism ι does not preserve the units. Stillι induces a chain map C(A)→ C(Mn(A)) which will again be denoted by ι.Conversely, define the trace map τ : C(Mn(A))→ C(A) by

τ(A0 ⊗A1 ⊗ · · · ⊗Ak) =

n∑i0,...,ik=1

A0i0i1 ⊗A

1i1i2 ⊗ · · · ⊗A

kini0

in degree k where Aij denotes the (i, j)-th entry of the matrix A. It can be easilychecked that τ is a chain map. This also follows from the following excercise.


Excercise 3.60. Let Mn(A) be the algebra of n × n-matrices over a unitalalgebra A and let P = An and Q = An be the natural Morita context relating Aand Mn(A). Then the maps φ and ψ constructed above are equal to ι and τ .

7. The Chern character in K-theory

In this section we define the K-group K0 of an algebra A and a natural homo-morphism ch0 : K0(A)→ HP0(A).Let A be a unital algebra. In order to be defininite, we shall work with unitaryleft modules in the sequel. However, we will see that one could equally well workwith right modules. Recall that a unitary A-module P is finitely generated andprojective iff it is a direct summand in An for some n. In particular, if P and Q areunitary finitely generated projective modules then the direct sum M ⊕N is againfinitely generated and projective. It follows that isomorphism classes of unitaryfinitely generated projective modules form an abelian semigroup P (A) with directsum as addition and neutral element the zero module. Remark that, in contrastto isomorphism classes of arbitrary modules, isomorphism classes of finitely gener-ated projective unitary modules form a set since every finitely generated projectiveunitary module is isomorphic to a submodule of An for some n.

Definition 3.61. Let H be an abelian semigroup with neutral element. Anabelian group G(H) together with an semigroup homomorphism ι : H → G(H) iscalled a Grothendieck group of H if for every abelian group M and every semigrouphomomorphism f : H →M there is a unique group homomorphism F : G(H)→Msuch that Fι = f .

As usual, it is easy to see that a Grothendieck group G(H) of H is uniquelydetermined up to isomorphism.

Lemma 3.62. For every abelian semigroup with neutral element there exists aGrothendieck group.

Proof. Consider the free abelian group F (H) generated by H and let ι : H →F (H) be the natural map. We let G(H) be the quotient of F (H) by the relationsι(x + y) = ι(x) + ι(y) for all x, y ∈ H. Then the induced map ι : H → G(H) is asemigroup homomorphism and it is easy to verify the universal property.However, it is often important to work with a more concrete realization of theGrothendieck group. More precisely, an alternative definition is G(H) = H×H/ ∼where (a1, b1) ∼ (a2, b2) iff there exists c ∈ H such that a1 + b2 + c = b1 + a2 + c.It is straightforward to check that componentwise addition defines turns G(H) intoan abelian group. The neutral element is (0, 0) and the inverse of an element (a, b)is given by (b, a). The natural map ι : H → G(H) is defined by ι(a) = (a, 0).We leave it as an excercise to verify the universal property. One should think ofelements (a, b) as formal differences a− b. As an example consider the semigroup N0 of nonnegative integers with addition.

Excercise 3.63. The Grothendieck group G(N0) is isomorphic to Z and ι :N0 → Z is the obvious inclusion in this case.

We now define the K-group of a unital algebra.

Definition 3.64. Let A be a unital algebra. The K-group K0(A) of A is theGrothendieck group of the semigroup P (A).

First we shall discuss the functoriality of this construction.

Excercise 3.65. Let A and B be unital algebras and let AM be an A-module.If M is finitely generated, then B ⊗A M is finitely generated as well. If M isprojective, then B ⊗AM is projective as well.

7. THE CHERN CHARACTER IN K-THEORY 47

Every unital algebra homomorphism f : A → B induces a semigroup homo-morphism P (A)→ P (B) by sending a finitely generated projective (left) A-moduleP to the B-module B⊗A P . By the universal property of the Grothendieck group,this induces a group homomorphism K0(f) : K0(A)→ K0(B).

Excercise 3.66. Let A and B be unital algebras. Then the natural projectionsπA : A ⊕ B → A and πB : A ⊕ B → B induce an isomorphism K0(A ⊕ B) →K0(A)⊕K0(B).

In order to extend the definition of K0 to arbitrary algebras we have to proceedas follows.

Lemma 3.67. Let A be a unital algebra. Then K0(A) is naturally isomorphicto the kernel of the augmentation homomorphism K0(A+)→ K0(C).

Proof. Since A is unital we have an isomorphism A+ ∼= A ⊕ C of unitalalgebras. Now the assertion follows easily from excercise 3.66.

Definition 3.68. Let A be an algebra. The group K0(A) is the kernel of thenatural map K0(A+)→ K0(C) induced by the augmentation homomorphism.

According to lemma 3.67 this definition is compatible with the previous one forunital algebras.Consider for instance the case A = C. Since every finitely generated projectivemodule over C is isomorphic to Cn for some n we obtain P (C) = N0. Hence we getK0(C) = Z.We will now define an additive map ch0 : K0(A) → HP0(A) for an augmentedalgebra A. For an idempotent e ∈Mn(A) set

ch0(e) =

∞∑k=0

(−1)k(2k)!

k!tr

((e− 1

2

)(dede)k

)

viewed as an element in the periodic cyclic complex PC(A) where tr : PC(Mn(A))→PC(A) is the trace map defined by

tr(M0dM1 · · · dMk) =∑

1≤i0,...,ik≤n

M0i0i1dM

1i1i2 · · · dM

kiki0

tr(dM1 · · · dMk) =∑

1≤i1,...,ik≤n

dM1i1i2 · · · dM

kiki1

for differential forms of degree k. Here Mij denotes the (i, j)th entry of a matrixM ∈ Mn(A). Note that tr is actually just the trace map occuring in the proof ofMorita invariance for matrix algebras over unital algebras. Remark also that themap tr : PC(Mn(A))→ PC(A) is a chain map for arbitrary algebras A.

Lemma 3.69. The element ch0(e) is a cycle and defines a class in HP0(A).

Proof. We compute

B

((−1)k

(2k)!

k!

(e− 1

2

)(dede)k

)= (−1)k(2k + 1)

(2k)!

k!de(dede)k


and

b

((−1)k+1 (2k + 2)!

(k + 1)!

(e− 1

2

)(dede)k

)= (−1)k+1 (2k + 2)!

(k + 1)!

(ede(dede)k−

− 1

2ede(dede)k +

1

2de(dede)k − 1

2ede(dede)k

)= (−1)k+1 (2k + 1)!2(k + 1)

(k + 1)!

1

2de(dede)k

= (−1)k+1 (2k + 1)!

k!de(dede)k.

Hence these terms cancel and since tr is a chain map we deduce (B+ b) ch0(e) = 0.It follows that ch0(e) is a cycle and hence defines an element in HP0(A). Let e ∈ Mn(A) and f ∈ Mm(A) be idempotents and consider their direct sume⊕ f ∈Mn+m(A). By the definition of the trace map we see

ch0(e⊕ f) = ch(e) + ch(f).

It follows that ch0 defines an additive map from P (A) to HP0(A). By the universalproperty of the Grothendieck group we therefore obtain the following result.

Proposition 3.70. Let A be a unital algebra. The Chern character ch0 definesa natural transformation K0(A)→ HP0(A).

It remains to extend the Chern character to arbitrary algebras. This is doneusing the commutative diagram

K0(A) // //

K0(A+) // //

ch0

K0(C)

ch0

HP0(A) // // HP0(A+) // // HP0(C)

which is obtained using lemma 3.53.

CHAPTER 4

The Hochschild-Kostant-Rosenberg theorem

In this chapter we calculate the Hochschild and cyclic homology of the Frechetalgebra C∞(M) of smooth functions on a smooth manifold M . First we discusssome background material from functional analysis in section 1. More precisely, weexplain the concept of a locally convex vector space and the projective tensor prod-uct. In section 2 we discuss how to adapt the tools of homological algebra to locallyconvex spaces. Section 3 contains a review of standard constructions with differ-ential form on smooth manifolds including the exterior derivative, Lie derivativesand interior products. In section 4 we formulate the Hochschild-Kostant-Rosenbergtheorem which computes the Hochschild homology of the Frechet algebra C∞(M).We prove this theorem first in the special case of an open convex neighborhood ofzero in Rn. The proof for arbitrary manifolds is carried out in section 5 using anappropriate localization procedure. Section 6 contains the computation of cyclicand periodic cyclic homology for C∞(M). Finally, in section 7 we recall the clas-sical Chern-Weil construction of characteristic classes using connections on vectorbundles. If M is compact, the Chern character from K-theory to periodic cyclichomology for the algebra C∞(M) is identified with the classical Chern characterwith values in de Rham cohomology.

1. Locally convex vector spaces and tensor products

Let M be a smooth manifold. For the purposes of cyclic homology it is notappropriate to consider C∞(M) as a complex algebra without further structure.Actually, the purely algebraic Hochschild and cyclic homology groups of C∞(M)as defined in chapter 3 are not known in general. The main problem is that, apartfrom trivial cases, the algebraic tensor product C∞(M)⊗C∞(N) is not isomorphicto C∞(M ×N) for smooth manifolds M,N .It is more natural to consider C∞(M) as a locally convex algebra. Accordingly, thealgebraic tensor product is replaced by the completed projective tensor product.The completed projective tensor product has the property that C∞(M)⊗πC∞(N)is naturally isomorphic to C∞(M ×N).In this section we explain some of the concepts and results from functional analysisinvolved here. For more details we refer to [10], [15], [7].The complex numbers are always equipped with the natural topology coming fromthe metric d(λ, µ) = |λ− µ|.

Definition 4.1. A topological vector space is a vector space V which is equippedwith a Hausdorff topology such that the addition V × V → V, (v, w) 7→ v + w andthe scalar multiplication C× V → V (α, v) 7→ αv are continuous.

In a topological vector space the translation maps Tv : V → V given by Tv(w) =v + w are homeomorphisms for every v ∈ V . As a consequence, to describe thetopology of a topological vector space it suffices to specify a basis of neighborhoodsof the origin.Let V be a vector space. A seminorm on V is a map p : V → R+ such that

p(λv) = |λ|p(v), p(v + w) ≤ p(v) + p(w)

49

50 4. THE HOCHSCHILD-KOSTANT-ROSENBERG THEOREM

for all v, w ∈ V and λ ∈ C. Note that p(0) = 0 for every seminorm. A seminorm isa norm if p(v) = 0 implies v = 0.Assume that p is a seminorm on V . By definition, the (open) ball Bp(v; r) withradius r around v ∈ V consists of all vectors w ∈ V such that p(w − v) < r. If theseminorm is clear from the context we simply write B(v, r).We are interested in the following class of topological vector spaces.

Definition 4.2. A locally convex vector space is a topological vector space Vwith topology given by a family (pi)i∈I of seminorms pi : V → C. That is, a basisof neighborhoods around the origin is given by the balls Bpi(0, r) with r > 0 andi ∈ I.

Accordingly, the basis of neighborhoods around an arbitrary point v in a locallyconvex vector space V is given by the balls Bpi(v, r) with r > 0 and i ∈ I. Since Vis assumed to be Haussdorff there exists for every nonzero vector v ∈ V a seminormpi such that pi(v) > 0.A subset K of a vector space V is called convex if λv+(1−λ)w ∈ K for all v, w ∈ Kand 0 < λ < 1. We remark that locally convex vector spaces can be characterizedas those topological vector spaces V in which every point v ∈ V has a neighborhoodbase of convex sets.Examples of locally convex vector spaces are normed spaces or Banach spaces.These spaces are special in the sense that the topology is determined by a singlenorm.As for normed spaces, the concept of completeness plays an important role forlocally convex spaces. A net (vj)j∈J in a locally convex space V is called a Cauchynet if for every defining seminorm pi and every ε > 0 there exists k ∈ J such thatp(vm − vn) ≤ ε for all m,n ≥ k. A net (vj)j∈J in V is convergent to v ∈ V iff forevery ε > 0 and every defining seminorm p there exists k ∈ J such that p(v−vn) < εfor all n ≥ k. Note that the limit v is uniquely determined since V is Hausdorff.Clearly every convergent net is a Cauchy net. A locally convex vector space V iscalled complete if every Cauchy net in V is convergent.

Definition 4.3. Let V be a locally convex vector space. A completion of Vis a complete locally convex vector space V c together with a continuous linear mapι : V → V c such that for every complete locally convex vector space W and everycontinuous linear map f : V → W there exists a unique continuous linear mapF : V c →W such that the diagram

V V c-ι

f@@@@RW?

F

is commutative.

Being defined by a universal property, the completion is uniquely determinedup to isomorphism.

Theorem 4.4. For every locally convex vector space there exists a completion.

An important class of locally convex vector spaces is the class of Frechet spaces.

Definition 4.5. A Frechet space is a complete locally convex vector space Vsuch that the topology can be defined by a countable family of seminorms.

We remark that a locally convex vector space V is metrizable iff its topology isdefined by a countable family of seminorms.

1. LOCALLY CONVEX VECTOR SPACES AND TENSOR PRODUCTS 51

Many general constructions with vector spaces extend easily to the setting of (com-plete) locally convex spaces. For instance, direct products, direct sums, projectiveand inductive limits are defined by the analogous universal properties.A more subtle point is the notion of a tensor product. Similarly to the algebraicsetting, tensor products of locally convex vector spaces are determined by consider-ing bilinear maps with certain continuity properties. The most evident continuityproperty for a bilinear map b : V ×W → X is to require b to be continuous for theproduct topology on V ×W . We say that b is jointly continuous in this case.

Definition 4.6. Let p be a seminorm on V and let q be a seminorm on W .The tensor product p⊗ q : V ⊗W → R+ is defined by

(p⊗ q)(z) = inf

( n∑j=1

p(vj)q(wj) | z =

n∑j=1

vj ⊗ wj).

The following result summarizes basic properties of the tensor product of twoseminorms.

Proposition 4.7. Let p and q be seminorms on V and W , repectively. Thenp⊗ q is a seminorm on V ⊗W . Moreover

(p⊗ q)(v ⊗ w) = p(v)q(w)

for all simple tensors v ⊗ w ∈ V ⊗W .

Proof. It is easy to check that p ⊗ q is a seminorm. From the definition ofp⊗ q it is immediate that

(p⊗ q)(v ⊗ w) ≤ p(v)q(w)

for v ∈ V and w ∈W . For the other inequality choose v′ ∈ V ′ such that v′(v) = p(v)and |v′(x)| ≤ p(x) for all x ∈ V . Here V ′ denotes the space of linear maps fromV to C which are bounded for the seminorm p. The existence of v′ follows fromthe classical Hahn-Banach theorem for the normed space V ′/ ker(p). In the sameway one obtains w′ ∈W ′ such that w′(w) = q(w) and |w′(y)| ≤ q(y) for all y ∈W .Consider the linear form v′⊗w′ on V ⊗W and let v⊗w be represented as a linearcombination

∑xi ⊗ yi with xi ∈ V and yi ∈W . Then we have

|v′ ⊗ w′(v ⊗ w)| ≤∑|v′(xi)w′(yi)| ≤

∑p(xi)q(yi)

for every such representation. By the definition of p⊗ q we thus obtain

p(v)q(w) = v′(v)w′(w) = |v′ ⊗ w′(v ⊗ w)| ≤ (p⊗ q)(v ⊗ w)

which yields the claim. If V and W are locally convex vector spaces with defining seminorms (pi)i∈I and(qj)j∈J the projective topology on V ⊗W is the locally convex topology defined bythe seminorms pi ⊗ qj for all i ∈ I and j ∈ J . We write V ⊗π W for the algebraictensor product equipped with the projective topology. It can be shown that theprojective topology is again Hausdorff. Moreover it follows from proposition 4.7that the canonical bilinear map V ×W → V ⊗π W is jointly continuous.

Definition 4.8. Let V and W be locally convex vector spaces. The completedprojective tensor product V ⊗πW is the completion of V ⊗π W .

The completed projective tensor product is determined by the following uni-versal property.

Proposition 4.9. Let V and W be locally convex vector spaces. For everycomplete locally convex vector space X and every jointly continuous bilinear map


f : V ×W → X there exists a unique continuous linear map F : V ⊗πW → X suchthat the diagram

V ×W V ⊗πW-

f@@@@RX?

F

is commutative.

Proof. Let f : V ×W → X be a continuous bilinear map and let F : V ⊗W →X be the associated linear map. For every defining seminorm q on X there existdefining seminorms pV and pW on V and W , respectively, such that q(f(v, w)) ≤pV (v)pW (w) for all v ∈ V and w ∈ W . Consequently, for x =

∑vi ⊗ wi ∈ V ⊗W

we have

q(F (x)) = q

(∑F (vi ⊗ wi)

)≤∑

pV (vi)pW (wi)

and hence q(F (x)) ≤ (pV ⊗ pW )(x). This shows that F is continuous. By theuniversal property of the algebraic tensor product the map F is uniquely determinedby f . The proof is finished using the universal property of the completion. In the sequel we write V ⊗W instead of V ⊗πW for the completed projective tensorproduct of two locally convex vector spaces. Moreover, we will assume for simplicitythat all locally convex spaces are complete.Let us carry over some definitions of chapter 1 to the setting of locally convex vectorspaces.

Definition 4.10. A locally convex algebra is a locally convex vector space Atogether with a continuous bilinear map µ : A×A→ A such that

a(bc) = (ab)c

for all a, b, c ∈ A. A unital locally convex algebra is a locally convex algebra with anelement 1 ∈ A such that 1a = a1 = a for all a ∈ A.An algebra homomorphism f : A→ B between locally convex algebras is a continu-ous linear map such that f(ab) = f(a)f(b) for all a, b ∈ A. A unital homomorphismf : A → B between unital locally convex algebras is a homomorphism such thatf(1) = 1.

Note that the multiplication in a (complete) locally convex algebra A can equiv-alently be described by a continuous linear map µ : A⊗A→ A.It is clear that every locally convex algebra is in particular an algebra in the sense ofchapter 1. The standard constructions with algebras described in chapter 1 extendeasily to locally convex algebras.

Definition 4.11. Let A be a locally convex algebra. A locally convex (left)module over A is a locally convex vector space M together with a continuous bilinearmap A×M →M such that

(ab)m = a(bm)

for all a, b ∈ A and m ∈ M . A unitary locally convex (left) module over a unitallocally convex algebra A is an A-module M such that 1m = m for every m ∈M . AnA-module homomorphism f : M → N between locally convex (unitary) A-modules isa continuous linear map which satisfies f(am) = af(m) for all a ∈ A and m ∈M .In a similar way one defines locally convex (unitary) right modules, (unitary) bi-modules and their homomorphisms.

We will frequently speak of modules instead of locally convex modules for sim-plicity.

1. LOCALLY CONVEX VECTOR SPACES AND TENSOR PRODUCTS 53

Let us discuss the projective tensor product of locally convex modules. Assume MA

and AN are locally convex modules over the locally convex algebra A and let V bea locally convex vector space. A jointly continuous bilinear map f : M ×N → V iscalled A-bilinear if f(ma, n) = f(m, an) for all m ∈M,n ∈ N, a ∈ A.

Definition 4.12. Let MA and AN be locally convex A-modules. A completelocally vector space M⊗AN together with a jointly continuous A-bilinear map ⊗ :M×N 3 (m,n) 7→ n⊗n ∈M⊗AN is called tensor product of M and N over A if forevery complete locally convex vector space V and every jointly continuous A-bilinearmap f : M ×N → V there exists a unique continuous linear map F : M⊗AN → Vsuch that the diagram

M ×N M⊗AN-⊗

f@@@@RV?

F

is commutative.

As in the algebraic case, the tensor product M⊗AN is uniquely determined upto isomorphism by MA and AN . It is constructed as the quotient of the projectivetensor product M⊗N by the closed linear subspace generated by all tensors of theform ma⊗ n−m⊗ an.Let A be a locally convex algebra. A surjective continuous A-module homomor-phism π : M → N is called strict if there exists a continuous linear map σ : N →Msuch that πσ = id.

Definition 4.13. Let A be a locally convex algebra. A locally convex module

AP is called projective if for every strict epimorphism π : M → N of A-modulesand every A-module homomorphism f : P → N there exists an A-module homo-morphism F : P →M such that the diagram

M N-π

F

P

?

f

is commutative.

As in the algebraic case one has the following result.

Excercise 4.14. For every locally convex algebra A the A-module A+ is pro-jective. Direct sums of projective modules are projective.

An A-submodule M of an A-module P is called a direct summand if thereexists an A-submodule N in P such that the natural map M ⊕ N → P is anisomorphism. Equivalently, there exists an A-module homomorphism π : P → Msuch that πι = id where ι : M → P is the natural inclusion.

Excercise 4.15. If M is isomorphic to a direct summand in a projective mod-ule, then M is itself projective.

We need some more terminology. An epimorphism π : M → N of A-modulesis called split if there exists an A-module homomorphism σ : N → M such thatπσ = id. If V is any locally convex vector space, then A+⊗V with the obvious leftA-module structure is called the free A-module over V . In general, a locally convexA-module of this form for some locally convex space V is called free.


Proposition 4.16. Let AP be a locally convex module. The following areequivalent:

a) P is projective.b) Every strict epimorphism π : M → P splits.c) P is isomorphic to a direct summand in a free module.

Let us now have a look at the locally convex vector spaces we are interested in,namely spaces of smooth functions on manifolds. First let K ⊂ Rn be a compactsubset. We define seminorms on C∞(K) by

||f ||Kα = supx∈K|Dαf(x)|

where α = (α1, . . . , αn) is a multiindex and Dα denotes the derivative

Dα(f) =∂α1

∂xα11

· · · ∂αn

∂xαnn.

In other words, a sequence (fj)j∈N of smooth functions on K converges to f withrespect to ||−||Kα iff the functions Dαfj converge to Dαf uniformly on the compactset K.Now let M be a (second countable) smooth manifold. Choose a sequence (Kj)j∈Nof compact subsets Kj of M contained in chart domains Uj such that

⋃j∈N Ij = M

where Ij is the interior of Kj . The locally convex topology on C∞(M) is given byall seminorms ||f ||iα = ||f|Ki ||Kiα where f|Ki denotes the restriction of f to Ki and

the seminorms ||f|Ki ||Kiα are those for C∞(Ki) where Ki is viewed as a compactsubset of Rn. In this way C∞(M) becomes a locally convex topological vectorspace. It is not difficult to show that the topology does not depend on the choiceof the compact subsets Kj .

Excercise 4.17. Let M be a smooth manifold. Then C∞(M) is a Frechetspace. A sequence (fn)n∈N in C∞(M) converges to f ∈ C∞(M) iff D(fn) con-verges uniformly on compact subsets to D(f) for all differential operators D onM . Moreover C∞(M) is a locally convex algebra with pointwise multiplication offunctions.

One may generalize the construction of the locally convex topology on C∞(M)to vector bundles as follows. Let M be a smooth manifold and let E be a smoothcomplex vector bundle over M . Then the space C∞(M,E) of smooth sections ofE is a unitary locally convex module over C∞(M). The topology is defined byrequiring uniform convergence of all derivatives on compact subsets. For this oneuses the fact that locally E|U = U × Ck for some k ∈ N.Let E be a locally convex vector space and let U ⊂ Rn be open. A functionf : U → E is called differentiable at x0 ∈ U if there are vectors e1, . . . , en ∈ E suchthat

f(x)− f(x0)−∑nj=1(xj − x0

j )ej

|x− x0|converges to 0 in E as |x − x0| converges to zero. The vectors ej are then calledthe first partial derivatives of f at x0 and one writes

ej =∂f

∂xj(x0)

for j = 1, . . . , n. As usual, a function is called differentiable on U if it is differentiableat every point in U . Note that a differentiable function is continuous. A functionf : U → E is called smooth if all iterated partial derivatives of f exist.More generally, let M be a smooth manifold. A function f : M → E is calledsmooth if for every coordinate domain U ⊂ M the induced mapping f|U : U → E

2. HOMOLOGICAL ALGEBRA WITH LOCALLY CONVEX SPACES 55

is smooth in the sense explained before. We denote by C∞(M,E) the linear spaceof all smooth maps from M to E. The space C∞(M,E) becomes a locally convexvector space with the topology of uniform convergence on compact subsets of theiterated partial derivatives.

Proposition 4.18. Let M and N be smooth manifolds. Then there are naturaltopological isomorphisms

C∞(M ×N) ∼= C∞(M,C∞(N)) ∼= C∞(N,C∞(M)).

Proof. It clearly suffices to prove C∞(M ×N) ∼= C∞(M,C∞(N)). Define alinear map φ : C∞(M × N) → C∞(M,C∞(N)) by φ(f)(x)(y) = f(x, y). To seethat this map is well-defined observe first that φ(f)(x) ∈ C∞(N) for all x ∈M sincef is smooth. Moreover it follows easily from the definitions that φ(f) is a smoothmap from M to C∞(N). Conversely, define ψ : C∞(M,C∞(N)) → C∞(M × N)by ψ(f)(x, y) = f(x)(y). Again, it is straightforward to check that ψ(f) is indeed asmooth function. The maps φ and ψ are obviously inverse to each other. Moreoverone checks that both maps are continuous for the natural topologies.

Theorem 4.19. Let M be a smooth manifold and V be a complete locally convexvector space. Then there is a natural topological isomorphism

C∞(M)⊗V ∼= C∞(M,V ).

We will not discuss the proof of theorem 4.19. Let us only note that a combi-nation of this theorem with proposition 4.18 yields the following result.

Theorem 4.20. Let M and N be smooth manifolds. Then there is a naturaltopological isomorphism

C∞(M)⊗C∞(N) ∼= C∞(M ×N).

As a consequence, the abstractly defined completed tensor product has a veryconcrete realization for spaces of smooth functions on manifolds.

2. Homological algebra with locally convex spaces

In this section we explain how the homological algebra developped in chapter 2may be adapted to the framework of locally convex spaces. Again, we shall assumefor simplicitly that all locally convex spaces are complete.

Definition 4.21. Let A be a locally convex algebra. A chain complex of A-modules is a sequence C = (Cn)n∈Z of locally convex A-modules Cn together withA-module homomorphisms dn : Cn → Cn−1 such that dndn+1 = 0 for all n ∈ Z.A chain map f : C → D between chain complexes is a family fn : Cn → Dn ofA-module homomorphisms such that the diagrams

Cnd //

fn

Cn−1

fn−1

Dn

d // Dn−1


There are two possibilities to define the homology of a chain complex of locallyconvex vector spaces. Namely, one may divide the space of cycles Zn by the spaceof boundaries Bn as in the algebraic case or by the closure of Bn. Observe that thespace of cycles, being the kernel of a continuous linear map, is always closed.

Definition 4.22. The n-th homology group of a chain complex C is the spaceHn(C) = Zn/Bn.


If V is a topological vector space and W is a linear subspace one may equipV/W with the quotient topology. It is not hard to show that the quotient topologyon V/W is Hausdorff iff the subspace V is closed. Hence, according to definition4.22, the homology groups of a complex of locally convex vector spaces may fail tobe separated for the quotient topology.Although this does not happen in the case we study below, we shall view thehomology of a complex of locally convex vector spaces always as an abstract vectorspace without topology. Since we forget the topology of a complex when consideringhomology, the machinery of exact sequences can be applied without change.Homotopies and homotopy equivalences for complexes of locally convex modulesare defined in the obvious way by requiring continuity of the involved maps. Thesame applies to bicomplexes and their associated total complexes.Let us now discuss the appropriate notion of a projective resolution.

Definition 4.23. Let A be a locally convex algebra and let M be a locallyconvex A-module. A projective resolution P of M consists of a long exact sequence

M oo εP0oo P1

oo P2oo P3

oo · · ·of locally convex A-modules which is split exact as a sequence of locally convex vectorspaces such that all Pj are projective.

Recall that projective locally convex modules were introduced in the previoussection. As in the algebraic setting one proves that every locally convex modulehas a projective resolution.

Lemma 4.24. Let M be a locally convex A-module. Then there exists a projec-tive resolution for M .

Similarly, the comparison result for projective resolutions holds.

Proposition 4.25. Let M and N be locally convex A-modules and let P andQ be projective resolutions of M and N , respectively. If f : M → N is an A-modulehomomorphism there exist A-module homomorphisms fj : Pj → Qj for all j suchthat the diagram

M oo ε

f

P0oo

f0

P1oo

f1

P2oo

f2

P3oo

f3

· · ·

N oo εQ0oo Q1

oo Q2oo Q3

oo · · ·

is commutative. Moreover, if (gj)j≥0 is another family of such homomorphisms,then the chain maps f and g thus defined are continuously homotopic.

We leave the proof of this assertion to the reader. As a consequence, one hasthe following result.

Excercise 4.26. Two projective resolutions of a locally convex module M arecontinuously homotopy equivalent.

Let us proceed to and define the derived functor of the tensor product.

Definition 4.27. Let MA and AN be complete locally convex modules over alocally convex algebra A and choose a projective resolution P of AN . Then

TorAn (M,N) = Hn(M⊗AP ).

Using excercise 4.26 we see that, up to natural isomorphism, the definition ofTor(M,N) is independent of the resolution P .Finally let us discuss how Hochschild and cyclic homology are defined for locally

3. DIFFERENTIAL FORMS AND DE RHAM COHOMOLOGY 57

convex algebras. For a locally convex algebra it is natural to consider the space ofcompleted noncommutative differential forms.

Definition 4.28. Let A be a complete locally convex algebra. For n > 0 we let

Ωn(A)c = A+⊗A⊗n be the space of completed noncommutative n-forms over A. Inaddition we set Ω0(A)c = A.

All operators on noncommutative differential forms defined in chapter 3 andtheir algebraic relations carry over to the locally convex setting. In particular weobtain the following statement.

Proposition 4.29. Let A be a complete locally convex algebra. The spaceΩ(A)c of completed noncommutative differential forms together with the operatorsb and B is a mixed complex.

As a consequence, the definition of Hochschild and cyclic homology is straight-forward.

Definition 4.30. Let A be a complete locally convex algebra. The continuousHochschild (cyclic, periodic cyclic) homology of A is the Hochschild (cyclic, peri-odic cyclic) homology of the mixed complex Ω(A)c of completed noncommutativedifferential forms over A.

We denote by HHn(A) the n-th continuous Hochschild homology group of Aand accordingly for the other theories. Of course this notation is imprecise sinceone should distinguish the continuous homology groups from the purely algebraicones defined in chapter 3. For simplicity we shall not do this since we are onlyinterested in the continuous homology groups in this chapter. Similarly, we willalso write Ω(A) instead of Ω(A)c for the space of completed differential forms inthe sequel.

3. Differential forms and de Rham cohomology

In this section we review some constructions and results related to differentialforms on a manifold.Let M be a smooth manifold. We denote by Ak(M) the space of smooth complex-valued k-forms on M and write A(M) for the direct sum of the spaces Ak(M). SinceAk(M) = 0 for k > n = dim(M) this is a finite direct sum. An element of A(M) isa section of the complexified exterior algebra bundle of the cotangent bundle of M .In particular, there is a natural C∞(M)-module structure on the spaces Ak(M). Asmooth map φ : M → N induces a linear map φ∗ : A(N)→ A(M). Explicitly, onehas

φ∗(ω)(X1, . . . , Xk) = ω(T (φ)X1, . . . , T (φ)Xk)

if T (φ) : T (M)→ T (N) denotes the corresponding map of the tangent bundles.Locally in a coordinate domain U , every differential form ω ∈ Ak(M) can be writtenas

ω = f(x)dxi1 ∧ · · · ∧ dxikfor some smooth function f ∈ C∞(U). If η = gdxj1∧· · ·∧dxjl is another differentialform expressed locally in this form, the exterior product ω ∧ η ∈ Ak+l(M) is givenby

ω ∧ η = fgdxi1 ∧ · · · ∧ dxik ∧ dxj1 ∧ · · · ∧ dxjlon U . The exterior product is graded commutative, that is,

ω ∧ η = (−1)klη ∧ ω


for ω ∈ Ak(M), η ∈ Al(M). One has the explicit formula

(ω∧η)(X1, . . . , Xk+l)

=∑

σ∈Sk+l

(−1)sign(σ) 1

k! l!ω(Xσ(1), . . . , Xσ(k)) η(Xσ(k+1), . . . , Xσ(k+l))

for the exterior product.The exterior differential d : A0(M)→ A1(M) is the linear map defined by

d(f)(X) = X(f)

for all vector fields X on M . The map d is extended to a linear map d : A(M) →A(M) of degree 1 in a unique way such that d2 = 0 and such that the Leibniz rule

d(ω ∧ η) = dω ∧ η + (−1)|ω|ω ∧ dηholds for homogenous forms ω and η. If ω ∈ Ak(M) and X0, . . . , Xk are vectorfields on M one has the explicit formula

(dω)(X0, . . . , Xk) =

k∑i=0

(−1)iXi(ω(X0, . . . , Xi−1, Xi+1, . . . , Xk))

+∑

0≤i<j≤k

(−1)i+jω([Xi, Xj ], X0, . . . , Xi−1, Xi+1, . . . , Xj−1, Xj+1, . . . , Xk)

and in local coordinates the exterior differential is given by

d(f(x) dxi1 ∧ · · · ∧ dxik) =

n∑j=1

∂f

∂xjdxj ∧ dxi1 ∧ · · · ∧ dxik .

According to the relation d2 = 0 one obtains a cochain complex

A0(M)d // A1(M)

d // A2(M)d // A3(M)

d // · · ·

which is called the de Rham complex of M .

Definition 4.31. Let M be a smooth manifold. The de Rham cohomology ofM is the cohomology of the de Rham complex A(M) and denoted by H∗dR(M).

A smooth map f : M → N induces an algebra homomorphism f∗ : A(N) →A(M) which commutes with the exterior differential. Hence one also obtains in-duced maps HdR(f) : H∗dR(N)→ H∗dR(M).For later reference we note the homotopy invariance of de Rham cohomology. Twosmooth maps f0, f1 : M → N between manifolds are smoothly homotopic if thereexists a smooth map f : M × [0, 1] → N restricting to f0 and f1 at 0 and 1,respectively.

Proposition 4.32. Let f0, f1 : M → N be smoothly homotopic smooth mapsbetween manifolds M and N . Then the induced maps f∗0 , f

∗1 : H∗dR(N)→ H∗dR(M)

are equal.

Let X be a vector field on M . There is a unique operator ιX : A(M)→ A(M)of degree −1 such that

ιX(ω) = ω(X)

for all ω ∈ A1(M) and

ιX(ω ∧ η) = (ιXω) ∧ η + (−1)|ω|ω ∧ (ιXη)

for all homogenous forms ω and η. The operator ιX is called contraction with thevector field X and one has the explicit formula

ιX(ω)(X1, . . . , Xk) = ω(X,X1, . . . Xk)

3. DIFFERENTIAL FORMS AND DE RHAM COHOMOLOGY 59

for ω ∈ Ak+1(M). In particular ιX(f) = 0 and ι2X = 0. In local coordinates theinterior product is given by

ιX(f(x) dxi0 ∧ · · · ∧dxik) =

k∑j=0

(−1)jX(xij )f(x) dxi0 ∧ · · · dxij−1∧dxij+1

· · · ∧dxik .

Finally we want to discuss the Lie derivative. If X is a vector field on M then theLie derivative LX : A(M)→ A(M) is the linear operator of degree zero defined by

LX(ω) =d

dτexp(τX)∗ω|τ=0

where exp(τX) denotes the flow of X. Since pull-back of differential forms com-mutes with the exterior differential one easily obtains

LXd = dLX .

Moreover the Lie derivative is an even derivation on A(M) in the sense that

LX(ω ∧ η) = LX(ω) ∧ η + ω ∧ LX(η)

for all ω, η ∈ A(M). An important relation between the operators LX , ιX and d isthe following Cartan homotopy formula.

Proposition 4.33. Let X be a vector field on M . Then

LX = dιX + ιXd

on A(M).

Proof. Since both sides define even derivations on A(M) it suffices to provethis formula in degree zero and one. In degree zero one has LX = ιXd and dιX = 0.Since d commutes with LX we also have LX(df) = (dιX + ιXd)(df) for all exactone-forms df . Hence the claim in degree one follows from the observation that lo-cally every element in A1(M) can be expressed as a sum of one-forms f0df1 withf0, f1 ∈ C∞(M). We now come to an explicit calculation that will be needed in the proof of theHochschild-Kostant-Rosenberg theorem. Let U ⊂ Rn be a convex open neighbor-hood of zero. The Euler vector field on U is defined by

E =

n∑j=1

xj∂

∂xj.

The corresponding flow ΦEt on U is given by

ΦEt (x) = exp(t)x.

For fixed x ∈ U we consider ΦEt (x) for all t such that exp(t)x is contained in U . Ina similar way we define another flow Φt on U by Φt(x) = tx. By definition, one hasΦEt (x) = Φexp(t)(x) provided t is small enough.

Proposition 4.34. Let U ⊂ Rn be a convex open neighborhood of zero. Then∫ 1

0

Φ∗tLE(ω)dt

t= ω − i∗ω

for all ω ∈ A(U) where i : U → U denotes the constant map with value 0.

Proof. We calculate∫ 1

0

Φ∗tLE(ω)dt

t=

∫ 0

−∞Φ∗exp(s)LE(ω)ds =

∫ 0

−∞(ΦEs )∗LE(ω)ds


using the coordinate change t = exp(s) and the relation between the flows Φ andΦE . By definition of the Lie derivative the last expression is equal to∫ 0

−∞(ΦEs )∗

d

dσ(ΦEσ )∗(ω)|σ=0ds =

∫ 0

−∞

d

dσ(ΦEσ )∗(ω)|σ=sds = ω − i∗ω

which yields the claim. Observe that i∗f = f(0) for f ∈ A0(U) and i∗ω = 0 for ω ∈ Ak(U) and k > 0.

4. The Hochschild-Kostant-Rosenberg theorem

In this section we formulate the Hochschild-Kostant-Rosenberg theorem de-scribing the Hochschild homology of the algebra of smooth functions on a manifoldM . We prove this theorem in the special case where M is a convex open neighbor-hood of zero in Rn.Let M be a smooth manifold. We view A(M) as a mixed complex with b-boundaryequal to zero and B-boundary equal to the exterior differential d. The Hochschild-Kostant-Rosenberg map α : Ω(C∞(M))→ A(M) is defined by

α(a0da1 · · · dan) =1

n!a0da1 ∧ · · · ∧ dan

on elementary tensors. It is easy to check that this formula induces a map on thecompleted tensor products used in the definition of Ω(C∞(M)).

Lemma 4.35. The Hochschild-Kostant-Rosenberg map α : Ω(C∞(M))→ A(M)is a map of mixed complexes.

Proof. We compute

αb(a0da1 · · · dan) =

n−1∑j=0

(−1)jα(a0da1 · · · d(ajaj+1) · · · dan)

+ (−1)nα(ana0da1 · · · dan−1)

=1

(n− 1)!

( n−1∑j=0

(−1)j a0da1 ∧ · · · ∧ d(ajaj+1) · · · ∧ dan

+ (−1)nana0da1 ∧ · · · ∧ dan−1

)= 0

using the Leibniz rule. Moreover we have

αB(a0da1 · · · dan) =

n∑j=0

(−1)njα(dan−j+1 · · · danda0 · · · dan−j)

=1

(n+ 1)!

n∑j=0

(−1)nj dan−j+1 ∧ · · · ∧ dan ∧ da0 ∧ · · · ∧ dan−j

=1

n!da0 ∧ · · · ∧ dan = dα(a0da1 · · · dan)

which shows that α commutes with the boundary operators as claimed. The goal is to show that this natural map induces an isomorphism in Hochschildhomology.

Theorem 4.36 (Hochschild-Kostant-Rosenberg). For every smooth manifoldM the Hochschild-Kostant-Rosenberg map

α : Ω(C∞(M))→ A(M)

induces an isomorphism in Hochschild homology.

4. THE HOCHSCHILD-KOSTANT-ROSENBERG THEOREM 61

The proof of theorem 4.36 is divided into several steps. We define a continuousmap β : C∞(M)n+1 → Ωn(C∞(M)) by

β(a0, a1, . . . , an) =∑σ∈Sn

(−1)sign(σ)a0daσ(1) · · · daσ(n)

where Sn is the symmetric group on n elements. A straightforward calculationshows that bβ(a0, . . . , an) = 0 for all a0, . . . , an ∈ C∞(M). Moreover we have

αβ(a0, a1, . . . , an) =∑σ∈Sn

(−1)sign(σ)α(a0daσ(1) · · · daσ(n)) = a0da1 ∧ · · · ∧ dan

and these relations imply that the map α : HH∗(C∞(M)) → A(M) is surjective.

Consequently, in order to prove theorem 4.36 it suffices to show that α is injective.We will first consider the special case where M is a convex open neighborhood ofzero in Rn. The general case will be treated in section 5.

Theorem 4.37. Let U ⊂ Rn be a convex open neighborhood of zero. TheHochschild-Kostant-Rosenberg map

α : Ω(C∞(U))→ A(U)

induces an isomorphism on the homology with respect to the Hochschild boundary.

Proof. We construct a projective resolution of the C∞(U)-bimodule C∞(U)as follows. Let Λk(Rn)∗ be the space of complex-valued alternating k-linear mapson Rn. We set

P k = C∞(U)⊗Λk(Rn)∗⊗C∞(U)

and equip this space with the obvious C∞(U)-bimodule structure

(f · ω · g)(x, z) = f(x)ω(x, z)g(z)

using the identification

C∞(U)⊗Λk(Rn)∗⊗C∞(U) ∼= C∞(U × U,Λk(Rn)∗).

The differential ∂ : P k+1 → P k is defined by

∂(ω)(x, z)(y1, . . . , yk) = ω(x, z)(z − x, y1, . . . , yk)

and it is clear that ∂2 = 0. If we let µ : P 0 → C∞(U) be the multiplication mapwe obtain a complex

C∞(U) ooµ

P0oo ∂

P1oo ∂

P2oo ∂

P3oo · · ·

which we will call the Koszul complex for U .We want to show that the Koszul complex is a projective resolution of C∞(U). Inorder to do this we need a different description of the differential ∂. Consider theisomorphism

P k ∼= C∞(U × U,Λk(Rn)∗) ∼= C∞(U,C∞(U,Λk(Rn)∗) = C∞(U,Ak(U))

given by γ(ω)(x)(z) = ω(x, z). Fix an element x ∈ U and consider the vector fieldXx on U defined by Xx(z) = z − x. If ι(x) : A(U) → A(U) denotes contractionwith Xx we have

∂(ω)(x) = ι(x)ω(x)

for all ω ∈ P k and k > 0. Let us also write L(x) for the Lie derivative with respectto Xx.The flow Φt(x) : Rn → Rn given by Φt(x)(z) = (1−t)x+tz preserves U for t ∈ [0, 1].In particular there are induced maps Φt(x)∗ : A(U) → A(U) for all t ∈ [0, 1]. Letus define h : P k → P k+1 by

h(ω)(x) =

∫ 1

0

Φt(x)∗(dω(x))dt

t


for ω ∈ C∞(U,Ak(U)) and k ≥ 0. In addition we define h : C∞(U)→ C∞(U ×U)by h(f)(x, z) = f(x). Let us check that the map h is well-defined. This is evidentin degree −1. Since the differential form dω(x) has degree at least one it is easy tocheck that, when viewed as a function of t, the integrand

1

tΦt(x)∗(dω(x))

is bounded on [0, 1] for every ω ∈ P k. For instance, if ω(x)(z) = f(z) is a functionon U we have

1

tΦt(x)∗(dω(x))(z) =

1

t

n∑j=1

∂f

∂zj((1− t)x+ tz)d(tzj) =

n∑j=1

∂f

∂zj((1− t)x+ tz)dzj .

Hence the integral defines indeed an element in P k+1. Moreover one checks that his a continuous map.

Proposition 4.38. The map h defines a contracting homotopy for the Koszulcomplex P .

Proof. Under the coordinate change ψx(z) = z−x we have Φt(x) = ψ−1x Φtψx

where Φt = Φt(0) is the flow considered in proposition 4.34 and Xx correspondsto the Euler vector field E. Using proposition 4.33 and proposition 4.34 we thuscompute

(h∂+∂h)(ω)(x) =

∫ 1

0

Φt(x)∗(dι(x)ω(x))dt

t+ ι(x)

∫ 1

0

Φt(x)∗(dω(x))dt

t

=

∫ 1

0

Φt(x)∗(dι(x) + ι(x)d)ω(x))dt

t

=

∫ 1

0

Φt(x)∗(L(x)ω(x))dt

t= ω(x)− i∗xω(x) = ω(x)

on P k for k > 0 where ix is the constant map with value x. In addition we have

(hµ+ ∂h)(f)(x, z) = f(x, x) + f(x, z)− f(x, x) = f(x, z)

on P 0 and (µh)(f)(x) = f(x) on C∞(U) which yields the claim. We may thus compute the Hochschild homology of C∞(U) using the resolution P .First observe that there are natural isomorphisms

C∞(U)⊗C∞(U×U)Pk = C∞(U)⊗C∞(U×U)C

∞(U × U)⊗Λk(Rn)∗

= C∞(U)⊗Λk(Rn)∗ ∼= Ak(U).

It remains to determine the boundary operators of this complex. Using the identi-fication

C∞(U)⊗C∞(U×U) Pk ∼= C∞(U)⊗Λk(Rn)∗

one sees that this map is given by restriction of the boundary operator ∂ to thediagonal ∆ in U × U , that is,

∂(ω)(x)(y1, . . . , yk) = ω(x)(x− x, y1, . . . , yk) = 0

for all ω ∈ C∞(U,Λk+1(Rn)∗). As a consequence we obtain an isomorphism

HHn(C∞(U)) ∼= An(U)

for all n. However, theorem 4.37 claims slightly more, namely, that this isomorphismmay be realized using the Hochschild-Kostant-Rosenberg map.From the general theory we know that there exists a C∞(U × U)-linear chainmap f : P → Bar(C∞(U)), unique up to homotopy, which induces the aboveisomorphism after tensoring over C∞(U × U) with C∞(U). Let us explicitly write

5. THE PROOF IN THE GENERAL CASE 63

down such a map. In degree zero we have P0 = Bar0(C∞(U)) = C∞(U × U) andwe let f0 be the identity map. Define fk : Pk → Bark(C∞(U)) for k > 0 by

fk(ω)(x, y1, . . . , yk, z) = ω(x, z)(Xy1(z), . . . , Xyk(z))

for all ω ∈ C∞(U × U,Λk(Rn)∗). We compute

b′f(ω)(x,y1, . . . , yk, z) = ω(x, z)(Xx(z), Xy1 , . . . , Xyk(z))

+

k∑j=1

(−1)jω(x, z)(Xy1(z), . . . , Xyj (z), Xyj (z), . . . , Xyk(z))

+ (−1)k+1ω(x, z)(Xy1(z), . . . , Xyk(z), Xz(z))

= ω(x, z)(Xx(z), Xy1(z), . . . , Xyk(z)) = f∂(ω)(x, y1, . . . , yk, z)

using that ω(x, z) is alternating and Xz(z) = 0. Hence f defines a chain map. Wemay also view an element ω ∈ C∞(U×U,Λk(Rn)∗) as a smooth function on U withvalues in A(U). Since U is an open subset of Rn such an element may be writtenin a unique way as a linear combination of terms of the form

η(x)(z) = a0(x, z)dzj1 ∧ · · · ∧ dzjk = a0da1 ∧ · · · ∧ dakwhere zj denotes the j-th component of z, a0 is a smooth function on U × U andai(z) = zji for i > 0. Now observe that for the function a given by a(z) = zj − xjfor some j we have

da(Xy)(x) = da(x)(x− y) = xj − yj = −a(y).

Moreover dzj = d(zj − xj) if xj is viewed as a constant function of the variable z.Hence for η in the form above we get

f(η)(x,y1, . . . , yk, x)

=∑σ∈Sk

(−1)sign(σ)a0(x, x)da1(x)(x− yσ(1)) · · · dak(x)(x− yσ(k))

=∑σ∈Sk

(−1)k(−1)sign(σ)a0(x, x)a1(yσ(1)) · · · ak(yσ(k)).

Consequently, the induced chain map F : A(U)→ C(C∞(U)) is given by

F (a0da1 ∧ · · · ∧ dak)(x0, . . . , xk) =∑σ∈Sk

(−1)sign(σ)+ka0(x0)aσ(1)(x1) · · · aσ(k)(xk)

and we obtain

αF (a0da1 ∧ · · · ∧ dak) = (−1)ka0da1 ∧ · · · ∧ dak.Since we know that F : A(U) → C(C∞(U)) is a quasiisomorphism it follows thatα induces an isomorphism HH∗(C

∞(U)) ∼= A(U). This finishes the proof of theHochschild-Kostant-Rosenberg theorem 4.37.

5. The proof in the general case

In section 4 we formulated the Hochschild-Kostant-Rosenberg theorem com-puting the Hochschild homology of C∞(M) and proved it in the special case whereM is a convex open neighborhood of zero in Rn. We shall now treat the generalcase of an arbitrary smooth manifold M . The idea is to reduce the problem toconvex open subsets of Rn by an appropriate localization procedure. We follow theproof of Teleman [11], [12].Choose a Riemannian metric on M and let d : M ×M → [0,∞) be the associateddistance function. For every k > 0 we let ρ : Mk+1 → [0,∞) be the smooth map

ρ(x0, . . . , xk) = d2(x0, x1) + d2(x1, x2) + · · ·+ d2(xk−1, xk) + d2(xk, x0)


which measures the square of the distance to the diagonal. Moreover we choose asmooth function λ : [0,∞)→ [0, 1] with support in [0, 1] which takes the value 1 onthe intervall [0, 1/2]. Let us define

ρt(x0, . . . , xk) = λ

(ρ(x0, . . . , xk)

t

)for every k > 0 and t > 0. In addition let ∆t be the set of all points in Mk+1

with ρ(x0, . . . , xk) ≤ t. We call ∆t the ((k + 1)-dimensional) t-neighborhood ofthe diagonal. By construction, the support of the function ρt is contained in thet-neighborhood of the diagonal.Let us identify Hochschild chains in Ck(C∞(M)) with smooth functions on Mk+1.If a Hochschild chain f ∈ Ck(C∞(M)) is zero on the (k + 1)-dimensional t-neighborhood of the diagonal, then the boundary b(f) ∈ Ck−1(C∞(M)) vanisheson the k-dimensional t-neighborhood of the diagonal. For t > 0 let C(C∞(M))t bethe subcomplex of C(C∞(M)) consisting in every dimension of all chains which arezero on ∆t. Moreover let C(C∞(M))0 be the union of the complexes C(C∞(M))tfor all t > 0. Then C(C∞(M))0 is a subcomplex of C(C∞(M)) and we obtain ashort exact sequence

C(C∞(M))0// // C(C∞(M)) // // C(C∞(M))∆

of complexes where C(C∞(M))∆ denotes the corresponding quotient complex. Wewill call C(C∞(M))∆ the complex of germs around the diagonal.For t > 0 we define an operator Et : C(C∞(M))→ C(C∞(M)) of degree one by

Et(f)(x0, . . . xk+1) = λ

(d2(x0, x1)

t

)f(x1, . . . , xk+1)

which has the following property.

Lemma 4.39. Let ε > 0. Then the support of Et(f) is contained in ∆3t+3ε

provided f is supported in ∆ε. Moreover, the operator Et maps C(C∞(M))ε intoC(C∞(M)) ε

3.

Proof. Assume ρ(x0, . . . , xk+1) > 3t+ 3ε and d2(x0, x1) ≤ t. By the triangleinequality we have

d2(xk+1, x0) ≤ d2(xk+1, x1) + 2d(xk+1, x1)d(x1, x0) + d2(x1, x0)

which implies

d2(xk+1, x0) ≤ 3d2(xk+1, x1) + 3d2(x1, x0).

Hence we have

ρ(x1, . . . , xk+1) = d2(x1, x2) + · · · d2(xk, xk+1) + d2(xk+1, x1)

≥ 1

3(d2(x1, x2) + · · · d2(xk, xk+1)) + d2(xk+1, x1)

≥ 1

3(d2(x1, x2) + · · · d2(xk, xk+1)) + d2(xk+1, x1) + d2(x1, x0)− t

≥ 1

3(d2(x1, x2) + · · · d2(xk, xk+1) + d2(xk+1, x0))− t

=1

3ρ(x0, . . . , xk+1)− t > t+ ε− t = ε

which yields the first claim. The estimate

d(xk+1, x1)2 ≤ d(xk+1, x0)2 + d(x0, x1)2 + 2d(xk+1, x0)d(x0, x1)

shows

ρ(x1, . . . , xk+1) ≤ ρ(x0, . . . , xk+1) + 2ρ(x0, . . . , xk+1)


which easily implies the second assertion. We define an operator Nt : C(C∞(M))→ C(C∞(M)) of degree zero by

Nt(f)(x0, . . . , xk) = (−1)kλ

(d2(x0, x1)

t

)(f(x1, . . . , xk, x0)− f(x1, . . . , xk, x1))

for t > 0. We can rewrite this as

Nt(f)(x0, . . . , xk) = (−1)k(Et(f)(x0, x1, . . . , xk, x0)− Et(f)(x0, x1, . . . , xk, x1))

using the map Et.

Lemma 4.40. The operator Nt is a chain map and we have

bEt + Etb = id−Ntfor every t > 0. The support of Nt(f) is contained in ∆3t+3ε provided f is supportedin ∆ε for some ε > 0 and Nt maps C(C∞(M))ε into C(C∞(M)) ε

3. Moreover

(Nt)k = 0 on Ck(C∞(M))(k+k2)t.

Proof. A straightforward calculation yields the relation bEt + Etb = id−Nt.This relation also shows that Nt is a chain map.Assume that f is supported in ∆ε. Then the first term in the definition of Ntis again supported in ∆ε. For the second term observe that the argument givenin lemma 4.39 shows ρ(x1, . . . , xk, x1) > ε provided ρ(x0, . . . , xk) > 3t + 3ε andd2(x0, x1) ≤ t. It follows that the second term is supported in ∆3t+3ε. Hence thesupport of Nt(f) is contained in ∆3t+3ε as well. The fact that Nt maps C(C∞(M))εinto C(C∞(M)) ε

3is proved in the same way as the corresponding assertion for Et.

From the explicit formla for Nt it follows that the operator (Nt)k is of the form

(Nt)k(f)(x0, x1, . . . , xk) =

k−1∏j=0

λ

(d2(xj , xj+1)

t

)F(f)(x0, x1, . . . , xk)

where F(f) is a linear combination of functions constructed out of f by permutationof the arguments and restriction to certain diagonal subsets. For the first factor inthis expression to be nonzero at (x0, . . . , xk) we necessarily have d2(xj , xj+1) < t

for 0 ≤ j < k. The triangle inequality implies d(x0, xk) < kt12 in this case. Hence

ρ(x0, . . . , xk) < kt + k2t at such a point. As a consequence we have (Nt)k(f) = 0

for f ∈ C(C∞(M))(k+k2)t. Lemma 4.40 implies the following result.

Proposition 4.41. The natural map HH∗(C∞(M)) → H∗(C(C∞(M))∆) is

an isomorphism.

Proof. It suffices to show that C(C∞(M))0 is acyclic. According to lemma4.39 and lemma 4.40 the operators Et and Nt define maps from C(C∞(M))0 intoC(C∞(M))0 and we have (Nt)

k = 0 on Ck(C∞(M))(k+k2)t. Since Nt is a chainmap this implies

id = id−(Nt)k =

k−1∑j=0

(Nt)j − (Nt)

j+1 = (id−Nt)k−1∑j=0

(Nt)j

=

k−1∑j=0

bEt(Nt)j + Et(Nt)

jb = bht + htb

on Ck(C∞(M))(k+k2)t where

ht =

k−1∑j=0

Et(Nt)j .


Applying this formula to a cycle f ∈ Ck(C∞(M))(k+k2)t yields [f ] = 0 for the corre-sponding homology class in Hk(C(C∞(M))0). Since every cycle f ∈ Ck(C∞(M))0

is contained in Ck(C∞(M))ε for some ε > 0 this yields the claim. The assertion of proposition 4.41 may be rephrased by saying that the homologyclass of a Hochschild cycle only depends on its germ around the diagonal. We shallrefine this statement and show that the homology class actually depends only onthe infinite jet at the diagonal.In order to make this precise we need some preparations. For a smooth functionh ∈ C∞(N) on a smooth manifold N let us consider locally the iterated partialdifferentials

∂|α|h

∂xα=

(∂

∂x1

)α1

· · ·(

∂

∂xp

)αp(h)

in some coordinate system (x1, . . . , xp) where α = (α1, . . . , αp) is a multiindex.Clearly these partial differentials depend on the choice of the coordinate system,but the statement that all iterated partial differentials are zero on a fixed subset ofN is independent of the coordinates.Let us call two Hochschild chains f and g in Ck(C∞(M)) = C∞(Mk+1) equivalentif all iterated partial derivatives of f − g vanish on the diagonal ∆ ⊂ Mk+1. Bydefinition, the space J∞C(C∞(M)) of infinite jets at the diagonal is the quotientof C(C∞(M)) under this equivalence relation. The infinite jet at the diagonalof a Hochschild chain f ∈ C∞(Mk+1) is the class of f in J∞C(C∞(M)). It iseasy to check that the Hochschild boundary induces a map b : J∞C(C∞(M)) →J∞C(C∞(M)). Hence J∞C(C∞(M)) is a complex in a natural way and the pro-jection J∞ : C(C∞(M))→ J∞C(C∞(M)) induces a short exact sequence

KC(C∞(M)) // // C(C∞(M)) // // J∞C(C∞(M))

of complexes where KC(C∞(M)) is the kernel of J∞. Note that we have canonicalchain maps C(C∞(M))0 → KC(C∞(M)) and C(C∞(M))∆ → J∞C(C∞(M)).

Proposition 4.42. The map J∞ : HH∗(C∞(M)) → H∗(J

∞C(C∞(M))) isan isomorphism.

Proof. It suffices to show that the complex KC(C∞(M)) is acyclic. Let[f ] ∈ Hk(KC(C∞(M))) be a homology class and set c = 2(k + k2). Accordingto proposition 4.41 we may assume that [f ] is represented by a cycle f which issupported in the c/2-neighborhood of the diagonal. Consider the chain ft given by

ft(x0, . . . , xk) =

(d

dτρcτ (x0, . . . , xk)

)|τ=t

f(x0, . . . , xk)

for t > 0. It is straightforward to check that ft is again a cycle. Moreover, thesupport of ft is contained in ∆ct \∆ct/2. Since ct/2 = (k + k2)t we may apply thehomotopy formula id = bht+htb obtained in proposition 4.41 to ft. If we integratefrom ε to 1 and use the assumption that f is supported in the c/2-neighborhood ofthe diagonal this yields

f − ρcεf = blε(f) + lεb(f)

for every ε such that 0 < ε < 1 where

lε(f) =

∫ 1

ε

ht(ft)dt.

Since f vanishes on the diagonal the limit

limε→0

ρcεf

exists pointwise and is zero.Using lemma 4.39, lemma 4.40 and the definition of ht we see that the support of


ht(ft) is contained in ∆rt \∆st for some positive constants s < r independent of tand f . In particular, the function ht(ft) vanishes on the diagonal for all t > 0. Itfollows that the limit

l(f) = limε→0

lε(f)

exists pointwise and defines a function on Mk+2 which is smooth outside the diag-onal ∆.Let us show that l(f) is in fact a smooth function on Mk+2. Fix a point y ∈ Mand choose a compact neighborhood K ⊂ M of y. We let Kk+2 ⊂ Mk+2 be thecorresponding neighborhood of y∆ = (y, . . . , y). Moreover let r be as above anddenote by µ the supremum norm of the derivative of λ. According to the chain ruleand the definition of ht we see that there exists a constant C > 0 such that

|ht(ft)(x)| ≤ C µ

t2sup

v∈Kk+1∩∆ct

|f(v)|

for all x ∈ Kk+2. Since the support of ht(ft) is contained in ∆rt we may assumet2 ≥ ρ(x)2/r2 and obtain

|ht(ft)(x)| ≤ C µc2

ρ(x)2sup

v∈Kk+1∩∆ct

|f(v)|

for all x ∈ Kk+2. After possibly shrinking K appropriately, we may apply theTaylor formula to f in a local coordinate system and obtain for every p > 0

f(v) =1

p!

∑|α|=p

(v − v∆)α∂|α|f

∂xα(v∆ + θ(v − v∆))

for v ∈ Kk+1 where θ is a real number between zero and one. Here v∆ denotesthe euclidean projection of v onto the diagonal and wα = wα

1

1 · · ·wαm

m for w =(w1, . . . , wm) and every multiindex α. Note that for the above description of f weuse that all partial derivatives of f vanish on the diagonal by assumption.Let s be chosen as above. In order to estimate |ht(ft)(x)| we may assume in additionst ≤ ρ(x) and obtain

supv∈Kk+1∩∆ct

|f(v)| ≤ 1

(2p)!

∑|α|=2p

supv∈Kk+1∩∆ct

∣∣∣∣∂|α|f∂xα(v)

∣∣∣∣|(v− v∆)α| ≤ cp tp ≤cpspρ(x)p

for every p and some constant cp. In particular, using this estimate for p = 3 andour previous considerations we compute for x ∈ Kk+2

|l(f)(x)| ≤ limε→0

∫ 1

ε

|ht(ft)(x)|dt ≤ R∫ 1

0

ρ(x)dt = Rρ(x)

for some constant R > 0. This shows that l(f) is continuous in a neighborhood ofy∆. Similarly, one sees that the partial derivatives of l(f) exist and are continuousfunctions vanishing on the diagonal. We deduce that l(f) is actually infinitely oftendifferentiable and that all higher partial derivatives of f at the diagonal ∆ are zero.Hence l(f) defines an element in KCk+1(C∞(M)).The remaining part of the proof is straightforward. The relation f = (bl + lb)(f)implies [f ] = 0. Hence the complex KC(C∞(M)) is acyclic. We will now finish the proof of theorem 4.36. Let (Uj)j∈J be a locally finite opencovering of M . Restriction of functions defines homomorphisms C∞(M)→ C∞(Uj)and chain maps pj : C(C∞(M)) → C(C∞(Uj)) for all j ∈ J . These maps deter-mine a chain map p from C(C∞(M)) into the direct product of the complexesC(C∞(Uj)).


Proposition 4.43. Let (Uj)j∈J be a locally finite open covering of M . Thenthe natural map

p : HH∗(C∞(M))→

∏j∈J

HH∗(C∞(Uj))

induced by restriction of functions is injective.

Proof. Let [z] ∈ HHk(C∞(M)) be a homology class such that p([z]) = 0.This means that pj([z]) = 0 in HHk(C∞(Uj)) for all j ∈ J . Let (χj)j∈J be apartitition of unity subordinate to the covering (Uj)j∈J of M . That is,

∑j∈J χj = 1

and the support Kj of χj is contained in Uj for all j. For every j we choose tj > 0

such that ∆tj ∩ (Kj ×Mk) is contained in Uk+1j . Since pj([z]) = 0 there exists

dj ∈ C∞(Uk+2j ) such that b(dj) = z|Uk+1

j. Define cj ∈ C∞c (Uk+2

j ) by

cj(x) = ηj(x)hj(x)dj(x)

where hj is a function in C∞c (Uk+2j ) such that hj = 1 on Kk+2

j and ηj ∈ C∞(Uk+2j )

is the pull-back of χj along the projection onto the first factor. Observe that∑j∈J ηjz = z and that the chains ηjz are again cycles. By construction, the

functions b(cj) and ηjz coincide on the set Kk+1j . Moreover cj can be viewed as a

chain in Ck+1(C∞(M)) if we extend it by zero outside Uk+2j . We define an element

c ∈ Ck+1(C∞(M)) by

c =∑j∈J

cj .

Note that this infinite sum is well-defined since locally only finitely many summandsare nonzero. By construction of c the element d = z − b(c) is contained in thekernel of J∞ : Ck(C∞(M)) → J∞Ck(C∞(M)). It follows that [J∞(d)] = 0 inHk(J∞C(C∞(M))). Since J∞ is a quasiisomorphism according to proposition4.42 we deduce [d] = 0. Hence [z] = [d] = 0 in HHk(C∞(M)) which yields theclaim. Choose a locally finite open covering (Uj)j∈J of M by coordinate domains suchthat all charts φi identify Ui with some convex open neighborhood of zero in Rn.Consider the commutative diagram

HH∗(C∞(M)) //

α

∏j∈J HH∗(C

∞(Uj))

∏α

Ω(M) // ∏

j∈J Ω(Uj)

where the horizontal maps are induced by restriction to the open sets Uj . Theupper horizontal arrow is injective by proposition 4.43. The right vertical arrowis an isomorphism according to theorem 4.37. Hence the left vertical arrow α isinjective. We have seen in section 4 that the map α : HH∗(C

∞(M)) → Ω(M) issurjective. Hence the Hochschild-Kostant-Rosenberg map for M is an isomorphism.This finishes the proof of the Hochschild-Kostant-Rosenberg theorem 4.36.

6. Cyclic homology and periodic cyclic homology

In this section we calculate the cyclic homology and periodic cyclic homology ofC∞(M). Using the Hochschild-Kostant-Rosenberg theorem 4.36 this is quite easy.We begin with cyclic homology.

7. THE CLASSICAL CHERN CHARACTER 69

Theorem 4.44. Let M be a smooth manifold. Then the cyclic homology ofC∞(M) is given by

HCn(C∞(M)) ∼= An(M)/dAn−1(M)⊕⊕j>0

Hn−2jdR (M).

Proof. According to theorem 4.36 and lemma 3.33 the cyclic homology ofC∞(M) is isomorphic to the cyclic homology of the mixed complex A(M) withb = 0 and B = d. The cyclic homology of this mixed complex is equal to the righthand side of the above formula. It is instructive to determine the explicit form of the maps S,B and I relatingHochschild and cyclic homology. For I and S this can immediately be read off fromthe mixed complex A(M). The map I : HHn(C∞(M)) → HCn(C∞(M)) is givenby the natural projection An(M)→ An(M)/dAn−1(M). The periodicity operatorS : HCn(C∞(M))→ HCn−2(C∞(M)) kills the first summand An(M)/dAn−1(M),is the obvious map Hn−2

dR (M) → An−2(M)/dAn−3(M) on the second componentand the identity on the remaining summands. Finally, for B : HCn(C∞(M)) →HHn+1(C∞(M)) we apply lemma 3.32 and obtain that this homomorphism can beidentified with the map d : An(M)/dAn−1(M)→ An+1(M).Let us now consider periodic cyclic homology.

Theorem 4.45. Let M be a smooth manifold. The periodic cyclic homology ofC∞(M) is given by

HP∗(C∞(M)) ∼=

⊕j∈Z

H∗+2jdR (M).

Proof. According to theorem 4.36 and proposition 3.55 the periodic cyclichomology of C∞(M) is isomorphic to the periodic cyclic homology of the mixedcomplex A(M). The latter is easily seen to be equal to the right hand side of theabove formula. As a consequence, one may view periodic cyclic homology as a noncommutativeanalogue of de Rham cohomology. Indeed, in the general framework of noncom-mutative geometry, cyclic homology plays a role similar to the one of de Rhamcohomology in differential geometry.

7. The classical Chern character

In this section we recall the classical Chern-Weil construction of the Cherncharacter and compare it with the noncommutative Chern character introducedin chapter 3. Throughout this section we assume that M is a compact smoothmanifold and that all modules over A = C∞(M) are unitary. Moreover, we tacitlyview A-modules as left, right or bimodules using that A is commutative.The K-group K0(M) of the manifold M is equal to K0(C∞(M)) provided M iscompact. According to the following classical result, the group K0(C∞(M)) may beviewed as the group of stable isomorphim classes of smooth complex vector bundlesover M .

Proposition 4.46 (Serre-Swan). Let M be a compact smooth manifold. Thenthe category of smooth complex vector bundles over M is equivalent to the categoryof finitely generated projective modules over C∞(M).

Proof. If V is a smooth vector bundle over M then the space C∞(M,V )of smooth sections of V becomes a unitary C∞(M)-module by pointwise multi-plication. Clearly every vector bundle morphism φ : V → W induces a modulehomomorphism C∞(M,φ) : C∞(M,V ) → C∞(M,W ). Since every vector bundleover M is a direct summand in a free bundle M × Cn for some n it is easily seenthat C∞(M,V ) is actually a finitely generated projective module.


Conversely, assume that the unitary projective C∞(M)-module P is representedas P = C∞(M)n · p ⊂ C∞(M)n for some idempotent matrix p ∈ Mn(C∞(M)).Define V ⊂ M × Cn by V =

⋃m∈M Vm where Vm is the image of the evaluation

map evm : C∞(M)n · p → Cn at m. If the dimension of the vector space Vm isk then there exists a small neighborhood U of m such that dim(Vx) ≥ k for allx ∈ U . Applying the same argument to the projective module corresponding to1− p we see that the dimension of the fibers is locally constant. Choosing elementse1, . . . , ek ∈ P such that around e1(m), . . . , ek(m) form a basis for V (m) yieldsa trivialization of V in a neighborhood of m. It follows that V defines indeed asmooth vector bundle over M . The module C∞(M,V ) of sections of this bundle isnaturally isomorphic to P . Recall that A = C∞(M) denotes the algebra of smooth functions on the manifoldM . We will write Ak(A) for the space Ak(M) of differential k-forms. This notationis motivated by the fact that parts of the discussion in the sequel may be general-ized to arbitrary commutative algebras. If B is a commutative algebra, one mayactually define a space Ak(B) of (commutative) differential k-forms over B. In thecase A = C∞(M) one reobtains the space of differential forms in the usual sense.Although we will not discuss this more general approach here, it is remarkable sinceit provides a very algebraic description of (ordinary) differential forms.Let P be an A-module. Then P⊗AA(A) is a graded vector space where the gradingis induced by the degree of a differential form. Recall that a linear map f : V →Wof graded vector spaces has degree k if f(Vn) ⊂Wn+k for all n.

Definition 4.47. Let P be an A-module. A connection on P is a linear map∇ : P ⊗A A(A)→ P ⊗A A(A) of degree 1 which satisfies

∇(sω) = ∇(s)ω + (−1)nsdω

for all s ∈ P ⊗A An(A) and ω ∈ A(A).

Here P ⊗A A(A) is viewed as a right A(A)-module in the obvious way. IfP = C∞(M,V ) for a complex vector bundle V over M we also say that ∇ is aconnection on V .Let us first show that connections exist for all finitely generated projective modules.According to proposition 4.46 this is equivalent to showing that every vector bundleover M admits a connection.

Proposition 4.48. Let P be a finitely generated projective A-module. Thenthere exists a connection on P .

Proof. If P = An is a free module of rank n we have P ⊗A A(A) = An ⊗AA(A) = A(A)n. In this case the map d⊕n defined by

d⊕n(ω1, . . . , ωn) = (dω1, . . . , dωn)

is a connection where d is the exterior derivative. In general, P is a direct summandof An for some n. Hence there exist A-module maps ι : P → An and π : An → Psuch that πι = id. We define a map ∇ : P ⊗AA(A)→ P ⊗AA(A) of degree 1 usingthe commutative diagram

An ⊗A A(A)d⊕n //

OO

ι⊗id

An ⊗A A(A)

π⊗id

P ⊗A A(A)

∇ // P ⊗A A(A)

It is straightforward to check that ∇ is indeed a connection. We shall now define the curvature of a connection.


Definition 4.49. Let ∇ : P ⊗A A(A) → P ⊗A A(A) be a connection on anA-module P . The curvature of ∇ is the linear map

∇∇ : P = P ⊗A A0(A)→ P ⊗A A2(A).

We will write R or R∇ for the curvature of a connection ∇.

Lemma 4.50. Let ∇ be a connection on the A-module P . Then the map ∇∇ :P ⊗A A(A) → P ⊗A A(A) is A(A)-linear. In particular, the curvature R of ∇ isan A-module map.

Proof. We compute

∇∇(sω) = ∇(∇(s)ω + (−1)nsdω)

= ∇∇(s)ω + (−1)n+1∇(s)dω + (−1)n∇(s)dω + s⊗ d2(ω) = ∇∇(s)ω

for s ∈ P⊗An(A) and ω ∈ A(A). This shows that ∇∇ is A(A)-linear. In particularR is A-linear. Let P be an A-module. Then there is a natural linear map Φ : EndA(P )⊗AA(A)→HomA(P, P ⊗A A(A)) defined by

Φ(φ⊗ ω)(s) = φ(s)⊗ ω.Observe that since A is commutative it does not matter if we view P as a left orright module and wether we use the A-module structure of EndA(P ) = HomA(P, P )coming from the first or second variable.

Proposition 4.51. Let P be a finitely generated projective A-module. Thenthe natural map

Φ : EndA(P )⊗A A(A)→ HomA(P, P ⊗A A(A))

is an isomorphism.

Proof. Let f1, . . . fn ∈ HomA(P,A) and p1, . . . , pn ∈ P be elements satsifyingthe conditions of the dual basis lemma 1.33. We define a map Ψ : HomA(P, P ⊗AA(A))→ EndA(P )⊗A A(A) by

Ψ(φ) =

n∑i,j=1

db(pi ⊗ fj)⊗ (fi ⊗ id)φ(pj).

Then one computes

ΦΨ(φ)(s) =∑

pifj(s)(fi ⊗ id)φ(pj) =∑

(db(pi ⊗ fi)⊗ id)φ(s) = φ(s)

andΨΦ(f ⊗ ω) =

∑db(pi ⊗ fj)⊗ fi(f(pj))ω = f ⊗ ω

using the dual basis lemma. Hence Ψ is inverse to the natural map Φ. Assume that ∇ is a connection on the finitely generated projective module P .Using proposition 4.51 we may define a linear map ad(∇) : EndA(P ) ⊗A A(A) →EndA(P )⊗A A(A) by

ad(∇)(α) = ∇α− (−1)|α|α∇for α ∈ HomA(P, P ⊗A A(A)) ∼= HomA(A)(P ⊗A A(A), P ⊗A A(A)) equipped withthe natural grading. To check that ad(∇)(α) is indeed A(A)-linear we compute

ad(∇)(α)(sω) = ∇α(sω)− (−1)|α|α∇(sω)

= ∇(α(s)ω)− (−1)|α|α(∇(s)ω)− (−1)n+|α|α(sdω)

= ∇α(s)ω + (−1)|α(s)|α(s)dω − (−1)|α|α∇(s)ω − (−1)|α(s)|α(s)dω

= ad(∇)(α)(s)ω


for a homogenous element α ∈ HomA(A)(P ⊗A A(A), P ⊗A A(A)) and elementss ∈ P ⊗A An(A), ω ∈ A(A).Moreover we may view the curvature R of ∇ as an element in EndA(P ) ⊗A A(A)using lemma 4.50 and proposition 4.51.

Lemma 4.52. The curvature R of the connection ∇ satisfies ad(∇)(R) = 0 inEndA(P )⊗A A(A).

Proof. Since R has degree 2 we compute

ad(∇)(R) = ∇R−R∇ = ∇∇2 −∇2∇ = ∇3 −∇3 = 0

in HomA(A)(P ⊗A A(A), P ⊗A A(A)) which proves the claim. The dual basis lemma 1.33 yields an isomorphism EndA(P ) ∼= P ⊗ P ∗ for everyfinitely generated projective module P . One may thus define a map tr : EndA(P )→A by tr(p ⊗ f) = f(p). It is easy to check that tr is indeed a trace on the algebraEndA(P ) and that it coincides with the natural trace on Mn(A) if P = An is freeof finite rank. Moreover tr : EndA(P )→ A is A-linear.

Lemma 4.53. Let P be a finitely generated projective A-module. Then there isa commutative diagram

EndA(P )⊗A A(A)ad(∇) //

tr⊗ id

EndA(P )⊗A A(A)

tr⊗ id

A(A)

d // A(A)

where d is the exterior differential.

Proof. Observe that the assertion holds for a direct sum P ⊕ Q iff it holdsfor P and Q. Thus it suffices to consider the case of a free module of finite rankwhich in turn reduces to the case P = A. Using that ∇ satisfies the Leibniz rulethe calculation

ad(∇)(Ω)(1) = ∇(ω)− (−1)nΩ∇(1) = ∇(1)ω + dω − (−1)nω∇(1) = dω

yields the claim where ω ∈ An(A) is identified with a right A(A)-linear map Ω :A(A)→ A(A) in the obvious way. Observe that EndA(P )⊗AA(A) is an algebra in a natural way. Moreover let ∇ bea connection on P with curvature R. Since Ak(A) = 0 for k > n = dim(M) and Ris homogenous of degree 2 the expression

exp(−R) =

∞∑j=0

(−1)jRj

j!

reduces to a finite sum and defines an element exp(−R) ∈ EndA(P )⊗AA(A). Usinglemma 4.52 and lemma 4.53 one obtains

d(tr(exp(−R)) = tr(ad(∇)(exp(−R))) = 0

for this element where we have written tr instead of tr⊗ id. It follows that

ch(P,∇) = tr(exp(−R))

defines a cohomology class in the even de Rham cohomology HevdR(M) of M . If

P = C∞(M,V ) for a complex vector bundle V over M we also write ch(V,∇)instead of ch(P,∇).Let us show that this cohomology class does not dependent on the choice of theconnection ∇.


Lemma 4.54. Let ∇0 and ∇1 be connections on a complex vector bundle V .Then

ch(V,∇0) = ch(V,∇1)

in H∗dR(M).

Proof. We denote by P = C∞(M,V ) the projective module correspondingto V . Consider the compact manifold M [0, 1] = M × [0, 1] and let A[0, 1] =C∞(M [0, 1]). There is an obvious homomorphism A → A[0, 1] induced by thecanonical projection M [0, 1] → M . Consider the A[0, 1]-module P [0, 1] = P ⊗AA[0, 1]. Geometrically, P [0, 1] corresponds to the pull-back bundle of V alongthe map M [0, 1] → M . Let us define a linear map ∇ : P [0, 1] → P [0, 1] ⊗A[0,1]

A(A[0, 1]) = P ⊗A A(A[0, 1]) by

∇(s⊗ f) = (1− t)∇0(sf(t)) + t∇1(sf(t)) + s∂f

∂tdt

for s ∈ P and f ∈ A[0, 1]. Here ∇i(sf(t)) is viewed as an element of P⊗AA(A[0, 1])using the natural map A(A)→ A(A[0, 1]). One has

∇(s⊗ fg)(t) = (1− t)(∇0(sf(t))g(t) + sf(t)dg(t))+

t(∇1(sf(t))g(t) + sf(t)dg(t)) + s∂(fg)

∂t(t)

= ∇(s⊗ f)(t)g(t) + sf(t)(dg)(t)

for all s ∈ P [0, 1] and f, g ∈ A[0, 1]. The map ∇ can be extended to a connection∇ : P [0, 1]⊗A[0,1] A(A[0, 1])→ P [0, 1]⊗A[0,1] A(A[0, 1]) using the Leibniz rule.Now let ιt : M →M [0, 1] be the inclusion of M into M× [0, 1] at the point t ∈ [0, 1].The image of ch(V [0, 1],∇) under the map H∗dR(M [0, 1]) → H∗dR(M) induced byιi is equal to ch(V,∇i) for i = 0, 1. According to proposition 4.32, that is, byhomotopy invariance of de Rham cohomology, the maps H∗dR(M [0, 1])→ H∗dR(M)induced by ι0 and ι1 are equal. Hence we obtain ch(V,∇0) = ch(V,∇1). We may now define the classical Chern character.

Definition 4.55. Let M be a compact manifold and let V be a complex vectorbundle over M . The (classical) Chern character of V is the cohomology class

ch(V ) ∈ HevdR(M)

defined as above using an arbitrary connection on V .

Lemma 4.56. Let V be a complex vector bundle over M determined by theidempotent e ∈Mn(A) according to C∞(M,V ) = eAn. Then

ch(V ) =

∞∑k=0

(−1)k

k!tr(e(dede)k)

in H∗dR(M).

Proof. The Levi-Civita connection ∇ : eAn → eAn ⊗A A(A) is computed by

∇(a1, . . . , an) = e · (da1, . . . , dan) =

(∑e1j1daj1 , . . . ,

∑enjndajn

)for a1, . . . , an ∈ Ane and e = (eij). It follows that the curvature of this connectionis given by

R(a1, . . . , an) =

(∑e1i1dei1j1daj1 , . . . ,

∑enindeinjndajn

).


We compute for every r∑i,j

erideijdaj =∑i,j,k

erideijdejkak + erideijejkdak

=∑i,j,k

erideijdejkak + erideikdak − erjdejkdak

which implies ∑i,j

erideijdaj =∑i,j,k

erideijdejkak

and thus

R(a1, . . . , an) =

(∑e1i1dei1j1dej1k1ak1 , . . . ,

∑enindeinjndejnknakn

).

If de ∈ Mn(A) ⊗A A(A) is the matrix with entries (deij) and tr denotes the tracemap the relation

(edede)k = e(dede)k

yields the assertion. The latter is easily proved by induction taking into accountthat e is idempotent.

Proposition 4.57. The classical Chern character determines an additive mapK0(M)→ Hev

dR(M).

Proof. Using that tr : EndA(P )→ A is invariant under conjugation one easilychecks that ch(P ) depends only on the isomorphism class of the finitely generatedprojective module P . The assertion that ch is additive with respect to direct sumsfollows easily from lemma 4.56 and the additivity of tr.

Proposition 4.58. Let M be a compact smooth manifold. Then there is acommutative diagram

K0(C∞(M))ch //

ch0

H∗dR(M)

HP0(C∞(M))α // H∗dR(M)

Hence the Chern character in cyclic homology coincides with the classical Cherncharacter.

Proof. It suffices to compare the images of an idempotent e ∈ Mn(A) underthe maps α ch0 and ch. Composition of the Chern character in cyclic homologywith the Hochschild-Kostant-Rosenberg map yields the class

α ch0(e) =

∞∑k=0

1

(2k)!(−1)k

(2k)!

k!tr

((e− 1

2

)(dede)k

)=

∞∑k=0

(−1)k1

k!tr(e(dede)k)

in H∗dR(M) where we use the fact that the differential form tr((dede)k) ∈ A2k(M)is closed. According to lemma 4.56 this is precisely the class defining the classicalChern character of e.

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Trans. Amer. Math. Soc. 102 (1962), 383 - 408[9] Loday, J-L., Cyclic Homology, Grundlehren der math. Wissenschaften 301, Springer, 1992

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