Cyclic Homology of Algebras 9971504685, 9971504707

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CYCLIC HOMOLOGY OF ALGEBRAS

CYCLIC HOMOLOGY OF ALGEBRAS

Peter Seibt CNRS, Centre de Physique Theorique Marseille, France

\bWorld Scientific Singapore 0 New Jersey 0 Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P.0. Box 128, Farrer Road, Singapore 9128 U. S. A. office: World Scientific Publishing Co., Inc. 687 Hartwell Street, Teaneck NJ 07666, USA

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CYCLlC l-lOMOLOGY 0F ALGEBRAS Copyright © 1987 by World Scientific Publishing Co Pte Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photo— copying, recording or any information storage and retrieval system now

known or to be invented, without written permission from the Publisher. ISBN 9971 -50-468 -5 9971 ~50-470-7 pbk

Printed in Singapore by Utopia Press.

To Sebastian, Julie

Anna and Willy

vii

TABLE OF CONTENTS

Introduction.

CYCLIC (CO)HOMOLOGYIAND HOCHSCHILD (CO)HOMOLOGY.

Chapter

|.1 |.1.1 |.1.2 |.1.3 |.1.4 |.2 I.2.1 l.2.2 |.2.3 l.2.4 |.2.5 |.2.6 |.2.7

Preliminaries: Spectral Sequences.

Filtered Complexes and Exact Couples. The Spectral Sequence associated with an Exact Couple. Convergence of 3 Spectral Sequence. Double Complexes and their Spectral Sequences.

Cyclic (Colhomology and Hochschild (Colhomology. The Double Complex C(A). The Cyclic Homology of an Associative Algebra. Generalities about Mixed Complexes. Cyclic Homology and Hochschild Homology. Nonunital and Reduced Cyclic Homology.

Cyclic Cohomology. Morita- Invariance of Hochschild Homology and of Cyclic Homology.

Comments on Chapter I. References to Chapter I.

Chapter ".1 ll.1.l l|.1.2

85

PARTICULARITIES lN CHARACTERISTIC ZERO.

86

Relation to de Rham Theory.

86 86

A First Approach: Noncommutative de Rham Complexes.

Cyclic Homology and de Rham Cohomology of Commutative Algebras.

104

viii

11.2.1

Preliminaries around Invariant Theory.

121 121

ll.2.2

Cyclic Homology and the Lie Algebra Homology of Matrices.

133

".2

Relation to Lie Theory.

Comments on Chapter II.

156

References to Chapter II.

157

Further references.

158

List of symbols and notations.

159

Index.

160

Introduction.

These lectures are an extended version of my contribution to a seminar

on cyclic cohomology, held at the University of Marseille Luminy, in 1985.

They are essentially based on a paper of J.L. Loday and D.

Quillen:

Cyclic homology and the Lie algebra homology of matrices

ment. Math. Helvetici, 59

(Com-

(1984), 565-591), and contain also ideas and

results of M. Karoubi and C. Kassel. The exposition is purely algebraic, according to my own background, and thus concentrates rather on cyclic homology (than on cohomology), the former being a more natural starting point for an algebraist.

But many

of the leading ideas of the theory, more apparent in cohomology, come from topology and differential geometry, in the language of operator algebras. cyclic

Thus it should be clear that this is ESE an introduction to

(co)homology, but only the attempt to single out the basic alge-

braic facts and techniques of the theory.

The reader who wants more

motivations should imperatively consult the fundamental article of Alain Connes: 62

Noncommutative differential geometry, I.H.E.S. Publ. Math. vol.

(1985), 41-144.

The lectures are organized in two chapters. The first chapter deals with the intimate relation of cyclic theory to ordinary Hochschild theory, which is at least not surprising by the parallel definition of both theories.

There are some important quasi-

isomorphisms, proving the equivalence of different approaches to cyclic

homology, and spectral sequence techniques areconvenientto establish these facts.

Thus a comforting paragraph on spectral sequences opens

the exposition.

Fortunately, I could already take in account the ex-

tremely elegant mixed complex approach to cyclic homologyofIL Burghelea which streamlines a lot of arguments.

The first climax is the funda-

mental long exact sequence

-- —> Hn(A) -> HCn(A) -> HCn_2(A) -> Hn_1(A) -> .. relating the Hochschild homology groups and the cyclic homology groups (analogously in cohomology), a cornerstone for all structural trans-

mission between both theories.

Normalized mixed Hochschild complexes

and reduced theory are treated in order to invest conveniently differential ideas:

Our operator

B

becomes a good candidate for a non-

commutative outer derivative.

Finally, Morita-invariance of Hochschild

homology and of cyclic homology are treated, following closely an exposition of K. Igusa. The second chapter deals with cyclic homology as a typical characteristic zero theory.

sidered.

First,

its relation to de Rham cohomology is con-

It comes out that (noncommutative) de Rham cohomology in the

sense of M. Karoubi can be embedded in (reduced) cyclic homology.

For

smooth commutative algebras this can be made more precise by a sort of inverse limit constellation, which is formulated via a decomposition of cyclic homology into ordinary de Rham cohomology: _

n

c(A) — n /dn

n—1

n-2

n—4

0 HDR (A) e HDR (A) e

This result of J.L. Loday and D. Quillen has a dual version in continuous cyclic cohomology, due to A. Connes

(with

A = C°(X,C), the m-alge-

bra of smooth complex-valued functions on acompact manifold

x).

The

final sections of the second chapter deal with cyclic homology as "additive K—theory", in the following sense: A

over a field

k

For an associative algebra

of characteristic zero, cyclic homology

HC,(A)

is, up to a dimension shift, isomorphic to the space of primitive elements

Prim H*(gl(A)) of the Lie algebra homology of

gl(A) = lim glr(AL

This result should be appreciated in the light of D. Quillen's "multiplicative"version: with of

Rational algebraic K-theory

Prim H*(GL(A),@), the primitive part of GL(A) = lim GLr(A).

sions:

K*(A) 0 Q

identifies

(discrete) group homology

I have to admit two important algebraic omis-

First, I did not treat products

I neglected a bit) and Kfinneth-formulas

(essential in cohomology, which (since I got afraid of the co-

structure invasion). Then, which is perhaps more serious, I did not treat the relations to algebraic K—theory via Chern-characters.

On a

certain level of arguments this motivates half of the existence of the theory ("create a range for invariants“), but whenever you make the first step towards topology and geometry ...

A few words about the use of spectral sequences in these lectures.

We

only need them in order to establish some fundamental quasi-isomorphisms, by an approximation argument. involved

calculations.

This could also be done via explicit and

But, since we aim at furnishing the necessary

material for further reading, there is no reason to avoid spectral se-

quence techniques (look at the literaturel).

At any rate, there is a

coherent approach to the basic skeleton of the theory, avoiding all

spectral sequence arguments.

You begin with I.2.3; then you define

cyclic homology of a unital associative k-algebra homology of the mixed complex

C(A)

(cf. I.2.4.2).

A

as the cyclic

You get all of the

xi material from I.2.4.6 to 1.2.4.12. (I.2.5.1 to 1.2.5.13).

Reduced theory remains unchanged

Cyclic cohomology is treated analogously.

As

to Morita-invariance, the spectral sequence argument in 1.2.7.9 is eas-

ily replaced by a direct reasoning. clean.

de Rham theory (11.1) is already

Thus you cover rapidly all of the basic material.

It should

be pointedout,however,that the equivalence of the different approaches to cyclic (co)homology is an essential part of its handiness. Finally, I would like to express my thanks to all those who helped me to finish these lectures:

first of all, to Daniel Kastler, whose stim-

ulating enthousiasm for the subject and clever support (on many levels) pushed me across this experience.

Then,

to Joachim Cuntz and Georges

Zeller—Meier, who taught me the essentials of the subject, and finally to Philippe Blanchard, Sergio Doplicher, Rudolph Hang and Daniel Testard, whose hospitality and interest at different stages of thewbrk I shall never forget. This paper was written while the author was guest of the Research Center Bie1efeld—Bochum—Stochastics

(31305) at the University of Bielefeld.

I would like to express my thanks for its kind hospitality.

Thanks

also to Mrs. Aoyama-Potthoff for the excellent and competent typewrit— ing.

Rome, May 1986

Peter Seibt

Chapter I.

Cxclic

gcgzhggologx 33d

Hochschild (cglhomolggx.

The fundamental result relating Hochschild and cyclic

(co)homology

splits in a spectral sequence formulation and a long exact sequence formulation. Spectral sequence techniques reveal essential, so we begin with an exposition of the relevant material about (a rather special type of) spectral sequences.

1.1

Preliminaries:

I.1.1

Filtered Complexes and Exact Couples.

Definition 1.1.1

unitary ring).

plexes of

C

Remark 1.1.2 C:

..

Let

C

be a chain complex

A filtration

such that

(c)p€Z

FP-1CCFPC

of

for all

(of left R-modules,

C

R

a

is a family of subcom-

p E 22.

More explicitely, the situation is as follows:

—'

Cn+1

u FPC:

Spectral sequences.

en

u

-- —» £4”a U

4

—~ FPCn U

—->

cn-1

_+

H

—+

--

u —» cn_1 U

FP'1C: -- —-» 155’"a —+ Fp'1cn -+ 1:"P'1cn_1 _. The arrows are the differentials Definition 1.1.3 D

and

E

(compatible with the inclusions).

An exact couple is a quintuple

' are bigraded (at least

E - )modules :

(D,E,a,B,y), where D = (Dp,q)p,q E 2’

E = (E ) and where a, B, Y are homomorphisms of bigraded q Pig 5 E modules such that the following diagram

E

is exact.

2 Remark 1.1.4 B

and

y

Let

(a,a'),

respectively.

(b,b')

and (c,c')

be the bidegrees of

a,

Then our exact couple consists of an infinity

of long exact sequences Y

a

#Ep-crq-C'

Dprq

B

—‘

Dp+a.q+a'-’ Ep+a+b.q+a'+b' _’

(1:0 q 18:8

(Y = Yp_c’q_cll

p+a , q+a I)

Conversely, any such family of long exact sequences defines an exact couple.

Proposition 1.1.5

Every filtration (FPC)p€2z

of a chain complex

C

defines an exact couple

D

a

-—-—-+

D

where

Y\ fl E

Proof.

For every

0 —»

FP'1c —>

p 6 u

a

is of bidegree

(1,-1)

8

is of bidegree

(o, o)

Y

is of bidegree

(-1,0).

there is an exact sequence of chain complexes

FPC —>

FpC/FP-1C —>

0

Consider now the long exact homology sequences

_§

Hp+q(F p-‘l C)

CI ——~

Hp+q_1 (FP’1 C) —. (The decomposition

filtration index

Hp+q(F p C)

Y Hp+q(FPC/Fp-1 C) —>

...

n = p+q

p

B —>

of the grading index

n

relative to the

will reveal pretty when dealing with spectral se-

quences of double complexes) a

is induced

6

is induced by natural surjection of chain complexes,

Y

is the connection homomorphism.

Define

D

Prq

EP I Q

by

inclusion of chain complexes,

H

P“!

H

P”;

(FPC) (FpC/Fp_1C),

p, q e z.

The long exact homology sequences can be rewritten as

..

_’ DP-1.q+1

__.°t

D

_L

plq

E

w;

__L,

D

p-1 .q —*

..

which establishes our exact couple. a

is of bidegree

(1,-1),

B is of bidegree

(0,0),

Y is of bidegree

(-1,0), as desired. 1.1.2

The Spectral Sequence associated wiuianExact Couple.

Construction 1.2.1

exact couple

The derived exact couple

(D2,E2,a2,82, yz)

of an

(D,E,a,B,Y) = (D1E11,B1.Y1)-

D——°‘—»D

Consider an exact couple

Y\/B E

with

a, B, Y of bidegrees

(1,-1),

(0,0),

We shall construct an exact couple

the derived exact couple of

a2

is of bidegree

(1,-1)

82

is of bidegree

(-1,1)

Y2

is of bidegree

(—1,0)

(a)

Definition of

Consider

.

d1

1

smce dp.q (-1,0), and

D2 -—--+ D2

(D,E,a,B,Y)

(as

a)

(as

y).

given by

such that

Y

d

1

= By.

B

we have :d1

EPA! _’ Dp—1.q _’ E p-1rq' d1d1 = 0 (since

32 = H(E,d1) = Ker d1/Im d1,

Y8 = 0).

i.e.

E2Prq = Ker d1q /Im :11P+1Iq for p,q e z. (b)

Definition of 2 D

=

In a,

.

1 e

respectively.

E2:

: E + E :

(-1,0)

D2: .

D

2 q

=

Im aP'1:q+1

c

D

Prq

.

is of bidegree

(e)

Definition of

a2, 82

and

D2

72:

a

2

D2

u/ég

Y;\\\ U

H

9

9

E2

(of (bidegree

82 : D2 + E2

(1,-1), as

is defined as follows:

2 Bp.q(“p-1.q+1(xp-1,q+ 1)) where

[---1

2 e E p-1,q+1

= [B p-1,q+1(xp-1,q+ 1)]

means residue class.

82

is well-defined:

*

B

p-1.q+1(xp-1.q+ 1) 1

a)

E Ker d1

p-1,q+1

since

_

dp-1,q+1 ' Bp-2,q+1Yp-1,q+1 as

“9-1.q+1(xp-1.q+1) = “9-1.q+1(yp-1,q+1 E Ker a

xp-1.q+1 ' yp-1.q+1 B

9-1.q+1(xp-1.q+1 )

E

B

)

implies hence

p-1,q+1 = I” Yp.q+1’

p-1,q+1(yp- 1,'_1+1)mod

Im d

1 q+1

32 is of bidegree (-1,1). yz : E2 + D2

is defined by

y:

Y:.q[zp.q] = vq(zp.q) E D:-1,q Y2 is well-defined: *

z

p.q

E Ker d1

p.q

,

d1

p.q

= B

y

p-1,q p.q

,

hence

Yp,q(zp,q) E Ker Bp-1,q = Im up-2,q+1 = D:_1’q

** zp.q 6 Im d;+1.q’

zp,q = 5p.qyp+1:q(up+1.q)

Yp,q(zp,q) = Yprq,q+1,q(up+1.q) = 0 y2

(d)

has bidegree

(-1,o).

Verification of exactness:

the“

(i)

Ker 62 = Im a2

(ii) Ker yz = In 82

(111)Ker dz = Im y2 The inclusions image c Kernel are trivial, since induced by

u, 8

and

y.

a2, 82

and

72

are

We have to show the reverse inclusions.

For notational convenience we shall suppress indices (1)

Ker

82 c Im uz:

x E Ker 82

c

D2 = Im a

can be written as

= o, i.e. 8(u) 6 Im d1. E Ker B = Im a = D2.

x = a(u), and

82(x) = [Bu]

There is w e E: 8(u) = By(w), hence u - y(w)

But

u2(u-y(w)) = u(u) - ay(w) = a(u) = x, i.e.

x 6 Im a2.

(ii)

Ker 72 c Im 62:

Consider

write (iii) For

x = [z] 6 B2

2 = 8(w).

y2(x) = y(2) = O.

z E Ker y = In 6;

x = [z] = [8(W)]= 82(u(w)) 6 Im 82.

Ker a2 c In 72: x 6 D2 = Im a

such that

c2(x) = u(x) = O

y E E

x = 7(y).

Ker u, i.e. there is 6 Im 72 hence

provided that

with

y E Ker d1 = Ker By.

we have:

We will have But

x E Im y =

x = y2[y]

x 6 Im a = Ker 6,

By(y) = 8(x) = 0.

Example 1.2.2

y)

such that

Then

Let

C

be a filtered chain complex, and let

(D,E,a,B,

be the exact couple associated with the filtration (FPC) 2 1 1 We s h all determine Ep,q = Ker dp’q/Im. dp+1,q . Cons i der

C.

Z

of

Hp+q(F p-1 C)-—* Hp+q(F p C) —> Hp+q(F p C/F p-1 C) — Hp+q_1(F p-1 C)

D

p-1Iq+1

where

__si

D

u

u

u

n

__E,

p.q

Yp,q : EP'q + D94,q

E

prq

.19

D

is the connecting homomorphism (cf.

We haveexact sequences of chain complexes

0 +

FP-1C/Fp-2C

+ FPC/FP'ZC

+

FPC/FP'1C

giving rise to connecting homomorphisms

a p+q :

Hp+q(F

Pc /F p“c )

*

p-1.q

Hp+q-1(

F

9" C/F P'2 C) .

+

0

1.1.5).

1 dp,q . Eq + Ep-1,q

Let us show that

. . . identifies with

ap+q.

To see

this consider the commutative diagram of homomophisms of chain complexes

0 —» FP'1C

——>

FPc

u

1

l o -»

—» FPC/FP'1C —> o

FP'1C/Fp'2c-—+

with exact rows.

FPC/FP'ZC——»

FPC/FP‘1C —+

o

We obtain the following commutative diagram relating

the two long exact homology sequences in question:

“I'm? ‘0 —°‘-» HP+q(FPC) —L up«ape/PP ‘m—L HP+q_1

D(C)

\/ f1“

D(D)

-——————»

D(D)

E(D) where

f(1)

and

f

1

are induced by

f

in homology.

It is immediate that we obtain thus a commutative diagram of the same kind between the derived couples by

f‘1)

and

f1, hence by

The functoriality of

Er

(where now

f(2)

and

f2

are induced

f); iteration gives all we want.

(arrows of filtered chain complexes + arrows

of bigraded differential modules) follows from the functoriality of

homology. Theorem 1.3.7 Let

(Approximation theorem)

f : C + D

be a homomorphism of filtered chain complexes, both with

bounded filtrations. Ifthereexists an Et(D)

is an isomorphism, then

H(C) + H(D)

f

r 2 1

such that

fr : Er(C) +

is a quasi-isomorphism (i.e. H(f):

is an isomorphism).

Proof. (a)

Recall the proof of 1.3.5

r

Dp+r_2'q_r+2 = Im(H

= Im(I-lp r Dp+r-1,q—r+1

Er

P"!

(2)(ii)

q(F P-1 C) + Hp+q(F p+r-2 C)) (F P C) + Hp+q(F p+r-1 C))

= 1mmP+q(FpC) -> HP+q'(FP+r'1C/FP 1cm

(which follows from the definition of the derived exact couples),

the arrows r Dp+r-2,q-r+2

at

r

r _Q* r Dp+r-1,q-r+1 Ep,q

where

11 are the obvious ones. For large

r (depending on p,q) we obtained the exact sequence

-1 i P °° o a ¢p HP"qm) ¢ Hp+q(C) + E p.q + o wher

Eon

e

= I

pyq

¢PHP+q/¢p_1fl

H

m‘ P+q (

F P C ) + H P+q( C / F P-1 C )) .

Th e 1 scmorphi smi

EP'q a

is thus induced by the commutative diagram (with exact

P+q

row)

Hp+q(

F pc

)

/1\

H

(b)(i)

p+q(

We have

duced by

F P-1 C

-——* H

r

_

(p(D))

pea

(ii)

)

f;’q: E;’q(C) + E;,q(0) inr

respects the filtrations

0n the

(¢PH(C))PER

(and the gradings), thus induces

P+q

with

(C)/¢ 9-1 Hp+q(C) , w pHp+q(D)/w p-1 HP+q (0)

By virtue of part (a) of our proof we obtain a commutative diagram

f

w Ep,q(C) I

("everythingis induced by Conclusion: for

If

s z r, hence

gr¢H(D)

fr

_

f

w Ep’q(0)

in homology“).

is an isomorphism, then f”

fS

is an isomorphism

is an isomorphism, i.e. gr(H(f)): gr¢H(C) +

is an isomorphism.

We have to show that (iii)

p,q

(H(f)) " ___Ef______Jaua_. w Plq(v)/w 9-1 Hp+q(0)

¢ PHp+q(C)/¢ P-1 Hp+q(C)

H(f): H(C) + H(D)

is an isomorphism.

The boundedness of the filtrations on

H(C)

the desired conclusion: H(f)

C F 9-1 C

p+q( /

fq . Ep’q(C) + Ep.q(v)' r 2 r°(p,q)).

gr(H(f)): gr¢H(C) + gr¢H(D)

gr(H(f))P'q.. ¢ PH

——+ H

with

r

other hand, H(f) : H(C) + 3(9) and

C

p+q( )

f” : E“(C) + 2°(v)

(actually

f

)

is injective since

gr(H(f))

is injective.

and

H(D)

now give

12 H(f)

is surjective since

gr(H(f))

is surjective.

Let us write

h = H(f), H = gr(H(f))-

Chcase

x

f it rs

61’ ¢ Hp+q( c )

P P'1 HP+q(D), in wHP+q(1))/-y Now iterate:

suc h that 1h(x) i.e. x 6 ¢

x E ¢p-KHp+q(C)

p-

for all

=. 0

Hp+q(C) K 2 0,

'= 0 E§’q(x)

Then .

(H

.

being injective).

hence

x = 0

(boundedness

of the ¢-fi1tration).

Take now

y E wPHP+q(D).

We have to find

x E Hp+q(c)

such that

h(x)

=Y‘

By

the

't'it surjec 1v y of

_t h here

-

. You will obtain

Hp+q(v). .

t

is con aine

d

.

in

¢

P

-2

HP

Iterating, you obtain

x1 6 ¢ (

D) .

xx 6 ¢

xo 6 ¢ PHp+q( c )

is

. = p-1 y, i.e. y h(x°)mod w Hp+q(D).

Continue with p-1

Hp+q(C)

‘ With

Y1

=

such that

_ '= hp,q(xo)

_

y h(xo) E w

p-1_

y2 _ — y1 _ h(x1)

K

p-

Hp+q(C)

such that

yK _: h(xK)mod w P-K-1 Hp+q(9). The boundedness of the w—filtration yields some

5 2 O

Then:

s y = h(i:oxi) = h(x°) + h(x1) +---+ h(xs).

1.1.4

Double Complexes and their Spectral Sequences.

Definition 1.4.1 (

)

“pn: 9,q

A double complex is a triple

is a bigraded module and where

with

Ys+1 =

(M,d',d“), when

d', d": M + M

M =

are homo-

morphisms such that (i)

d'

has bidegree

(-1.0), d'od' = 0

(ii)

d"

has bidegree

(0,-1), d"od" = 0

(iii) d'od" + d"od' = 0

Remark 1.4.2 —-——

d'p.q P,q

Every row

MP *

Mp,q + Mp-Lq M

M

pa + plq~1

gives rise to a chain complex

I

column (M,d")

M

(MP *,d; ‘), and every I

gives rise to a chain complex

* q are chain complexes

Definition 1.4.3

Let

I

,d' ). (M,d') *Iq ’,q (forget one of the differentials).

(M,d',d")

(M

be a double complex.

and

(Tot(M),d), the

13 (M,d',d")

(chain) complex of

total

' (1)

To t ( M )n = p+q=n 0 MP’q.

(ii)

an : Tot(M)n + Tot(M)n_1

is given by

n E 22

d M = d' + d" n/ p.q p.q p.q Remark 1.4.4 =

u

dn-1°dn d'

+

0’ dp-1.q° plq

Example 1.4.5

=

by Virtue of

O

I

all

=

dp.q-1° prq

Let

=

II II O, dp,q—1°dp,q

0

M = (MP q)p qEZZ I

M -> M

I l dp-1,q°dp,q

be a bigraded module, 6', d":

I

two differentials of bidegree

(-1,0)

and

(0,-1)

respectively

which commute. Define

3" = Fig

(-1)pd" . Prq

Definition 1.4.6

Let

Then

(M,d',d")

the second filtration of

(IFPTot(M))n =

0

Mn—j j

For every

Mi n-i

be a double complex.

The first and

Tot(M):

o M.1,n-i jsp

Let

isadouble complex.

15p

(:l:l:1"pTot(M))n =

Remark 1.4.7

(M,d',3")

(Attention: p

restricts the second index!)

’ p E Z,

be a component of

IFpTot(M)

is a subcomplex of

(IFPTot(M))n, La.

1 s p.

Tot(M).

Then

I

ai,n-i

=

I

II

di,n-i“i,n-i + di,n-iMi,n-i C Mi-1,n-i ° M1,n-i—1

i'e'

ai,n-i c ( I F P Tot(M))n_1. Anologously: Lemma 1.4.8

11FPTot (M) Let

( IFPT°t(M))P€E Z and

is a subcomplex of

(M,d',d")

Tot (M) .

be a double complex.

(IFPTot(M))p€2z

are bounded filtrations if and only if for every

there is only a finite number of

(p,q) E E2

such that

Trivial .

Remark—Corollary 1.4.9

Let

(M,d',d")

n E

p+q = n

M O. P"! 7‘

Proof .

and

be a double complex of the

first or of the third quadrant (obvious vanishing conditions on the

14 I r

I

(

and

( E )r21

“p,q)’ and let

E r)r21

be the spectral sequences

determined by the first and second filtration on (1)

IE“

= IEr

q

for

q

r

Tot(M).

Then we have

sufficiently large(depending on

p,q).

I 2 E Fig n 2 E Fig

Proof.

U‘

(2)

Hn(T0t(M))

WI

Analogous statement for the second spectral sequence.

Hn(Tot(M))

Immediate by 1.3.5 and 1.4.8.

Remark 1.4.10

We want to determine

IE: q (and

IE: q)'

I

the first filtration.

WehaeE V

B t u

Pig

=H

P+q(

Write

Let us attack

I

T = Tot(M), and drop the upper-I-index.

—1 FPTFPT. /

)

_ G M. _ M F p T F 9-1 T — 0 M — . ( / )p+q iSP 1:P+Q'1/isp_1 i,p+q-i Prq

Thus

FPT/Fp-1T

is the pth row

MP l I

d; * (actua11y, the differential of

of

M, with differential

FpT/FP-1T is induced by

d" =

d = d'+d",

I

but

d'

goes vertically, i.e.

zero~homomorphism on Finally: (where

E H"

plq

Hll

p.q

with differential

d'

induces the

(M)

denotes the homology of the chain complex

...+—— H" I,_1’q( M )

.q [ zplq ]

Fp—1T, hence

FpT/FP'1T).

= Ker d" /Im d" = q p,q+1

For every q E Z

5'

d'FPTc

(M,d")).

we get a chain complex

‘— lq( ) +— H II

3'

M

induced by

+—...

M

II

P+1:q(

)

d':

= [d'

p,qlq]

(this is well-defined,

since

d'od" = -d"od').

when passing to homology, we obtain a bigraded module

(Han; q(M))P qez I

associated with the double complex Proggsition 1 .4. 11 Proo f.

In the situation 1 . 4 . 10 we have

We a 1 tea d y R now th a t

I

(M,d',d").

E Pig = Hp,q( “ M ),

I 2

y

u Ep,q = HPHPrq( M) . 2

and that EPrq =

15

1 1 , Ker dp.q/Im dp+1.q

d1p,q : E p,q + E P_1’q

where

i den t ifies w ith th e

connecting homomorphism

aplq

: H"

M

"

of the long exact homology sequence associ-

P.q( ) + H9-1.q (M)

ated with

o-——+ FP'1T/FP'2T-——+ FPT/Fp_2T ——» FPT/FP'1T ——+ o The explicit description of

3P q

is as follows: Look at our exact

I

sequence of chain complexes in degree M

P'1rq

Let

z E Mp,q

E H;_1 q(M)'

——L- M

represent

and ‘IT

p,q

o Mp-1,q+1 I

o -—» M

n = p+q

P'1Iq

——» M

p.q

n-1 = p+q-1: -—»

0

dl+dll

e M

Pig-1

x E H;,q(M)'

We have to identify

8P'q(x)

By definition of the connecting homomorphism you have to

I

choose

(2,0) 6 n‘1(z), and to pass to

i'1d(z,o) = d'(z)

(d"(z) = 01)

3P'q(x) = [d'z] E H;_1'q(M).

Thus dglq = ap,q = 3'

(1.4.10), i.e.

E;,q = Ker d;’q/Im d;+1,q = Héflg’q(M), as desired. Corollary 1.4.12 HEHé'P(M) grggf.

(M,d',d“)

Reduction to the first filtration case.

double complex

Mtq = Mqrp , Then

For a double complex

we have

nEg’q =

(Attention: look at the subscripts!)

Tot(Mt)

We have thus

Define the transposed

(Mt,A',A") by:

A'q = d"‘LP , = Tot(M),

I t 2 E

q

A“1’"! = d'CL? ,

with the same differential.

=

H'H"

P q(

u

u

122

H"H'

Prq

P QrP

M

t

(M)

)

16 Notational convention: Let

M

be a double

(chain) complex of the third

quadrant.

Set

Mp,q := M'P -q’ p 2 0, q 2 0, analogously for the differentials. I

BE"! := Efp’_q, dr : Er + Er decreasing:

p 2 o, q 2 o,

has now bidegree Fp+1 c F I

r 21 (r,1-r).

All filtrations now become

We thus may treat spectral sequences of double

cochain complexes of the first quadrant as double chain complexes of the quadrant. Propgsition 1.4.13

Let

(Er,dr)r21

pose

E: q = O

Then

w _ 2 Ep,q EPIC!

for

be one of the two spectral se-

of the first or third quadrant, and sup-

M

quences ofadouble complex

q f 0 (the spectral sequence "degenerates").

I

Proof .

= 0

dr = 0 Prq

for

r 2 2

and thus M

2

and all

or Er = 0), hence p—r,q+r-1

r 2 2 Let

(p,q) E Z

and

Hn(Tot(M))

M.

2 En,o’ n 6 22.

(p,q) E 32

(since either

Er = Ker dr = Ker dr/Im dr = Er+1

Er q for

E” = E2.

now be of the first quadrant, and let

complex of

=

r r r aplq : E q + E p-r,q+r-1

Consider

we have

for all

T = Tot(M)

be the total

Then we have for either filtration:

(Mo,*'d3,*’

for the first filtration

FOT = POT/F-1T= (M

(F n T)n

=

dl

*,o' 1,0

)

for the second filtration

Tn.

Recall that

¢pfin(T) = Im(Hn(FpT) + Hn(T)).

Hence we obtain the finite

chain o _ — ¢ -1 Hn(T) c---c ¢ n Hn(T) _ — Hn(T) (since

Hn(FnT) + Hn(T)

We know already that But only 3: 0

is surjective).

E2

q

a ¢pH (T)/¢P-1H (T), n

n

does not necessarily vanish.

(n = p+qL

This implies

¢pHn(T) =

I

op-1Hn(T) = O

for

p < n, and consequently

This third-quadrant case is treated similarly.

E:,o = ¢a(T) = Hn(T).

17 1.2

Cyclic

1.2.1 Let

(co)homology and Hochschild (co)homology.

The double complex k

C(A).

be a unitary commutative ring, A an associative k-algebra (with

unit), Ae = A a AOP

the enveloping algebra of

the opposite algebra of that an A-A-bimodule

A, where

A (with multiplication

AOp

means

a°b° = (ba)°).

Note

M (mixed associativity‘forthe left and right ac-

tions, symmetric action of

k)

is equivalently a left or right Ae-mod-

ule by the formulas

(a e b°)m = (am)b = a(mb) = m(b a a°). In particular, A

is naturally a left Ae-module, and the mapping

3 a 9 b0 » ab 6 A

Notation:

Ae

is an Ae-epimorphism.

A“ := A o A e---e A k

k

(n

times), n 2 1.

k

(a1,---,an):= a1 8---@ an We shall consider every way:

A“, n 2 1, as an A-A-bimodule in the following

a(a1,---,an)b = (aa1,az,---,an_1,anb)

(left and right action on

the external factors). Definition 2.1.1

The operators n-1

b'(a°.~'°,an)

1

n + A .

1

i:o(-1)

n-1 b(aot"'ran)

b',b: AP+

(aor"'laiai+1l'°'lan)

i

iio('1)

n (aor"‘raiai+1l"'ran)

+

('1)

(aha OI...Ian_1)

Remark 2.1.2 (1)

. The chain complex

of Ae-homomorphisms

A?),

since

b' 3 b' 2 b' ——» A ——+ A ——+ A

(the standard Hochschild resolution of

s : An + An+1, defined by

is a homotopy operator (satisfying When

A

is flat over

Notation:

(2)

(A

t+1

,b'),

The chain complex

by Ae + A.

A

over

s(a1,---,an) = (1.al."‘.8n):

b's + sb' = id).

(cf.

k, we get an Ae—flat resolution of

[C.E., p.1741L A.

the acyclic Hochschild complex.

-2» A3 —2» A2 -2# A

morphisms, which may be identified with More explicitely:

. . is an acyclic complex

(A

is a complex of k-homo-

9 A‘+2, 1 0 b'), augmented

he

(A 0 Aall+2)

___ A 0

An+2 = A

e

A

A

e

8 Ae

n

n

n+1

with the identification :

a

O Ae

(30,81,---,an,an+1) = (an+1aao) 8 (a1,---,an) k

(which gives immediately Notation:

(A‘+1,b),

b = 1 0 b').

the Hochschild complex.

H, (A) = H(A“+1,b), the Hochschild homology of A. Note that Hn(A)

is a subquotient of A“ 1 .

particular, when

k

Remark 2.1.3

e Hn(A) = Tor: (A,A), n 2 0.

is a field), we get

Homology of finite cyclic groups.

clic group of order

with generator

Let

Gn

be "the" cy-

n (think multiplicatively: nth roots of unity),

t = tn.

the group algebra of

There are two distinguished elements in

Gn

over

ZIGn] ,

Z:

N=1+t+t 2 +---+t“'1

D=1—t,

e D zIGnlt— N Monk—D z 4—Z[Gn]‘—

(1) is a

Z[Gn]-free resolution of

Z

(where

n-1 11-1 e( E ziti) = E zi, and where i=0 i=0

D, N).

(cf.

(2)

When A is k-flat (in

[C.E.,

D, N

means multiplication by

p.251] or [R0, p.296])

For a left Gn-module

M

we set

Z[Gn]

Hm(Gn,M) = Torm Hm(Gn,M)

ma]

is thus the mth homology group of the chain complex

0

“ ZIG] n

MfiLmG]

e

“ me] n

M (3)

(2,14), m z o.

D

twig-male

M

n me] n

N

For an associative k-algebra

A

M

—

( - 1)

n-1

(anla1l

I.-

D

we shall always consider

a left Gn-module by letting the generator

t-(a1r'°'lan)

144%

Ian-1).

t

An

act as the operator

as

19 Suppose now

Q c k, and let

sists of the cyclic

(n = %N

on

A:

be the Gn-submodul of

(i.e. Gn-invariant) tensors.

is the projection on

fl' = 1 - %N

A2, and

A“

which con-

Then An = A: 0 DAn

is the projection

DAn = (1-t)An).

Definition 2.1.4 Let

The double complex

C(A) = (cp,q)p,q20

first quadrant

C(A)2 *:

C(A).

be the following double chain complex of the

(with differentials as indicated):

A3 (BM,d)

0

—>

(N,b) —->

1"

—»

(BM[2] ,d[2]) -—-v 0

—->

(BN[2] ,d[2]) ——v 0

13”

13““

(BN,d)

and thus to a commutative diagram relating the long exact sequences

—» anon —I—+ cwn —S—> c_2 (M) —B-» un_1(m JHn(F)

JHCn(F)

l

—

j

—» Hn(N) —I-. acnm) —S—» c_2 (N) —-B—+ Hn_1(N) We want to discuss under which milder assumptions of

F

— we still obtain

such a diagram.

Definition 2.3.12

Let

(M,b,B)

and

(N,b,B)

be two mixed complexes

(d.g. A—modules). A stronglx homotopx A—map from graded maps

G 1 : M + N i.e.

M

to

of degree 6(0)

N 2i

(1)

G(°)b = bG(°),

(2)

c‘i)3 + G(i+1)b = so”) + bow”)

M

\\

G(1)

\

\

\

\\

G(0)

G \

\\

to

(N,b)

for all i 2 o. G(°)

and

G(1)

‘\\ \\\

\

Mn

of

. G(O)

\\\

As maps from

(M,b)

(6(1))izo such that

(1)Mn+1

\\\

N

i 2 0

is a morphism from

Visualization of the connection between

n-l

is a sequence for all

\

‘

b

Nn+1

'—’ to

Nn+1

\

5

Nn+2

~—

the two following maps are equal:

G(°)B + 6(1)}: = BG(°) + be”) Proposition 2.3.13

Let

(M,b,B)

and

(N,b,B)

be two mixed complexes;

assume that there exists a strongly homotopy A-map

N.

Then there exists a map of complexes

ing diagram is commutative

G: BM + BN

(Gun)120 from M to

such that follow-

31

o —> (M.b) — (Ema) —— (BM[2].d[2]) —— 0 16(0) 0

—’

(Nib)

JG —’

(BN'd)

[G[2] _’

Proof.

Look at the situation in degree

(BM)n

=

Mn

9

G (o)

Mn_2

e

(1)

Mn_4

a)

Nn_4

a

(BN[2] rd[2])

(2)

=

Define

_ (o) +G (1) +...+G (i) , Gn|M_21—G

0

Nn_2

The compatibility of

0

n:

(3N)n

Nfl

—’

G

e

(o)

with the operators

n 051s[§]. b, and the relations

c (0) 3+G”)b=BG (o) + m G(1)B + 6(2)]: = BG”) + bG(2)

G(i)B+G

(1+1)b

= 36(1) + bG

(i+1)

yield immediately the commutation of d = B+b (on Hence

and

G

G[2]

B“

and on

G

with the total differentials

BN).

is a complex homomorphism, and trivially compatible with G(°)

as asserted.

Conseguence 2.3.14

In the situation of 2.3.13 we have a commutative

diagram

—> anon

—Ia HCn(M) —S> c_2(m L Hn_1(m —

1 an“: (o) ) ——> Hn(N)

l ncnm)

l

l

I s s — acnm) — c_2(N)——» Hn_1 (N) —

Proposition 2.3.15

In the situation of 2.3.13 the following holds:

32 G

(o) : M + N

G

: BM +BN

is a quasi-isomorphism if and only if is a quasi-isomorphism

Proof.

Recall the five-lemma (cf.

Lemma:

Given a diagram with exact rows

M1

M2

Fl

M3

lfz

N1

[Bou, Ax.7]):

M4

F3

v N2

5

[£4

a N3

[£5

N4

N5

we have (1)

f2, f4

injective, f 1

(11)

f2, f4

surjective, f5

In particular:

surjective uv injective

f1, f2, f4, f5

»

f3

injective

f3

surjective

isomorphisms o f3

isomorphism

The assertion of the proposition follows by the five-lemma. cation “G (o) duction on

c_1(M)

l

HCn_1 (N)

quasi-isomorphism - G

The impli—

quasi-isomorphism" is seen by in-

n:

—> anon — cu/n ———> c_2(M)

l

l

l

—* Hn(N) — HCn(N) ——* HCn_2 (N)

whereas the implication "G

——.Hn_1 (M)

1

—’ Hn-1 (N)

quasi-isomorphism - G (O)

follows directly from the five-lemma (every

an

quasi-isomorphism“

in the long exact se-

quences has two HC-partners O

df(x)

(D-b',b+N)(sNx,x)

fd(x)

£(Bx,0,bx) — (sNBx,Bx,sNbx,bx) = (Bx,sb'Nx,bx)

(since

NB = 0, Nb

Proggsition 2.4.5

=

(Bx,Nx—b'sNx,bx)

f: BC(A) + Tot(C(A))

Write

C(A) = (Cs,t)s,t20

(where

C

A

We have for T

“

=

O

s+t=n

C

T

s,t’

, s,t 2 0).

Tot(C(A))

B

is a quasi-isomorphism, i.e.

is an isomorphism for all

Proof.

s+1

(Bx,sb'Nx,bx)

b'N (2.1.5)).

Hn(f): HCn(C(A)) + HCn(A)

3,1: =

=

C(A)

“

=

and for

c s+t=n t even

c s,t

BC(A):

n 2 0.

In this notation

B: Cs,t + Cs+1,t-2'

Let us consider the second filtration of filtration on

T = Tot(C(A)) and the induced

BC(A).

Explicitely: — ( F PB C ( A ))n _

0 jSp

— j$p fl C n-j,3 . . ( F P T )n _

C n-J,J’ . .

j even

FPBC(A)

is a subcomplex of

BC(A)

for all

p 2 0 (since

B

lowers

the second index). f: BC(A) * T = Tot(C(A))

becomes a homomorphism of filtered complexes. (since

f(cq.p) c Cq+1:P‘1 o Cq'P)

We have to look at the spectral sequences associated with these filtra-

tions on

BC(A)

and

T = Tot(C(A)).

We know already: H (A)

for

p

even

O

for

p

odd

E1Pig (T) = n'‘1: P(C(A)) = (see 2.2.4)

and that

2 _ Ep,q(T) E Hn(T) _ — HCn(A)’ n P + Q-

On the other hand, the filtration on

BC(A)

is clearly bounded, and

thus 2 _ — p + q. Ep,q(BC(A)) 3 Hn(BC(A)) — HCn(C(A)), n _

Let us calculate

1 Ep,q(BC(A))'

1 = p 9-1 Ep,q(Bc(A)) Hp+q(F BC(A)/F BC(A))

C Bu t

( FPB C(A)/FP-1 B C(A)) p+q -

p

even

p

odd

q,p 0

36 Since

B(FPBC(A)) c Fp-1BC(A), the induced differential on

FPBC(A)/FPD1BC(A)

is merely given by

H (A)

E1Figs (C(A))= Consider now

f1:

and

x E Cq'P,

f [Mr

even

p

odd

Thus

q 0

f

p

b.

+ E1(T).

f1

p

even, to

(sNx,x) E Cq+1,p-1 0 Cq,p'

Hq(A) + Hq(A)

p

even

O

9

odd

maps

E1(BC(A))

+ O

is induced by

f

in homology,

Hence

is the identity. By the approximation theorem 1.3.7 we can conclude that H(f): HC‘(C(A)) + HC,(A) is an isomorphism, which proves our assertion. Theorem 2.4.6

For every unital associative k-algebra

A

there is a

long exact sequence

+Hn(A) 4» scum) —S> c_2(A) —B> Hn_1(A) + Proof.

This is an immediate consequence of 2.3.6 and 2.4.5: the exact

sequence of chain complexes

0 "* (C(A),b)

"* (BC(A).d)

-+ (BC(A)[2]’d[2])

-+ 0

yields our long exact sequence when passing to homology and identifying H*(A) = H.(C(A)), HC*(A)

= HC,(C(A)).

The connecting homomorphism is induced by our operator Complement 2.4.7

(1)

Recall 2.3.7:

We have in lowest degrees

an isomorphism o + H°(A) —I-> ncom) + 0

(ii) an epimorphism

B.

H1(A) + HC1(A) + 0

37. Agglication 2.4.8

Let

Mr(k)

be the k-algebra of rXr-matrices with

coefficients in the commutative ring

Then

k

for

n

even

0

for

n

odd

k.

HCn(Mr(k)) =

We have in Hochschild homology:

H°(Mr(k)) = Mr(k)/[Mr(k),Mr(k)] = k Hn(Mr(k)) = O

for

n 2 1 k e _ r Hn(Mr(k)) Torn

is k—free, hence

Mr(k)

is Mr(k)e-projective:

Thus

(2.4.7):

The long exact sequence

(2.4.6) yields immediately

HCn_2(Mr(k))

for

n 2 2,

Remark 2.4.9

Let

A1

6(0):

[C.E., p.179l)

k

HC°(Mr(k))

HC1(Mr(k))

(Mr(k).Mr(k))

0

but

ll

(Mr(k)

c(Mr(k)) a

and thus our result.

and

(C(A1),b) + (C(Az),b)

A2

be two unital associative k-algebras,

a homomorphism of the Hochschild complexes

such that there exists a homomorphism

G:

(BC(A1),d) + (BctAz),d)

mak-

ing the following diagram commutative:

o

——» (C(A1),b)

la“) O

—D (C(AZ),b)

Then (2.3.15) G(°)

-—> (BC(A1),d) —> (BC(A1)[2],d[2])

1G —9 (BC(A2),d)

—* 0

lGIZJ —' (BC(A2)[2],d[2])

is a quasi-isomorphism if and only if

—§ 0 G

is a

quasi-isomorphism

(H,(A1) a H,,(A2) via H*(G(°)) .. Hc,(A1) a Hc_(A2) via HC,(G)) Caution:

The equivalence is only true as a statement in all degrees.

38 Note that every homomorphism of k-algebras (G (o) ,G)

such a couple

and

g: A

1

+ A2

gives rise to

in an obvious way (functoriality of

H*(-)

HC*(-)).

Proposition 2.4.10 which are k-flat.

Let

A1, A2

Consider

be two unital associative k-algebras,

A = A 1 x A2, their direct product, and the

projections

TrzA+A

1

Then

1'

1T2:

A+A

2'

HC(111 ) ac. (A) .—T;—)t Hc,..(1\1 ) a He. (A)

is an isomorphism. Proof.

It is well-known

([C.E., p.173]) that the result holds in

Hochschild homology (the flatness assumption allowstoidentifyfibchschild homology with a Tor-functor, cf. 2.1.2). Consider now the following commutative diagram (we have dropped the arguments; the meaning is obvious): —*HC

n-1

—*H

ll

‘—"HCn

11

'_*HC

11

n-2

_"H

11

n-1

—'

ll

(1)191mm (2) 3 Hn(1) on“(2) 2 c(1) men(2) 3 c__20c (1) (2)2 : Hn-10Hn-1 (1) (2) 3 2 c_ Once more, the five-lemma yields our result by induction on

Corollary 2.4.11

R1 0 A

Let

A

n.

be a unital associative flat k-algebra, A =

the unital k-algebra obtained by adjunction of a (new) unity.

Then

HCn(A)

= HCn(k)

Proof.

Let

6 HCn(A)

e E A

k 0 HCn(A)

n

even

HCn(A)

n

odd

=

be the unity of

A.

Then

~

¢: k x A

(aha)

+

->

A

(aha-Ge)

is an isomorphism of k-algebras.

2.4.8 and 2.4.10

now give our result.

39 AERlication 2.4.12

Cyclic homology of Clifford algebras.

ment , we shall only treat the nondegenerate case. of characteristic # 2, and let

Let

K

Let

For the mok

be a field

be an algebraic closure of

k.

(v.9)

be a nondegenerate finite-dimensional quadratic space over k, A = C(V,Q) the associated Clifford algebra. By extension of scalars one obtains

KA = Kc(vrQ)

= C(KVIKQ)

But, since

M

2m

K

is algebraically closed,

(K)

dim V = 2m

KAN

M2m(K) x M2m(K)

dim V = 2m+1

Thus

HCn(KA)

=

K

dim V

even

K 0 K

dim V

odd

0

But by 2.2.2:

n

even

n

odd

K G HCn(A),

HCn(KA)

n 2 O

Hence we obtain finally:

HCn(C(V,Q)) =

k

dim V

even

k 0 k

dim V

odd

0 Definition 2.4.13 Let

A

The double complex

Define a now double complex and rendering

B

n

odd

8(A).

B(A)

plq _

p-q,2q 0

C(A)

=

P 2 q 2 o otherwise

(C

)

q p,q20

the dou-

A.

be deleting the acyclic columns in

a horizontal differential:

c B(A)

even

be a unital associative k-algebra,

ble (Hochschild) complex associated with C(A)

n

40 Vertical differential:

b

Horizontal differential:

B

B: C p-q,2q = AP-q+1 + AP'q+2 = cp_q+1’2q_2, cf. 2.4.1)

(Recall:

Buns .3

A44 A3; A2; A

lb

1b

lb

B(A)2

A3¢i

Azqi

A

a—-

+—-—-

A

+—-—

0

«——- O

FpB/FP_ZB —-> FpB/Fp_18 —+ 0 (see

1.2.2)

But in 1.4.11 we saw that for either standard spectral sequence of a double complex, this connecting homomorphism is induced by the differential still alive after having taken first step homology. Let us write down once more the situation in degrees Aq-p+3 Q Aq-p+1

o ——> Aq

—p+2

No te tha t

__ Aq-p+1

p+q

and

p+q-1:

_ O

1/1

_..Aq- p+2

( FP B/ FP'1 B)p+q

0 Aq-p

= Aq—p+1

(F13-1 B/F13-2 mp“?1

=

Aq-p+2 .

This concludes the proof of our theorem.

1.2.5

Nonunital and Reduced gyclic Homology.

Remark-Definition 2.5.1

Let

A

consider the following subcomplex

be a unital associative k-algebra; (D*,b)

of the Hochschild complex

(231.1»): +

D

CAIN-lI

n for some

is spanned by all elements

i:

(note that

Dn

bDn c Dn_1,

the terms in

since for a typical element

b(a°,a1,...,an)

such that

31 = 1

(a°,a1,

,an)

of

where the argument 1 does not occur

cancel out).

(1)

(ao,a1,...,an)

1 s i s n.

(D,,b) is acyclic.

The verification can be done by hand.

42 First, in lowest degree we have

(30,1) = b(a°,1,1)

Then, consider an element of the form

(a°,1,a2,...,an)

E

Dn

such that

b(a°,1,a2,...,an) = (a°,1,a2 a3,...,an)-...+ 61)n(ana°,1,...,an_1) = O (the first two terms have cancelled out). I claim that

(a°,1,a2,...,an) = b(ao,1,1,a2,...,an) Now, b(a°,1,1,a2,...,an) = (a°,1,a2,...,an) -

+ But

b(a°,1,1,an,...,an) -

isomorphism of

(a°,1,1,a2a3,...,an)

+ (—1)n+1(ana°,1,1,a2,...,an_1).

(ao,1,a2,...,an)

is, up to a permutation

An+1, of the form

1 o b(a°,1,a2,...,an), hence equal

(a1 = 1

i 2 2)

to zero. The other cases

(2)

With

for some

3 = CoKer(k + A)

are treated similarly.

we have

A“+1/Dn=AoXe...0i=AeK“ (right exactness of the tensor product). we obtain a short exact sequence of chain complexes

0

+ (o)

where

+ (A*+1,b)

(A B 3*,b)

Notation:

+ (A e 3*,13)

+ o

is the normalized Hochschild complex.

(ao;a1,...,an) = (ao,a1,...,an)mod Dn

(in A e E“ = An+1/Dn) (a°;a1,...,an) = 0

whenever one of the Im(k + A)

(3)

Hn(A) = Hn(A a X‘,b),

n 2 o.

ai, 1 s i S n, lies in

This is an immediate consequence of the acyclicity of

(D*,b):

look

at the long exact homology sequence of

o — (o) — HC2m(A)

—’ HC2m(A)

-" 0, m 2 O.

The second long exact sequence of 2.5.9 yields in lowest degrees

(of. 2.3.7)

(1)

an isomorphism

§°(A) = §C°(A)

(ii)

an epimorphism

§1(A) + fiC1(A) + 0

Examgle 2.5.11

Let

k

be a commutative noetherian ring, A = Mr(k)

the k-algebra of rXI—matrices with coefficients in

Then (i)

§Cn(A) = 0

for all

n 2 o.

We first look at reduced Hochschild homology

for

n 2 1

(cf. 2.4.8), hence

for

n 2 2

(2.5.8)

l

H (A) = 0

E

(in all degrees).

SI:

§,(A) = O

0

R.

fi*(A).

I claim that

47 It remains to show that Write

A = k1 + X

where

§°(A) = H1 (A) = 0

(k-direct sum)

r-1 2 ke . 2 k(1-e ii ) + 1a ij i 1

K

(relative to the standard k-basis of

A)

Now, ‘A = [A,A] , and thus

fi°(A) = K/[A,A] = o.

The five—term exact sequence 2.5.8 (i) becomes

0 —>'fi1(A)—>k —.H°(A) —-» o and

H°(A) = A/[A,A] = k.

Since

k

is noetherian,

the surjective k-endomorphism

k -> k = Ho (A)

most be injective too.

Finally (ii)

331 (A) = 0.

We now pass to reduced cyclic homology.

By 2.5.1o(b) we have Since

finm = o

ion (A) = fiCn_2(A) Remark 2.5.12

Room) = fic1 (A) = 0.

for for

n 2 1, we obtain by 2.5.9 (2) n a 2.

Assume now that

(as in 2.4.8):

This proves our assertion. A = k e K

is an augmented k-algebra.

The commutative diagram —'

A

ii/

of k-algebra homomorphisms gives rise to a commutative diagram of mixed complexe s

O — E(k) —* EM)

1m

60:) i.e. to a splitting of the exact sequence of chain complexes

0 —> 360:) —> 36m) _. Banned —. o.

48 Hence the long exact sequence 2.5.9

(1) splits too, i.e. we have

HC, (A) = Hc,(k) 0 EC. (A) More explicitely:

k 0 ficn(A)

n

HCn‘A’ = {ficn(A)

even

n odd

Note that in example 2.5.11 we obtained the same result in a non-augmented setting. Example 2.5.13 k-module,

Cyclic homology of a tensor algebra.

A = T(V) =

0 Vm

the tensor algebra of

Let V

V

over

be a flat k.

m20

(Vm

means m-fold tensor product over

(1)

Hochschild homology of

(a)

The acyclic Hochschild complex

k)

A = T(V).

(A‘+1,b')

gives rise to an exact

sequence of A—A-bimodules I

o—+osA —b—#A@A

I

-b—>A—>o I

Recall ([C.E., p.168]) that J = Ker(A a A b» commutative differentials of

Forevery'A-A-bimodule

M

A)

is the module of non-

A:

and for every derivation

d: A e M

there is

a unique factorization

d

A -——* M

13/:

J where

j(a) = a e 1 - 1 0 a, a E A, and where

f

is an A-A—bimodule

homomorphism. Since derivations on

A = T(V)

are uniquely definable and determined

by their (k-linear) restrictions on

V, the A-A—bimodule

has the same universal factorization property.

Thus

A 0 V 0 A

A a V O A

N bl

(b)

The Hochschild homology of

H°(A) = o Vm/(1-o), mzo where

a : Vm + Vm

A = T(V)

H1(A) = e (vm)°,

is given by

Hn(A) = o

m21 is the cyclic permutation

for

n 2 2,

J.

0(v1,...,vm) = (vm,v1,...,vm_1). Proof.

Consider the long exact homology sequence

Ae

Ae

e

Ae

Ae

(A,A e v e A) ——> TorAO (A,Ae) —~ ToroAe (A,A)

...-+ Torn (A,A e V 8 A) -+ Torn (A,A ) -+ Torn (A,A)-+ ... ...-* Tor

0

e

Since A6

and A o v e A are Ae-flat, we obtain

(i)

Hn(A)

= Torn

(ii)

an exact sequence in lowest degrees:

—>

O

Ae (A,A)

0—PH1(A)—>A6e A

= O

for

n 2 2

(AOVGA)&>AOAe-—’HO(A)_’O

I

N

b AOV—>A Recall:

b: A o A + A

Spezializing

is given by

b(a°,a1) = aoa1 - a1a .

a0 = (V1,...,Vm) E Vm-1, a1 = vm E V:

b((v1,...,vm_1) 8 Vh) = (V1,...,vm) -

(vm,v1,...,vm_1) = (1-0)(v1,..,vm)

We obtain finally:

noun

CoKer(A o v bm) =

e vm/(1-a)

H1(A)

Ker(A a v b+A) = 0 Wm)“

mso

ms1 (0)

Reduced Hochschild homology of

A = T(V).

We have (2.5.8): (1)

an exact sequence

0+ 31m) +fi1(A) + k-> H°(A) +fi°(A) -> o

(11) finm = Hn(A) = o, Now, [A,A] c X =

n 2 2

a v”, and hence m21

50 o

—>

0 —~

k ——v k—>

fiom

Hoax) ——r ov'“/(1-a)-—>

mo

—>

o

ov’“(1-a)—->

0

m21

is exact.

This implies E1 (A) = H1 (A) = 9 Wm)". m21 (2)

(Reduced) cyclic homology of

(3)

Identification of

B: fiC°(A) + §1(A)

We have

HC°(A) = fio(A)

and

flow

(2.5.10)

= 31m.

is induced by

s:

K

A a X

-

3

a h+ Lemma:

A = T(V).

(17a)

The following square is commutative

Eco (A)

B—r E1 (A)

l

W

H

o v‘“/(1-a)-—m—> e (Vm)° m21 where

m21 v

m

=

m-1 i E a

i=0

(norm map).

(Note that we are dealing with operations of the various cyclic groups inside

A = T(V).

complexes of

Proof.

Note that

— _ 31(A) —

Ker(A 0 A

Consider

Don't confound with the operations on the Hochschild

A)

(x;yz) =

— b-’A)/b(A —

(xy;z) + (zx;y)

in

—2 ).

o A

a = v1...vm € VIn : A.

3(a) = (1;a)mod b(A e 32) But (recall (1)(b)): (1;v1...vm)

(v1;v2...vm) + (v2...vm;v1) (v1v2;v3...vm) +

(v3...vmv1;v2) + (v2...vm;v1)

51 m E i:1(vi+1...vi_1;vi) E H1(A) c A e V Identifying

Vm.1 8 V c A 8 V

v

as claimed.

(v ...v ) m 1 m

(b)

with

Vm c A, we obtain

Recall the long exact sequence 2.5.9

(1;v1...vm) =

(2) relating reduced Hoch-

schild homology and reduced cyclic homology.

We obtain in our special

case

(i)

EO(A) = fic°(A)

(ii)

a four—term exact sequence

(always true)

0 —-> ficzm) —-> ficom) —B-> E1 (A) — fiC1 (A) —> o ficn(A) a ficn_2(A), n z 3.

(111) isomorphisms

Conclusion:

fiC2(A) a Ker B EC1(A) u CoKer B

and consequently

Ker(fic°(A) —§» fi1(A))

n even

ficn(A) a

n 2 1 CoKer(§C°(A)-—§+ §1(A))

n

odd

0n the other hand, we have the following

Lemma:

Let

Gm =

module, v: M 4 M

be a finite cyclic group of order m-1 the norm operator v = 2 01.

m,

M

a Gm-

i=0

Then

H°(Gm,M) = M/(1—a)M

Hn(Gm'M)

Proof.

Ker v/Im(1—c)

n

even

Ker(1-a)/Im v

n

odd

= {

n 2

1

[H.St., p.201].

This 1emma,together with the above lemma identifying the operator as a sum of norm maps,give the

Proggsition:

fien (T(V))=

e H n (Gm ,vm), n 2 o

“‘21

B

52 where the cyclic group Proof.

(c)

Gm

Ker(1-a) = (Vm)°.

acts on

Put the two lemmas together.

c('r(v)) = c(k) e ficnvrm)

Assume now that

icon-(w) =

m c k.

Vm via a.

by 2.5.12.

Then

o vm/(1-a) m21

ficn('r(v)) = o

for

n 2 1.

and consequently

Hc°('1'(v)) =

e vm/(1-a) mZO

HCn(T(V))

k

n

even

O

n

odd

={

n 21

Remark 2.5.14

Cyclic homology of non-unital associative k—algebras.

A short inspection of the definitions 2.1.4 and 2.2.1

plex

C(A)

and

HC*(A) = H*(Tot(C(A))))

make sense for any associative k-algebra (Caution:

A, unital or not.

There is no longer a contracting homotopy degree columns of

Furthermore, 2.2.2

(the double com-

shows that these definitions

s

for the odd

C(A)).

(flat extensions of scalars) and 2.2.3

(direct lim-

its) remain valid in the non-unital setting. In particular, let

lim Mr(A)

A

be a unital associative k-algebra,

M(A) =

the algebra of infinite matrices with only a finite number

of nonzero entries in

A,

HC*(M(A)) = lim HC,(Mr(A))

then

M(A)

is non-unital,

but nevertheless

(= HC,(A): Morita-invariance of cyclic ho-

mology; cf. 2.7.14). At this stage, we only can prove that

HC*(M(k)) = lim HC*(Mr(k)) = HC,(k)

(2.4.8)

The connection between non-unital cyclic homology and reduced unital cyclic homology is simple:

Proposition 2.5.15

Let

A = k 6 I

be an augmented k-algebra.

Then

Hc,(I) = fiC*(A). Proof.

We want to establish an isomorphism of chain complexes

h: T = Tot(C(I)) + BC(A)red which will immediately give our result. For

r z 1

consider the ismorphisms

Ir+101r

—_——»A91r

((xo,...,xr),(Y1....,yr))

+ (x°;x1,...,xr) +

(1;y1,...,yr)

which yield isomorphisms of graded modules

Tn

= I

Nu (BC(A)

red)n

n+1

a I“ o In' 1 0 In' 2 a ... 2 2

(A o I“)e

(Note that for even

identity on

n

(A o In-z) o ...

the last component isomorphism is given by the

I = K)

It remains to verify that the d = b o (D—b') 0 N + b 0 ..

hn on

commute with the differentials T

and

d = b + B

on

BC(A)red'

This follows from the identities

(1) (both

bh(x°,...,xr) = b(xo;x1,...,xr) = hb(xo,...,xr) b

are given by the same formula,

(2)

Bh(y1,...,yr) = B(1;y1,...,yr) = 0

(3)

bh(y1,...,yr) = b(1;y1,...,yr)

"up to a semicolon")

(y17y2..--,yr) + (-1)r(yr:y1.---.yr_1) r-1 1 + 1Z1(-1) (1;y1,...,y1y1+1,...,yr) h((D-b')(y1,...,yr)

(4)

Bh(x°,x1,...,xr) = B(xo;x1,...,xr) r ir 1:0(-1) (17811...:xrrx°I-'-Ixi_1)

h(N(x°,...,xr)) A = k 6 I, a ring of dual numbers

Example 2.5.16

thisparticularsituation, the differentials

b

In

(i.e. 12 = O).

and

b'

on

C(I)

are

zero .

n

c(A) = c(I) = e Hn_K(GK+1,I K+1 )

Thus:

K=O

(group homology) Assume now that

We obtain

m c k.

ficn(A) = In+1/(1-t)I

Spezializing to

n+1

(cf. lemma in the proof of 2.2.6)

A = k[e], the usual ring of dual numbers over

k, this

gives k e k HCn(k[s])

(In+1 = k

n

even

n

odd

= {

for all

n 2 0, and t.1 = 1

for

n

even, t.1 = -1

for

n

odd) Remark 2.5.17

Let

as a k-module (with (1)

Write

A

be a unital associative k-algebra,

A = k 0 A

K = CoKer(k + A)).

A 9 Kr = K s Ar 6 1 e Kr

is the k-submodule of

A 0 Kr

(k—direct sum),

where

spanned by the elements

We have (as in the proof of 2.5.15)

1 0 ET

(1;a1,...,ar).

ismorphisms

xr+1 c 3r .____________Jl__. A 3 3r ((xo,...,xr),(Y1,...,yr))

Now, the operators

+ (x07x1,...,xr) +

D,N: Kr+1 + Kr+1

(1;Y1""’Yr)

make sense, and the formulas

and (4) in the proof of 2.5.15 read:

(3)

h(D(y1,...,yr+1)) = bh(y1,...,yr+1)mod 1 a Sr

(3)

(4)

h(N(x°,...,xr)) = B(h(xo,...,xr))

We thus obtain the following commutative diagram (note that

B(1 9 Ar)

= O):

heir/1931:4— 1ofir+14—B Ih

uh

—r+1

(2)

AGE/183”

D

uh

—r+1

N

—r+1

We have an exact sequence of k—modules

n 1 9 A

.

(Since

*

An+1

n

1 9 A

n+1 /(1-t)A

*

—n+1

_

n

—n+1 /(1 t)A

_

mod(1-t) — (1 a A

O

+

.

+ Dn)mod(1 t).

for

(a0....,an) 6 Dn

(2.5.1) there is x: 1 s K s n such that tK(ao,...,an) e1oA“, i. a. (a0 ,...,an ) = 9‘ (a0 ,...,an ) + (1-tK )(ao ,...,an ) e 1 9A“ + (1-t)An+1) Furthermore, b(1,a1,...,an) = (1-t)(a1,...,an)mod 1 0 An_1, i.e. the kernel of

A*+1/(1-t) + X*+1/(1-t)

Hence we have the chain complex

ential

(K*+1/(1-t),b)

A.

Assume that

Then the complexes

quasi-isomorphic, 1.e.

m c k, and that (A*+1/(1-t), b)

we have

fic,(A) = H (A—l+1 /(1-t).b)Proof.

(1)

Definition of a chain transformation

11: (36(A1red,a)+ (A—*+1 /(1-t).b) (which will reveal to be a quasi-isomorphism): -*+1

(A*+1/(1-t),b).

(with induced differ-

b).

Proggsition 2.5.18 mand in

is a subcomplex of

/(1-t),b)

n «— (BE(A)red,d)

k and

is a k-direct sum(35(A)red,d)

are

Ann/(ht)

1'

o (A 818-1) 6

(1/11,:-

1b -n+1

/(1- t)

...

bl/lb3

1b in/H—t)

0 (AG—n 2)$

(AOA

t—

(A a An

4—“- (A 9 An

Explicitely:

In degree

n,

1! is a composition of the natural surjec-

tions

(3cm) red) :1 -—> (A o A“) —-> An+1 —> 1!

—n+1

/(1-t)

is a homomorphism of chain complexes, since

(the k—submodule of

and thus

(2) that

A 0 K“

(17a1,. . .,an))

no]! = 0.

Definition of filtrations on 11

B(A 0 P4) : 1 o in

spanned by the elements

becomes

a

Haunted

and on

—*+1

A

/(1-t)

such

morphism of filtered chain complexes —n+1

/(1- t)

n s p

(FP("‘“/(1-t)))n = n > p

O

(BC(A)red)n

n S P

(FPBE(A)red)n = (1 a P“) + z (A e AP'ZK") K20

2 (A 0 KIFZK)

n — p odd n

n - p

K20

Let us draw a picture of the couple of subcomplexes

even

>

p

(A /(1—t)) F“n—*+1 0

— F n (Beaured)

and

¢————

0

l 4———-

9

(A9?)

e

6

lb /lb

" —n /(1-t)‘—(ADA)

1"

“

)

e

(A0?“ 2)

l”

4—(1e'in*1)e(AeA“'

1

(A093n

119/111“:

o

1 /l

1

o

A

(18AM

1

0

—n+1

e

9(Aoin'3)

1:3/11:

(AaAn'-2 )

e

(A193n

-4

)

lb/lb /lb e (AG—n- 3) a (AoAn5)

An/(1-t)‘—-(AeA“'1) As

B(A o 39) c a a SP

+1

, FnBE(A)red

is indeed a subcomplex of

BC(A)red'

(Convention: (3)

A 3 3° = K

in the reduced case).

Description of the quotient filtrations.

.

.

4+1 /(1-t),b)

Write

K

for the filtered chain complex

(A

and

E

for the filtered chain complex

(BE(A)red,d)

when passing to the quotient filtrations, we obtain

0

q < O

(FPK/FP‘1I)P+q = KPH/(Ft)

q = o

0

_

(FPE/FP ‘1')!” = ‘1

q > o

o

q < o

A a 39/1 9 KP _ +1 1 a AP

q = 0 q

odd }

A 0 39/1 9

q

even

_ AP

We are interested in the differential of Recall the commutative diagram of 2.5.17

q > 0

FPU/FP-1U. (1):

58

Aoi9/1oKPa—L1OKP+1¢—B In "h KP+1

i

D

Thus we obtain:

since

mension) for every (4)

n

l

N

KP+1

0 c k, FpU/FP_1?

is acxclic (in positive di-

p 2 O.

is a quasi-isomorphism:

Ema)

Hp+q(£PE/F

1 Eplqa)

“1:

KP+1

AeKP/1eKP In

=

Hp+q(FPI/Fp-1 I)

2H?) + 21(1)

KPH/(Pt)

E)

q = o

{o

q f o

KPH/(Pt) =

q = o

0

q ’4 0

is the identity.

By the approximation theorem 1.3.7 we obtain that

H*(n): H,(E) + H,(T)

must be an isomorphism.

1.2.6

Cxclic Cohomologg

Remark 2.6.1 k-algebra.

Let

k

be a commutative ring, A

a unital associative

Recall the definition of the double chain complex

C(A)

(2.1.4). Applying the functor

Homk(-,k), we obtain the double cochain complex

ctua.) = Homk(C(A) ,k) . (We write

( )t

vasion).

Explicitely:

t

2 *

C (A) ’

for "transposed“ in order to avoid a notational *-in-

3

Homkm .k)

Ibt ctm‘” t

o *

C (A) ’

Dt

3

—* Homkm k)

t

[-b't

Nt

3

— Homkm ,k)

t

Ibt

Homk(A2,k) Ibt

—D—» Homk(A2,k) -N——r Homk(A2,k) I-b't Ibt

Homk(A,k)

— Homk(A,k)

Nt Homk(A,k) —

ct(A)"°

c"(1s.)"'1

ct(A)*'2

Dt

59 C( )

is a covariant functor from the category of (unital) associative

k-algebras to the category of double chain complexes over phisms of bidegree

(0,0)), whereas

Ct( )

k (with mor-

is a contravariant functor

from the category of (unital) associative k-algebras to the category of double cochain complexes over (1)

The

(anti)commutativity of the relevant squares in

from 2.1.5 and the fact that

(2)

k.

Homk(-,k)

Ct(A)

follows

is a contravariant functor.

The explicit formulas:

btr(a°,...,an)

(tob)(ao,...,an) n-1

=

.

Z (-1)1T(ao,..,aiai+1,..,an)-+(-1)n1(anao,a1,..,a i=0 n-1)

b'tt(a°, . .,an) = (Tob') (a0, . . . ,an) n-1 _ i — .:o( 1) T(a°,...,aiai+1,...,an) _

Dtt(a°,...,an) = (toD)(a°,...,an)

— - T < ac....,an )

— (—1)“ T ( an,a°,...,an_1 )

Nt1(a o""’ an ) = (T°N)(a o""’ a n )

= 1(a ,...,a ) + (—1)n1(a ,a ,...,a ) o n n o n-1 + (-1)2nT(an_1,an,..,an_2) +.. +(-1)n 1(a1,a2,..,an,a°) (3)

The odd-degree columns

Ct(A)*’q, q

odd, are acyclic, since the

contracting homotopy operators dualizesltoo. (4)

For the even-degree columns

Ct(A)*’q, q

even, we obtain:

'Hp’qwtmn = HP(A,A*) (Hochschild cohomology of Whenever

A

A

with coefficients in

A* = Homk(A,k)).

is k-projective, then

HP(A,A*) = Ext:e(A,1-\*),

p 2 o.

In this case, the acyclic Hochschild complex

2

(A*+ ,b'

I

:..-2—+ A -—* A

60 is an Ag-projective resolution of the A-A-bimodule Note that our cochain complex

(Homk(A*+1,k),bt)

A. identifies anyway

with

(nomAeme g A", Homk(A,k)). Hom(b',1)) (cf.

[C.E., pp.174/175]).

(5)

Let us look now at the rows of

(cohomology of the cyclic groups with coefficients in the G

_ "

_

p+1

G

Ct(A).

[3+1

-modules

We get

=

of order

Homk(AP+1,k):

p+1, p 2 0, (tr)(a°,...,ap)

P

(1)1(ap'aol---Iap_1))-

Note that we have always

q n ,k)), Hq (Gn,Homk(An ,k)) = EXtEIGn](Z’H°mk(A

n 2 1..

Take the standard

Z:

Z +

8

D

Z[Gn]+

ZIGnJ-free resolution of

ZIGn1+

N

Z[Gn]+

D

...

and apply HomRIG“ ](-,Homk(An,k)), n 2 1. (6)

We had the total chain complex

double chain complex Tot(Ct(A))

C(A).

(cf. 2.1.3)

T = Tot(C(A)), associated with our

when passing to the total cochain complex

associated withthedouble cochain complex

precisely the dual complex

Tt.

Ct(A), we obtain

Explicitely:

(1‘)“ = (Tn)t = Homk(An+1,k) o Homk(An,k) 9 a: I

Ibt+Dt

(rt)“"— (Tn_1)t — aomk(A“,k) Definition 2.6.2 ative k-algebra

TNt—b't

e Homk(An-1,k) e

The cyclic cohomology A

is defined by

HCn(A) = Hn(Tot(Ct(A))),

n a o.

HC*(A)

of the unital associ-

61 Remark 2.6.3

This definition also makes sense in the non-unital case;

you should only observe that the odd-degree columns of

ct(A)

now don't

have a contracting homotopy (hence are not necessarily acyclic). This prevents from directly passing to comparison results relating Hochschild and cyclic cohomology in the non-unital setting.

The remedy will be,

as in the case of homology, reduced theory (cf. 2.5.14/15). Proposition 2.6.4

Let

ciative k-algebra.

k

be a field, and let

(a) Hn(A,A*) = (Hn(A))* (b)

A

be a (unital) asso—

Then

(k-dual),

n

n 2 o.

HC (A) = (HCn(A))*

Proof.

We need the following elementary

Lemma:

Let

k

(Homk(x,k),dt)

be a field,

its dual.

at) + Homk(H(X,d),k)

(x,d)

a differential k-module and

Then the canonical homomorphism

given by

[u] + ([x] + ux)

H(Homk(x,k),

is an isomorphism.

(see [G0, p.221). (a)

Apply the lemma to

(X,d)

(A*+1,b).

(b)

Apply the lemma to

(X,d)

(Tot(C(A)),d)

Remark 2.6.5 k).

and recall 2.6.1

Note that 2.6.4 only depends on the exactness of

If you consider (k,K)-cohomology in the following sense:

a homomorphism of commutative rings, A t

c(k,K)(A)

=

Homk(C(A),K), and

(6).

Homk(-, k + K

a unital associative k-algebra,

H7l)(A,Homk(A,K)),

.

He(k,x)(A)

defined

via this modified double cochain complex, then 2.6.4 reads: if k + K . . . . n _ is such that K is k—injective, the Hc(k,K)(A) — Homk(HCn(A),K),1120.

Remark 2.6.6 Let

k + K

(Flat extensions of scalars) be a flat homomorphism of commutative rings,

A

a unital

associative k-algebra, KA = K o A the K-algebra obtained by extension k Then Assume that A is a finitely presented k-module. of scalars.

HCn(KA) = K e c(A), Proof.

n 2 0.

We have without any extra-assumption

62

H°“x (( RA )n ' K) = HonK (K ok A“ ' K) = Ho“k (An ' K) But when

R

is k-flat and

A

Homk(An,K) = Homk(An,k) : x

finitely k-presented, then

(of. [Bou, A.X. 12]).

The rest of the proof is as in 2.2.2. Note that in the spirit of the remark 2.6.5 we have always

n

HC(k’K)(A)

_

— HC

Remark 2.6.7

n

(KA)'

n 2 0.

We have

"H“’°(Ct(A)) = H°(Gn+1,Homk(An+1,k)) = Ker(Homk(A

n+1 ,k)

Dt +

Homk(A

n+1 ,k))

= {T Eflomk(An+1,k): T(a°,..,an)==(—1)nt(an,ao,..,an_1),

31 E A}. OSiSn

Moreover the formula bt)

Dtbt = b'tDt

shows that

C;(A) = ("H*’°(Ct(A)),

is a subcomplex of the Hochschild cochain complex

(ct(A)*'°,bt) = (Homk(A It+1 ,k).bt). Put H§(A) = H“(c;(A)),

n 2 o.

It is immediate that the map

a:

c;(A)

+ Tot(Ct(A))

given by

a“: Ker Dt c Homk(An+1,k) c (Tot(Ct(A)))n,

n 2 o,

is a monomorphism of cochain complexes.

Proggsition 2.6.8 Assume that m c R. is a quasi-isomorphism, i.e. for every n 2 0.

Then a : c;(A) + Tot(Ct(A))

Hn(a): H:(A) + HCn(A)

is an isomorphism

63 Proof.

All arguments are dual to 2.2.6.

Recall 1.4.12:

Spectral se-

quence arguments apply to double cochain complexes of the first quadrant via visualization as double chain complexes of the third quadrant. In order to fix notational ideas, let us consider the first filtration

of Tt = Tot(Ct(A)): (IFPTt)n = e CW“j jZP

cp'q = Homk(CP’q,k) = Homk(Ap+1 ,k),

p 2 o, q 2 0.

We have:

H30!) = 'l-In"Hn’°(Ct(A)) = IE2”, and I Eglq

= "Hp’q(C t (A)) = Hq(GP+1,Homk(AP+ 1,k))p

Pig 2 0

Consequently:

IEp,q = 0

for

q > 0,

(once more (cf.

Q c k

[R0, p.2921):

arbitrary G-module;

A fortiori:

since

then

Ifig’q = O

G

a finite group of order

q(G,M)

for

= O

for

m, M

an

q > o)

q > o, and by 1.4.13 we obtain

H§(A) = IE§'° a HCn(A) = Hn(Tot(Ct(A))).

n 2 0.

Note that (as in 2.2.6) we should rigorously verify that our isomorphism is actually induced by Convention 2.6.9 suppress the s

for

The argument is the same.

In order to avoid notational clumsiness we shall often

( )t-superscript:

st, C(A)

Remark 2.6.10

a.

for

b

will stand for

bt, b'

for

b't,

Ct(A), and so on.

We have now to formulate the mixed cochain complex ap-

proach to cohomology. (1)

A mixed cochain complex

module

(Mn)n20

-1 endomorphism Thus

(M,b)

(M,b,B)

is a non-negatively graded k-

together with azdegree +1 endomorphism B

such that

b

is cochain complex,

b

and a degree

= 32 = b3 +Bb = 0. (M,B)

is a chain complex.

Morphisms

of mixed cochain complexes have to commute with both differentials.

64 Note that a priori mixed (chain) complexes and mixed cochain complexes are the same thing.

There is only a different emphasis on what should

be the primary differential and what should be the secondary differential. (2)

The associated cochain complex

(M,b,B)

(BM,d)

of a mixed cochain complex

is defined by

BM“ = M“ sun—2 014“": o

dn (m“,m“'2,m“'4, . . .) = (bm“,bm“'2+sm“,bm“'4+smn'2, . . .) (in short: (3)

6 = b + B)

We obtain the following exact sequence of cochain complexes

0 — (BM[2],d[2])—s> (Ema) A am) —> 0 which reads in degree

n

o — Mn'zenn'4 o. .. —S> MnOMn—ZQMn-4O. . . -—I> Mn —> o (5 (4)

means injection, I Let

(M,b,B)

H*(M) = H*(M,b)

HC*(M) = 3*(BM,d) (5)

means projection on the first factor).

be a mixed cochain complex.

the cohomology of

(M‘blB)

the cyclic cohomologyiof

(MLD,B)

There is a long exact cohomology sequence

.. -—+ H“(M) i Han" (M) —Sr HCn+1 (M) -—I-> Hn +1(M) where the connecting homomorphisnlisinduced by

B.

___# B

.

(this follows imme—

diately from (3); cf. 2.3.6). we have in lowest degrees:

(i)

an isomorphism Hc° (M) —I-> H° (M)

(ii) an exact sequence

(6)

0 + HC1(M) -l» H1(M)

A morphism of mixed (cochain) complexes

-§» HC°(M)

F:

(M,b,B) + (N,b,B)

gives rise to a commutative diagram of cochain complex homomorphisms

o

—» (BM[2],d[2]) — (Ema) — (M,b) — 0

13mm

1%

1F

-— (BN[2].d[2]) —— (Ema) — (N,b) — o

0

and thus to a commutative diagram relating the long exact cohomology

sequences

fl H“(M) — Hen"

lanm _» Hn(N)

(M)

-—-—* HC“+1

(N)

—* HCn +1

l

(M)

-——-*

__’ Hn+1 (N)

_

l c+1(F)

_ 11q

More generally, let

(M) —— an“

G(°):

(M,b) +

(N)

(N,b)

complexes which allows a prolongation

G:

l

be a homomorphism of cochain (BM,d)

(BN,d)

o

— (BM[2],d[2]) l

—-> (324,5) 1G

— (M,b) — o late)

0

—» (BN[2],d[2])

—+ (Ema)

— (N,b)

such that

—-> o

is commutative. Then we have the same commutative diagram relating the long exact cohomology sequences

(with

Hn(G(°)) and

HCn(G)

at the place of

Hn(F),

Hc“(r) ) . In this situation, the dual result to 2.3.15 is valid: 6(0):

(M,b) + (N,b)

is a quasi-isomorphism if and only if

G: (BM,d) + (BN,d)

is a quasi-isomorphism.

Definition 2.6.11

ctm) = (Homk(A*+‘,k),bt,Bt)

schild) cochain complex obtained by dualizing

shall write (cf. 2.6.9):

is the mixed (Hoch-

C(A) = (A‘+1,b,B).

We

C(A) = (Homk(A‘+1,k),b,B).

Remark 2.6.12

(1)

The cohomology

H*(C(A))

is the Hochschild cohomology

H*(A,A*)

(by definition). (2) of

The cyclic cohomology

HC*(C(A))

is the cyclic cohomology

HC*(A)

A, as is shown by dualizing 2.4.4 and 2.4.5:

As Tot(Ct(A))

is the k-dual of Tot(C(A)), and Bcm)

the k-dual of

66 ft:

BC(A), we get immediately the homomorphism of cochain complexes Tot(Ct(A)) + BC(A)

given by

ft = id + Ns

which is a quasi-isomorphism

(dualize the proof 2.4.5).

A

For every unital associative k-algebra

Theorem 2.6.13

there is a

long exact cohomology sequence

— Hn(A,A*)—B-> non—1(A)—S~ He“+1 (A) —I> Hn+1(A,A*) — Proof.

2.6.10(5) and 2.6.12.

Complement 2.6.14 (1)

Wehave in lowest degrees

an isomorphism

HC°(A) -3+ H°(A,A*)

(ii) a monomorphism

o + HC1(A)-—l» H1(A,A*)

Example 2.6.15

k

Let

be a commutative ring , and let

A

be a unital

commutative k-algebra such that ASA->18

a o b + ab is an isomorphism of k-algebras. Two typical (and frequent) cases: (i)

A = k/I, I

(ii) A

an ideal of

k;

S-1k, a localization of

Identifying

An+

1

with

chain complex

C(A)

C(A)3 ’

A J...

A

via

k. (a°,..., an)

+ aoa1

... an, our double

looks like this:

A ._o

A ._2

A .—

'

C(A)2 .

1°

A

o

1'1

A

3

1°

A

°

1'1

A

I

11

1°

11

1°

I

C(A) O,*

1°

A

o

1'1

A

1

1°

A

(Note that with our identification

(-1)n, hence

D = 1 — {-1)n

and

0 An+

1’1

A 1

= A,

n 2 0, we obtain

O

n

odd

(n+1)1

n

even

N = {

on

t

An+1

A)

67 Ct(A): replace

The double cochain complex

A

by

A“I = Homk(A,k)

and

reverse all arrows. We obtain for Hochschild (co)homology: {A

n=O

H(A)= o “

n21

Hn(A,A“‘) ={ n 2 1

O

(co)homology by the long exact (co)homology se-

which gives in cyclic

quences (2.4.6, 2.4.7 and 2.6.13, 2.6.14): A

n

even

n

odd

c(A) = {

n A“I HC (A) = { O

n

even

n

odd

Note that the eventual non—acyclic behaviour

of

the rows

(2-torsion,

(n+1)-torsion) does not affect the result. Let us generalize a little bit. Let

A

be as before, and consider

with coefficients in

Mr(A), the k-algebra of rXr-matrices

A.

Then ([C.E., p.172]):

Hn(Mr(A)) = Hn(Mr(k)'Mr(A))’

n 2 0

n _ n H (MI(A),Mr(A)‘) — H (Mr(k),Mr(A)*),

But

Mr(k)

n 2 0

is a separable k—algebra ([C.E., p.179]): hence

Hn(Mr(k)’_) = Hn(Mr(k).-) = o A

for

n 2 1.

This yields:

n = O

H (M (A)) = { n

n z

r

A* n H (M (A),M (A)*) = {

r

r

1

n = 0

n 2 1

By the same long exact sequence argument as before we thus obtain:

n

even

n

odd

A“ = no

acnmrm) = {

(A,k)

m"

0 For

A = k

A = (D.

n

even

n

odd

we have recovered and dualized 2.4.8.

Consider now

I:

We get

(I)

acnmrum = {o Whereas

n

even

n

odd

l-ICn(Mr(Q))

(Caution:

Mrm)

Analogously:

for all

n 2 o

is considered as a

Z-algebra:

Homm(m,z) = 0!)

A = E/p

Z/p

n

even

0

n

odd

n a 0

for all

HCn(Mr(ZZ/p))= 0

(once again:

O

k = E,

acnmrmupn = {

HomZ(Z/p, Z) = 0)

It is clear that this type of result holds for m integral domain (which is not a field) together with proper ideal of

k.

Thus

mology only depends on

ProEsition 2.6.16 k-algebras.

A = Quot (k)

or

A = k/I, I

k a

(at least in our special setting) cyclic ho-

A, whereas cyclic cohomology heavily depends on

the structure homomorphism

ections

F

A HC (M (A)) = { n r 0

I

68

Let

Consider

k + A.

A1, A2

A = A1

n1; A+ A1, 1r : A -> A . 2

be two unital associative k-projective

x A2, 2

their direct product,

and the proj—

Then

HC*(1r ) HC*(A1) $ HC*(A2)

#3 HC*(1r2)

HC" (A)

is an isomorphism. Proof.

Dualize 2.4.10 (see [C.E., p.173] for the result in Hochschild

cohomology) .

69

Remark 2.6.17 The suspension operator 5: ac“ (A) -> ac“+2 (A)

in explic-

it form. We shall assume that

Q c k.

Recall 2.6.7:

is the subcomplex of the Hochschild cochain com-

plex

(Homk(A

CX(A) 1+1

,k),b)

defined by

C2(A) = Ker(Homk(An+1 ,k)--:—+ 1 t

Homk(A“+1 ,k)),

n 2 0

. n _ _ n (1.2;11 E CA(A) n t(a°,..,an)—( 1) 1(an,a o""an-l) for all E A

(a°,..,an)

)

We have (by 2.6.8 and 2.6.12) 3 quasi-isomorphism

CK(A) + BC(A), which

is given by the inclusions

C?(A)¢—> Homkmn+1 .k)c—. 3cm“ Now,

5: acnm) -> nc‘1+ 2 (A)

is induced by the inclusion 13can)n 6—7

BC(A)n+2. We want to define

z n+2 (A) A

22(A) + z§+2(A)

such that

-—->BC(A)n+ 2

15" 2’; (A)

SA:

Is ——> Bc (A)n

commutes modulo coboundaries

(on the right side).

cyclic n n }) (ZA(A) = Z (A,A*) n Ker D = Ker b n Ker D = { n-cocycles

For

T e z’A‘m

I+(a°,...,a n+1)

define =

n 2

r

+

EHomkmn+2 ,k)

i 2 (-1)KT(a°,...,a KaK+1,...,an+1

i=0 K=O

_ _ 1 and SAT ‘ n+1 (n+2) 1“+ I claim that

(1)

SAT 6 zgflm)

by

for 'r e z§(A)

(11) SAT - T _ — (b+B)( _

(n+1 1 n+

(which will prove our assertion).

1+)

)

70 (n+2)1

sDt+

First stag.

1 € Z;(A)

for

(sD1+)(a°,...,an) = (Dr+)(1,a°,...,an)

= r+(1,a°,...,an) + (-1)n1+(an,1,a°,...,an_1) But n 1+(1,a°,..,an) + (-1) 1+(an,1,ao,..,an_1 ) n

i

i:o[(1(ao,..,an)-r(1,aoa1,..,an)+...+(-1) I(1,a°,..,ai_1ai,..,an))

+ (-1)n(t(an,a°,..,an_1)-T(an,ao,..,an_1)+I(an,1,a°a1,..,an_1)+...

+ ('1)iT(an,1,ao,..,a1_2a1_1,..,an))] n 21(a°,..,an) + i:1(1:(a°,..,an)+(-1)1I(1,30,..,a1_1ai,..,an)) (n+2)r(ao,..,an)+(bt)(1,a°,..,an)-T(ao,..,an)—(—1)

(n+2)r(a°,..,an)

Second steg. Now,

(since

T

n+1

r(an,ao,..,an_l)

E Ker b n Ker D).

31+ = (n+1)(n+2)r

31+ = NSDT+,

i.e.

(31+)(ao,..,an)

n nK =0 z (-1) (sDT+)(a n-K+1""an’ao""an-K) n

Z

(-1)

nK

(n+2)1(a

=0

n-K+1""an

,8,

o""an-K

)

(n+1)(n+2)1(ao,..,an)

(since

T E Ker D).

Finally: (ii).

(i)

SAT - T = (b+B)(—(n+1 un+2 is a consequence of

Remark 2.6.18 B(A) plex

(2.4.13).

1+)

for

T e z§(A)

which proves

(ii).

Recall the definition of the double

(chain) complex

We can dualize and obtain thus the double cochain com-

Bt(A), such that

(T0t(Bt(A)),d) = (BC(A),d),

i.e.

71

HCn(A) = Hn(Tot(Bt(A)),d). n 2 0Theorem 2.6.19

Let

(Er)r21

Eg'q =p-b cm.

be the spectral sequence associated with

T = Tot(Bt(A)).

the second filtration of

Then

(n = p+q)

and the following holds:

(1)

E?” = Hq'P(A,A*),

(2)

d$'q: Hq'P(A,A*) + Hq'P'1(A,A*)

is induced by Proof.

q 2 p

B.

The proof of 2.4.15 dualizes step by step.

Remark 2.6.20 projective.

Let

A

be a unital associative k-algebra which is k-

Then the projection

(A*+1,b) + (A o 3*,b) dualizes to an injection

(Homk(A o K‘,k),b)-*(Homk(A*+1,k)b) which is a Quasi-isomorphism. This is seen by the following argument ([E.C., pp.174/176]): acyclic Hochschild complex

(A*+2,b')

get an Ae-projective resolution of

projective).

Now, when passing to

in the form

A (since

A

Write the

(Ae 0 A‘,b').

We

is supposed to be k-

(Ae o K',b')(which makes sense),

acyclicity still is valid (since the contracting homotopy

3

passes

to the quotient), and we get thus another Ae-projective resolution of A.

we have

(Homk(A o i*,k),b) = (HomAe(Ae e K*,Homk(A,k)),Hom(b',1)) (Homk m“1 I k) I b) "- (HomAe (Ae e A* I Homk (A I k)) I Hom (b' I 1)) and hence our injection is a quasi-isomorphism by the standard homotopy equivalence argument for projective resolutions. Consider the normalized mixed Hochschild cochain complex

72

(Emma) = (Homkm o i*.k).b.s) where the operators on

B

b

and

C(A) = Homk(A*+1,k)

m

B

are merely restrictions of

b

and

(or, equivalently, induced by dualizing

B

b

and

am=(Ao?mJn.

Proposition 2.6.21 Proof.

HCn(A) = H“(BE(A),d),

n 2 0.

2.6.10(6) together with 2.6.20 (compare with the argument in

the proof of 2.5.3). Remark 2.6.22 ogy.

we shall not define nor discuss reduced cyclic cohomol-

The machineryisobviously available since the dualizing argu-

ments to be applied on 2.5.6 ...

1.2.7

(until 2.5.18)

should be clear.

Morita-invariance of Hochschild homology and of cyclic homology.

Example 2.7.1

Let

k

be an arbitrary commutative ring, and let

a unital associative k-algebra, rXr-matrices with coefficients in Consider

A, and

P

A.

is a right B-module, and

compatible: hence module

( )0

Q

is a

in a natural fashion.

Moreover, the left and right actions on

module;

(rows) with co-

Q = Ar, the left B—module of rX1-matrices (col-

umns) with coefficients in right A-module,

be

A.

P = tAr, the left A-module of 1Xr-matrices

efficients in

A

B = Mr(A)' r 2 1, the k—algebra of

P

is an

A-B

P

bimodule

and on

denotes opposite multiplication), and

(a left Be-module).

Note that

Q

are associatively

(equivalently: a left

P 8 Q a A

Q

is a B-A

A830bi-

as an A-A bimodule

B via scalar product multiplication of rows with columns,

and

Q 0 P a B A

as a 3-3 bimodule via Kronecker product multiplication of columns with rows

(identifying e1 0 tej

dard A-bases err)

of

(e1....,er)

with

of

Q,

eij'

1

s i,

t

t

j

( e1,..., er)

s r,

of

for the usual stan—

P

and

(e11,...,

B = Mr(A)).

Furthermore:

P

is A—projective

(since it is A-free) as well as B-pro-

jective (since it is a B-direct summand in AQBO-projective.

Similarly:

Q

B), but not necessarily

is projective over both rings, but

not necessarily Bv-projective. Definition 2.7.2

Let

A

and

B

be two unital associative rings

(unital

associative k-algebras for some commutative ring

R).

A

and

said to be Morita-eguivalent if there is an A-B bimodule bimodule

Q

such that

P 0 Q a A

as a B-B-bimodule.

Remark 2.7.3

Let

as an A-A bimodule,

B

P

and

Q 0 P a B

B

P

are

and a B-A A

be a left ABBo-module.

The following conditions are equivalent:

(-) 0 P: Mod-A + Mod—B A

(a)

is an equivalence of categories

(between right A-moduled and right B-

modules). (b)

There is a left BOAP-module

a B

Q

such that

as bimodules.

(c)

P a Q a A

and

Q 0 P

B

P 9 (-): B-Mcd + A-Mod

A

is an equivalence of categories.

B (cf.

[3a, p.601).

Comglement 2.7.4 (1)

Using the identification

pq

for the image of

we may assume that

p 0 q

(pq)p' = p(qp')

P; and similarly for

Q.

(2)

Q

Q

Let now

P

(P 0 Q) 6 P = P 0 (Q 0 P) and writing 3 A B A in A, qp for the image of q 0 p in B,

and

for the left and right actions on

be as in the definition 2.7.2.

Then

P

and

are necessarily finitely generated and projective as A-modules as

well as B—modules.

Let us show this for

P

as a B-module:

Write

N

1 =

E piqi i=1

in

A

(with the "scalar product" meaning of

pq

as indi-

cated in (1))Consider

a: P + 3“, P "

a

given by

(Q1PI0- - :qNP)

is a B-homomorphism.

Let

a: 3N + p be defined by

N 6(b1,...,bN) = 1:1Pibi (note that the unity in Then

p1,...,pN, q1,...,qN A

Bu = idP

are given by the fixed partition of

obove). by the associativity property of (1), hence

P

is

74 finitely generated and projective as a B-module. Definition 2.7.5 k-algebra, and

Let M

k

an

be a commutative ring, A

A-A bimodule.

For

cn(A,M)=MeA“=MoAsAo...eA b:

Cn(A,M) + Cn_1(A,M)

(n

copies of A)

is given by the formula

b(m o (a1,...,an)) = ma1 9

(a2,...,an)

n-1 1 + E (-1) m o 1-1

(a1,...,aiai+1,...,an)

+ (-1)“ anm e( a1....,an_1 ) The chain complex

(C,(A,M),b)

with coefficients in

a unital associative

n 2 0, set

M.

.

is called the Hochschild complex of

k-module), n 2 0, is called the n-th Hochschild homology of coefficients in

A

Hn(A,M), its n-th homology group (which is a A

with

M.

Remark 2.7.6

(1)

For A = M we have

complex of (2)

Assume

(C*(A,A),b) = (A‘“,b), our usual Hochschild

A.

to be k-flat. e H,(A,M) = Tore (A,M), by the same argument as in 2.1.2(2).

Then

Lemma 2.7.7

A

In the situation 2.7.5, let

M

be a left A-module, Q

a

projective right A-module.

fln(A,M s Q) = {

Then

Q o M

for

n = 0

for

n 2

A O

1

Proof. (1)

We shall first treat the particular case M

n = O

0

n 2 1

Hn(A,M O A) = {

Consider the augmentation map a: C°(A,M 0 A) = M 0 A + M

0 = A.

We have to show:

75 given by

e(m O a) = am.

We obtain a chain contraction for the augmented complex when defining

mm=mo1

5((mo a) o (a1.....an)) = (mo 1) e (a,a1,...,an) This yields the assertion. (ii)

The general case is easily reduced to

(1), since

Q 8 (-)

.

is

A

exact, and Since

C*(A,MeQ) coagulation A

via the isomorphisms

(MeQ)oA"~00 (MeAn) A

where we need only make explicit the left A-module structure on

M s A

n

9A: x.(m e a) o

(31,...,an) = (m e xa) a (31,...,an)

Complement 2.7.8

Let

tive left A-module.

M

M o P

for

n = 0

O

for

n 2 1

Hn(A,P o M) ={ A Theorem 2.7.9

be a right A-module, and let

Let

k

be any commutative ring, and let

unital associative k—algebras, P over both rings, and

P

be a projec-

Then

Q

A

and

B

be

an A-B bimodule which is projective

any B-A bimodule.

Then there is a natural sequence of isomorphisms

Fn:

Hn(A,P g Q) + Hn(B,Q : P).

n 2 0

which vary functorially with the 4-tuple Proof.

(A,B;P,Q).

Consider the following double complex

(CP q,d',d"): I

cpq=posqoooAP=cp(A,pesq)acq(B,QeAPeP),

pazo

I

where the last isomorphism is given by cyclic permutation of the relevant

76 terms .

dfi’q: cplq + Cp—1,q

is the boundary map

C‘(A,P O Bq 0 Q), whereas

ds’q: Cq

the boundary map (up to a sign)

b

for the Hochschild complex

+ Cplq-1

is equal to

for the Hochschild complex

(-1)Pb, C,(B,Q a AP

9 P). The columns of

C,“I

are Hochschild complexes for the homology of

A

with coefficients in certain A-A bimodules parametrized by

B, and the

rows of

B

C,,

are Hochschild complexes for the homology of

efficients in certain B-B blmodules parametrized by

with co-

A.

Let us draw a picture:

c3 *=

p s Q a A3 ¢-:9-- o B a Q s A3 4—29—— P o 32 o o e A3

c2*.

POQOA2+b—POBOQOA2 1, and suppose

on

already defined for

m < n

with the

required multiplicative property.

A Q n_1A, and define QnA = 9 1 A 8

Write syn:

91A X Qn-1A -> Zn

by

wn(w1,wn_1) = w1(w1)wn_1(mn_1).

We have

6n(w1a,mn_1) = $n(w1,awn_1), hence we get

¢:flA=fl 1 A + zn n n 1A 8 9n-

A

wn(w1mn_1) = w1(m1)¢n_1(wn_1) As to the multiplicative property =

wnmpmq)

wpmpmqmq),

it is immediate for For

p 2 2

w

=

n

= w1mp_1, and we use the inductive hypoth-

as a homomorphism of graded unital associative k-alge-

n z 0

d

9n+1A l¢n+1

a

n

———’

w

respects the differentials, i.e.

the following square is commutative:

n

———’

lwn

zn

9'"!

1.

w

We have to makesure that

that for QnA

mg 6 flqA I

P

We thus get bras.

p = O,

we decompose

esis.

up 6 OpA I

zn+1

Fot the first square (n = 0) this has been seen to be true. Let us look at the second square

Take

w1 = ad°b 6 91A, a,b e A.

Now,

d1(ad°b) = (d°a)d°b, and

(i.e. n = 1).

w2(d‘(ad°b)) = w2(d°ad°b) = w1(d°a)¢1(d°b) by definition of

$2.

0n the other hand,

¢1(ad°b)

= ¢°(a)w1(d°b),

hence

a1w1(ad°b) = 31(wo(a)w1(d°b)) 8°¢°(a)o1(d°b) + wo (a)a‘¢1(d°b) = w1(d°a)¢1(d°b). Since

1 o _ 1 o _ 3 ¢1(d b) - 3 a ¢°(b) - O, and we are through.

Let us look at

n >

We can decompose

1.

wn = w

Then we get by the inductive hypothesis:

n-1 _ 1 _ n ”n+1d ”n - ¢n+1((d w1)mn_1 w1d wn_1) 1 - «’2 (d “1)‘9n-1wn-1 - ¢1(w1)wn(d

_

1

3‘”1”1’¢n—1“n—1 _

_

”1‘“1)3

n-1

n-1

(”n-1)

wn-1wn-1

n

- 3 ((1421001 )wn_1wn_1) n a

q’nwn

The uniqueness of the extension of lows from the uniqueness of

$1:

A

Q,A

and

doA c 91A

Comglement 1.1.6

tension generate

generate

Let

w : 9(A) + z Z

we:

mo: A + Z0

91A + Z1

to

w : 9(A) + Z

fol-

together with the fact that

as a k-algebra.

A = 90A + 20

be surjective.

is surjective if and only if

20

Then its ex-

and

a°z°c:z

as a k-algebra.

Remark-Definition 1.1.7

Let

[9*A,9*A]

be the graded k-submodule of

1

94 9,A

which is generated by all graded commutators, i.e. by all elements upmq - ( - 1) pgwq, up 6 9 PA, wq E 9 qA I p,q 2 0.

of the form

We can

write for the n-th homogeneous component: 9 A Q A

[*’*]n

Define

=

2

Q A Q A

Malp'q]

M2(A) = n,A/[9,A,Q,A]

An(A)

is graded via the quotient grading:

A 9(A)

=

n

flnA/P+g=n[9PAIQqA]

Furthermore, we have

d(mpw

_

q

( _ 1)qq)

d[n,A,Q*A] c [9*A,Q*A]: =

(dwp)wq + ( _ 1) Pdwq

_

( _ 1) Pq (dwq)wp

- (-1)Pq(-1)qwqdwp ((dMq _ ( _ 1) (9+1)qwqdwp) +

for

( _ 1) P (mpduuq _ ( _ 1) P(q+1) (dwqhup)

A E Q qIq A 2 0. mp E Q lq

Hence we get an exact sequence of cochain complexes

0 —b ([Q,A,Q,A],d)—-—-D MA) --—§ (Afl(A).d)-—9 (A9(A),d)

is called the de Rham comglex of (noncommutative exterior)

differential forms on the unital associative k-algebra

Note that in general

[9*A,Q*A]

hence the projection

n(A) + An(A)

algebras. that

[A,A]

do).

Since

0

A.

is not a graded ideal of

Q‘A,

is not a homomorphism of graded k-

For example, take any unital associative k-algebra is not a two-sided ideal of

[A,A] = [Q‘A,9*A] n 90A, [9*A,Q*A] Remark 1.1.8

Let now

A

A

A

such

(matrix algebras will already

cannot be an ideal of

0*A.

be a unital commutative k-algebra.

Then

A°fl(A) = A/[A,A] = A, and Mom) = Q1A/[A,Q1A] identifies with

QA/k’

the A-module of

(Kahler)

k—differentials for

A

(cf. [Ma, pp.180-139]): First,

[A,01A]

91A/[A,Q1A]

is an A—A submodule of

a mere left A-module).

do: A + A1Q(A)

with the following property: bimodule)

91A, and thus

becomes a symmetric A-A bimodule

M,

A1Q(A) =

(i.e. can be treated as

is now a universal k-derivation

For every left A-module

and for every k-derivation

d:

A + M

(symmetric A-A

there isa unique

factorization

A

----—9 M

13° / A19(A)

with

f E HomA(A1Q(A) ,M).

This follows immediately from the universal factorization property of do: A + 91A, together with the fact that an A-A homomorphism M

with values in a symmetric A—A bimodule

hence factor through

91A + A1Q(A). flA/k;

universal property of

M

A1D(A)

f: 91A +

must vanish on

[A,91A],

has thus precisely the

consequently, we may identify both A-

modules. Consider now the Kahler-de Rham complex (RA/k'd)

9

ferential forms on A (of. [Bou,Ax. 43]). is the exterior algebra of the A-module -

derivative on forms

u

.

d.

flA/k + QA/k’

of a skew-commutative d.g. of

n(A) =

(9*A,d)

of cochain complexes ously an isomorphism.

o n“

=

A”

no A/k

which gives

(cochain) algebra.

OA/k

For

[9*A,Q*A]

Definition 1.1.9

the structure

(cochain) algebras

n:

n: AQ(A) + QA/k;indegree 0 and 1 this is obviAs to higher degrees, it is easily seen that [9*A,D*A]

fl

is

Q,A.

contains

A).

ring of dual numbers over [T,dT]dT 6 92A

Mk

[9*A,Q*A], hence induces an epimorphism

A = k[T], a polynomial ring in one variable over

means that

n20

By the universal property

is an isomorphism of cochain complexes if and only if a two-sided ideal of

= 9 Ann 1

QA/k, together with the "outer

we get a surjection of d.g.

9(A) -> (IA/k, which vanishes on

of (exterior) k-dif-

9 9nA

(since

n22n In a moment

9

k, this simply = A 6 As,

the

A/k (1.1.13)we shall see that

is not a k-linear combination of graded commutators.

of

(noncommutative) n

de Rham cohomologx n-1

3*(A9(A),d):

Hn(l\Q(A),d) = Ker(Anfl(A) 1» An+1n(A))/Im(An_1Q(A) ‘1 —~y A“n(A))

96 Complement 1.1.10

on graded traces.

Let us consider the dual complex 0

0(—

(A SHAH

(where

( )*

3

dt

1

(A SHAH

‘—

stands for k—dual

(A9(A)t,dt) a

at

of the de Rham complex:

(A 2 SHAH

‘—

It

(—

Homk(-,k)), and where

dt = Hom(d,1)).

We have — Homk(9nA/ (A n n(A)) a _

[9PA,QqA],k)

2 p+q=n

{T : 9n A + k: for a 11 Such k-linear

T:

1(wpwq) = ( _ 1) Pq 1(wqmp)

up 6 QPA, ”q 6 flqA, p += q n}

nnA + k

(which vanish on graded commutators)

called graded n-traces on (A9(A)t,dt)

k- linear and

T

A (or

will be

9(A)).

is thus the chain complex of graded traces on

A (on Q(A)).

Note that we have for the k—module of n—cycles:

Zn(An(A)t,dt) = {1: flnA + k,

1 is a closed graded n-trace on

(where closed means classically:

1(dw

n-1)

= O

for all

u

n-1

A} E n n-1 A)

We can summarize: The de Rham cohomology

H*(AQ(A),d)

measures the existence

(and the

amount) of nontrivial closed (noncommutative) differential forms on The de Rham homology

H*(AQ(A)t,dt)

amount) of nontrivial closed graded traces

Remark 1.1.11

The operator

Hochschild boundary operator Define

8(A) = 0, and

B: 9*A + 9*A. b

-1

A

8(wda) = (-1)|ml[m,a] = (-1)n-1(wa-aw) We have first to make sure that

(integrals) on

B

(n 2 1) for

tion

91A

as a subset of

0,A.

by the formula:

w € 9

n-1

is well-defined.

is a right Ae-module (cf. the beginning of I.2.1): Consider now

A.

We want to imitate the

as an operator on

B: nnA + 9n

A.

describes the existence (and the

A, a e A. Note that

finA

mn(x 0 yo) = ywnx

Ae = A 8 AOP, and define a composi-

97 9

n-1 A

X D A

+

1

n-1 A

n

9 ('1) wn_1-w1

(wn_1,w1) (where

Q

w1

operates as an elements of

It is immediate that through

(wn_1a)-w1 = wn_1-(aw1), hence we can factor 8: QnA + fln_1A, and we have actually

nnA = Qn-1A : 91A ;we get

“Wrflfl —(1)wm1(16a -

(

o

n

—

—

1)

n-1

Ae).

-

ae1)

=

(1)(mmflt%qm _

n

—

[tun-1’3]

as desired. By induction on

n

one sees easily:

“-1 i n 6(aoda1...dan) = i:o(-1) aoda1...d(aiai+1)...dan + (-1) anaoda1...dan_1 Thus we have got a representation of the Hochschild boundary operator b

as an operator on the differential envelope.

Note the trivial fact:

BQ*A c [9*A,Q,A],

more precisely:

BflnA = [A,9n_1Al, n 2 1

Thus we have a surjection of graded k-modules

n,A/Bn,+1A + AQ(A) which is simply given by 9 A/[A,9 A] + Q A/

n

n

Lemma 1.1.12 (1)

n

E

p+q=n

[Q A,Q A],

p

n 2 0

q

(Lifting of

A“n(A)

The cyclic group

Gn =

_ 0...0 wn -

mn e m1

inside

of order

nnA/Bnn+1A ). n

operates on

flnA/Bfln+1A

via t.w

(2)

1

Assume

(

_

1)

Q c k.

n-1

Then

8...0 wn_1,

Ker(l-t)

Ann () A = nn/P+q=n[P,q] A z 9 A n A .

mi E 91A

maps isomorphically onto

98 Proof.

(1)

Note first (since we are working with tensor products over

that the action of

Gn

on

nnA

A)

We have to divide

in order to get a well-defined action.

Bnn+1A = [A,nnA]

out precisely

is not well-defined.

We shall altogether suppress equivalence class notation. (2)

Kn =

With

Z- [nPA,9qA]mod 89n+1A p+q—n

it is immediate that

D(nnA/Bfln+1A) c Kn'

We want to show that

Kn

Consider

_

_

w1...wpw1...wq

and

D = (1-t)

(as usual),

Im D.

_

_

( 1)

Pq-

_

w1...wqw1...wp

with

_

wi’wj 6 91A,

1 s 1 s p, 1 s j s q. We have (everything taken q

—

—

t (u1...w w ...wq) p 1

mod 80

n+1

A, of course):

('1)(P+q-1)q51...5q w 1 ...w

— ( _ 1) Pq— m1...wqw1..wp and consequently _ _ _ w1...mpm1...mq

_ pg. _ ( 1) w1...wqw1...wp

-

' (1 tq )(w1...mpw1...wq)

— D(1+t+...+t Q'1 )(m1...mpw1...wq) 6 In D. since we have assumed that

Q c k, we get

nnA/Bnn+1A = Ker(1—t) 0 Im(1—t), i.e. Ker(1-t)

maps k—isomorphically onto

Conseguence 1.1.13

Assume

ring in one variable over

de Rham complex

A9(A)

Ann(A).

w c k, and let k.

A = l]

be the polynomial

We want to show that the noncommutative

does not coincide with the usual de Rham complex

QA/k' It suffices to show that

A29(A) # 0.

Taking in account 1.1.12, we shall consider show that

w = [T,dT]dT 6 92A, and

w :2 893A

(i)

(11) w 6 Ker(1-t), where = -w2m1,

(1):

8w = T[T,dT]

-

[T,dT]T

2

Indentify

A o A

with

dT = Y - X,

2

6 91A.

k[X,Y], where

_ 3 _ Bw — (dT) —

(Y-X)

3

where

Hence:

m = m1m2 - w2m1

m1

w2 = dT.

m

(1+t)m1m2, and consequently

zero element of

Remark 1.1.14 9(A)

9(A)l)

(1-t)m=0;w represents a non-

We want to consider more closely the case of an aug -

A = k 6 K, where

A

is the augmentation ideal of

A.

has an alternative description, as follows (cf. [C0, p.991):

finA = A a An

Set

whereas

A20(A), which shows our claim.

mented k-algebra Then

A e A, not in

We have: TdT,

x,

Y.

# 0

(the multiplication is now in (ii):

x = T o 1, Y = 1 O T.

and left T-action is multiplication with

right T-action is multiplication with Thus:

t(w1w2)

Bm # 0.

T dT - 2TdT.T + dT.T

Then

is given by

w1,w2 E 91A.

We show that

But

t: nzA/BQ3A + 92A/893A

(the tensor products are over

k).

We have a right

K—action defined by the formula n—1

.

(a°;a1,...,an).a = i§o(-1)n 1(a07a1,...,aiai+1,....,an,a) + (ao;a1,...,an_1,ana)

This X-action is associative, and extends to a unitary right action of

A

on

finA, which becomes thus a right A-module.

Define a composition

fimA x finA + fim+nA by the formula:

mm.(a°;a1,...,an) = (mm.a) 9 a1 @...0 an

100 (with the obvious identifications).

0 finA n20

§.A =

becomes a unital associative graded k-algebra.

d: fi*A + §*A, defined simply by satisfies trivially

d (wpyq) =

dwp . ”q

+

d2 = O,

and one checks that

( -1 ) Pmpd mg

f or

Consequently, §(A) = (§*A,d) The important fact is that 9(A)

(cf. 1.1.5).

mm

d(ao;a1,...,an) = (1;a°,...,an)

up 6

h PA ,

mg

e hqA

is a d.g. (cochain) algebra. 5(A)

has the same universal property as

Hence there is an isomorphism of d.g. algebras

9+ am)

which identifies a1,...,an

aoda ...dan 1

as elements of

with

A

or of

(ao;a1,...,an) A;

(you may think of

the notation will be coherent

in either case). Note that this isomorphism shows in particular

nition of d on the §(A)-side) that an augmented k-algebra A = k e A. Lemma 1.1.15

Let

A = k e A

(together with the defi-

9(A) = (9*A,d)

is acyclic for

be an augmented k-algebra.

For

consider the isomorphism e.

nnA 3 aoda1...dan

+

_ -n (a°,a1,...,an) e A 8 A

Then the image of a graded commutator is given by the formula:

0([aoda1...daK,

an+1 da n+2 ...dan+1])

K

Z (-1)K i(30;...aia 1+1 ...a n+1 )

=

i=0

n-K

-

2 (-1)

K(n-K)+n-K-i

i=0

(Convention:

Proof.

,

(aK+1'"'aK+1+iaK+1+i+1"'aK)

an+2 = a0)

It is immediate that

n 2 0

101 K

. 6((aoda1...daK)(ax+1dax+2...dan+1)) _ - 1:0( _ 1) K'i (30,...aiai+1... n+1) and that 6((ax+1daK+2...dan+1)(aoda1...daK)) n-K

=

' a K+1+i a K+1+i+1'.'. a K ) 2 (-1) n_K_j'(a K+1'..' i=0

which gives our assertion. Conseguence 1.1.16 ao,a1,...,an+1

on

In the situation 1.1.15, assume furthermore that

6 A.

Consider our standard operation of

Kn+1 I K a in c A o in.

Gn+1

=

Then

9([aoda1...daK, aK+1 da K+2"' dan+11) = (-1)Kb(a°;...an+1)mod Im(1-t). Proof.

K+1

t

We have

,

(aK+1,...aK+1+iaK+1+i+1,...aK) _

(K+1)n

_

(K+1)n

( 1) ( 1)

_

for

(ao""aK+1+iaK+1+i+1""an+1) ,

_

_

i - n K

for

(an+1ao,a1,...,an)

_

,

i < n K

Thus we obtain n-K

E (_1)K(n K)+n K 1

(aK+1;"'aK+1+iaK+1+i+1""aK)

i=0

_n—K-1 _

i n-K

— 1:0 ( 1) t

= -

n 2

_

_

(ao'"aK+1+iak+1+i+1'"an+1)+( 1)

. (-1)K-1tn-K(a°;..a iai+1"'an+1)

(‘1)

+

n-K n-K t

n-K n-K t

_

(an+1a°,a1,..an) '

(an+1ao,a1,..an)

i=K+1

From 1.1.15 we obtain finally

0([aoda1...dax,a

K+1

da K+2 ...da n+1 1) n

_

.

_ _ . - ( _ 1)Kb(a°,a1,...,an+1) (1 _ t n-K )(i=:+1( _ 1) K-l (a0...aiai+1,..an+1)

+(_1)n+1(a n+1a°;a1,...an))

102 which proves our claim.

Lemma 1.1.17 every 8:

Let

A = k e A

be an augmented k-algebra.

Then, for

n 2 1, the isomorphism

QnA 3 doda1...dan

-n

+ (a°,a1,...,an) E A e A

induces an isomorphism

e: A“n(A)/dA“"n(A) + [‘A“+1/(1—t)1mod m b. Proof.

(*)

In complement to out result 1.1.16 we have the formula

n (-1) andao...dan_1

aoda1...dan

n-1

= (-1)n[da°...dan_1,an] - i:o(-1)ida°...d(a1ai+1)...dan Now, by 1.1.16, 9

An9(A)/dAn-1S}(A)

is actually well—defined as an application from

to [Kn+1/(1-t)]mod Im b.

But, by virtue of the formula (*), combined with 1.1.16 other direction), the inverse application

6-1

(read in the

is well-defined, too.

This shows our assertion.

Theorem 1.1.18

Assume

m c k, and let

A = k 0 K

be an augmented k—

algebra. Then,

for every n 2

1, we have an exact sequence

0 —-> Hn(An(A),d)-—eb §Cn(A)—BO in“ (A) (i.e. non-commutative de Rham cohomology lies inside reduced cyclic homology). Proof.

In order to show the injectivity (and well-definedness!) of

Hn(Afl(A),d)

e

+ fiCn(A), we have only to exhibit a commutative diagram

A“n(A)/aA“"n(A) _°.._.

1.

An+19(A)

[PH/(puma Im b

1.,

———‘°—, in/u-t)

103 which, by I.2.5.18 and 1.1.17, will show our first claim. Now, for every upper triangular matrix eij E {0,+1,—1}

a = (eij)osisjsn+1

with entries

(and zeroes on the main diagonal) you can define

w 6 : “n+1A + A 9 ip-1 ws(a°da1...dan+1)

by the formula

=

E

e..(aiai+1;...ajaj+1,...ai_1)

i

and

is the k—algebra of

132 infinite matrices which have only a finite number of non-zero entries.

Then

(E(gl(A)),d)

+

(E(91(A))g.d)

is a quasi-isomorphism.

Proof. (1)

Some preliminaries:

Put

9r = g1r(k), with

1 S r S m (9°° =

g = lim 9r)' and let (sgn)

be the 1—dimensional yn-module on which

Tn

acts by signature: 0.1 =

e(a)1, a E yn. n n En(gr s A) H (gr 6 A )

Then

8

(sgn)

Yn (where

Yn

acts on

9: 0 AP = 9:“ 8 Asn

by parallel place-permutation:

[(x1,...,xn) s (a1....,an)]o = (“0(1)""’xo(n)) 0 (au(1),..-, 30(n)) This isomorphism is an immediate consequence of the very definition of

[\“(91" o A)

as a quotient of

k-linear span of all

(gr 8 A)n = (gr 8 A)”: divide out the

21 s...s zn - e(a)z°(1) 8...O za(n)’ a 6 yn.

Now, comparing the adjoint action

and the adjoint action

adx

of

9(x)

x 6 gr

of

x e 9r

on

On g: = 9r I we get, by trans-

on

En(gr s A)

port on the right side of our isomorphism:

6(x).[(x1,...,xn) a (a1,...,an) 8 1] = [adx.(x1,...,xn)] e (a1,...,an) o1 Consequences:

(i)

En(gr 0 A)

92 = 9:“

( ii )

is semi-simple under

is semi-simple under

E( n gr 0A) 9:

(") 9: a [gr

9: gr + Endk(En(9r g A)) provided

ad: 9: + Endk(g:)

9A“ ] Ynosn ( g)

(for the coinvariants) (2)

Let

r

be finite.

We shall first show that

(uglrmnm + (E(g1r(m)gr'd)

133 is a quasi-ismorphism.

(where

g1r(A) = gr 0 A = (Mr(A).[;]))

According to 2.1.11 and (1), consequence that

g:

is semi-simple under

equivalent to saying that

ad:

(i), we have only to make sure

gr + Endk(g:).

g: is semi-simple under

(cf. the arguments in the proof of 2.1.7).

from the linear reductivity of (3)

GL(r,k)

Note that this is Ad: GL(r,k) + GL(gE)

Our assertion follows now

(cf.

[Fo, p.146]).

We have now to pass to the direct limit.

The commutative squares

(of homomorphisms of chain complexes)

I

I

(E(glr+1(A)),d) —-————9 (E('§1r+1(1\))g

(E(91r(A)),d)

—_-9

,d)

(E(glr(A))gr,d)

allow to pass to the direct limits. By virtue of (1), the direct limit arrow identifies with

(E(91(A)).d)

+

(E(91(A))g,d)-

Since homology commutes with direct limits,

we get finally our asser-

tion.

11.2.2.

Cyclic homology and the Lie algebra homology of matrices.

Situation 2.2.1

k

a field of characteristic zero.

A

a unital asso-

ciative k-algebra. Mr(A)

the k-algebra of rXr matrices with coefficients in

M(A) = MD(A) = lim Mr(A)

A.

the k-algebra of infinite metrics which have

only a finite number of non-zero A—entries.

g1r(A) = (Mr(A),[,]) the Lie algebra of rXr matrices with coefficients in

A.

The standard inclusions

= (M(A),[,])

glr(A) c g1r+1(A)

define

g1(A) = lim g1r(A)

134 Notation:

gr = glr(k)

for

1 s r s m

9 = 9, = 91“(k) = (“(k),[,])

Note that glr(A) = 9r 8 A

1 S r S w, in particular

gl(A) = g 8 A. Lemma 2.2.2

Consider, for

1 s r s w, the sequence of maps

An: En+1(glr(A)) + c;(Mr(A)) defined by

An(x° A...A xn) = (-1)n )3 oeyn

e(c)(x°,xo1 ,...,x0(n) )mod(1-t)

1,: (E(q1r(A))[-1],d[-1]) + c§(Mr(A)) is a homomorphism of chain complexes. Proof.

Note first the dimension shift; furthermore, observe that for

r - a A

we are in the setting of non-unital cyclic homology.

is well-defined, thanks to the cyclic permutation relation

(a°,a1,...,an) = (-1)n(an,a ,...,an_1)mod(1-t), 0

on the right side. cycle

We shall consider our generator

t = (O,1,...,n)

of length

n+1

in

t E Gn+1

Yn+1 = Y{o 1 r

need

t

(x n ,x Go ,...,x

for the transcription of

(x°,xa1,...,xan)-tensors;

cf.

0(n-

bA(xo A...A xn)

1))-tensors to

the detailed arguments below).

Ad(x° A...A X“) (-1)

n

2 vEYn

in c"n-1 (Mr (m) .

e(v)(xv xv ,xv ,...,xv ) o

1

2

(we

I“'l

In the sequel we shall drop the "mod(1-t)"-notation.

Claim.

as the n}

n

135 For

t = (O,1,...,n) _

Decompose

we have

o

5(tK) =

1

(-1)Kn

n

Yn+1 _ Yn+1 U Yn+1 U"'U Yn+1

K — n+1: Yn+1 - 0}, O s K S n (Convention.n Yn+1 — {v E Yn+1'I v(K) — yn+1

where = Yn"

We get a bijection Now,

_ i n+1-i yn+1 3 a + v — act 6 Yn+1 .

let us write down

_ bA(x°A...Axn) -

_ n ( 1) Z €(°)b(xo’xo(1)""'xo(n)) aeyn

= (—1) n E o

e(a)( “-1 E (-1) i (xa(o),...xo(i)xo(i+1)...,xa(n))

a€yn+1

i—O

n + (-1) (xo(n)xo(o)’xo(1)""’xa(n-1))) We have (x

0(0)’

A . Cn_1(Mr(A)).

in ...,x

_

_

a(i)xo(i+1)""'xo(n))

i(n-1)

_| v

|

A

HA 5 I

‘ ( 1)

1)

(xa(i)xa(i+1)""’xa(n)’xo(o)""'xa(i-1)) (x

x

,...,x

ati(o) at1 (1)

,x ,...,x

at1(n-i)

°

Consequently:

a: o

e(o)(-1)

1

(xo(o)""’xo(1)xo(i+1)""'xo(n))

Yn+1

E o

e(o)(-1)

oEY n+1

in

(x

.

x

.

atl(o) ot1(1)'

...,x cti(n)

E 6(0ti)(x . x . ,x ,...,x ) o€y3+1 at1(o) ot1(1) oti(2) ot1(n) Z vEYn+1

_

e(v)(x i

x x ... x ) v(o) v(1)’ v(2)’ ’ v(n)

n+1 Analogously:

Z e(o)(-1)n(x a(n) x ,x ,...,x _ ) vn+1 0(0) 0(1) o(n 1)

)

oti(n)

136 =

E e(v)(x x ,x ,... I x ) VEYn+1 v(o) v(1) v( 2 ) v(n)

and finally _

_

bA(x°A...Axn) - ( 1)

n

2

e(v)(xv(°)xv(1),xv(2),...,xv(n)).

vEYn+1

0n the other hand

Ad(o...Axn)

=

_

( 1)

n-1

_

OsiEjsn( 1)

i+j

UéYi,j€(°)([xi’xj]’xa(o)""’xo(n)) n+1

where

Y;;¥ = {a 6 yn+1z o{i,j} = {ilj}}:

i < j

hence:

Ad(x°A...Axn) = (~1)“'1 Define y

n+1

_ —

(i’j) = {v E

Yn+1

z

osi

is the homology class of where,

for

x1,...,x

P

y1,...,yq E glr(A), say, the exterior product above is now in Ep+q(912r(A)).

H = H*(gl(A))

becomes thus a graded unital associa-

tive k-algebra. The verificationcafthe required properties for the graded algebra structure as well as for the graded coalgebra structure of straightforward.

H

is rather

As to the compatibility properties, which give finally

the graded Hopf—algebra structure, the only delicate point is the verification of the fact that the comultiplication

Azfl + H 0 H

is a homo-

morphism of graded k-algebras. Let us look at the situation. Take

u = [X1A-..AXP] E HP(gl(A)), v = [y1A...q] E Hq(gl(A)).

We have to show that A(u)A(v) = A(u(u 0 v))

in

(H 8 H)p+q‘

(the multiplication on the left side is in the graded tensor product H O H)

A(u)

[(x1 8 1 + 1 O x1)...(xP 8 1 + 1 8 xp)]

MV)

[(Y181+10y1)...(yq®1+16 q

(the product inside the brackets is the product of the graded tensor

product

E(91(A)) O E(91(A))

u(u 0 v)

is represented by

x1 0

( O

x 0 0 ) A...A ( P ) A ( 0 O 0 0

0

0 ) A...A ( y1 0

o

); yq

consequently

Wu M)=[((:1:)e1+1e(:1:))...((: °)e1+1e(: :))] Yq Now look at

A(u)A(v)

The multiplication in

in

(H e H)P+q.

H O H

is given by the following formula:

q

149 ([a1A...Aar] 0 [b1A...Abs])'([aiA..-Aa;,] 8 [biA...Ab;.]) =

. .

=

o

o o

b o

o o

(-1)5 r [(21 0)A...A(o 3;.)] 9 [(01 O)A...A(o bé,)]

This implies:

(i)

Au([x} 0 [Y])

A([x])A([y])

[C :)A(::)°1+1°C:)A(°0 °)] Y for (ii)

With

u = u1 u

A...A ur, 0

O

we have:

=

v = v1 u

O

)

O

(2 :j)

A...A vs

0

) A...A ( r

(5,0) = ( 1

(or?)

x, y E gl(A)-

O

(Z 35) x

0

0

0

)A ( (0,6) )1

Au([EAx]o[V]) = A([EAx])A([V]) = [A((E,0))A(

(where the last product inside the brackets is in

E(gl(A)) 0 E(gl(A))).

Analogously for the other side. Our result follows by induction on (4)

H = H*(gl(A))

n = p+q.

is a connected commutative and cocommutative graded

Hopf-algebra. (a)

Connectedness means merely that

(b)

Commutativity (in the graded sense) means that

h ph q = ( -1 ) tqh p

f or

a1l

hp

Ho = H°(gl(A)) = k.

e

H P' h q

E Hq, p,q

2

0.

This follows immediately from the definition of the multiplication (and from Lemma 2.2.11),

since,

0 conjugate by

for

x,y € glr(A),

Er = (

Cocommutativity means the following:

Let

T : H 8 H + H o H

h

o h

q)

=

(

—1 t

)

ur(y,x)

O

(c)

n( p

and

r) E GL(2r,k). -Idr

T

ur(x,y)

Id

q

be the twisting morphism given by

e h

p

f

°r

h

H

p E p’

h

H

+

q E q' P q

=

n

.

are

150 Then the following diagram is commutative:

H

.,.—A,.—

—'WH

9

H

1T

A\’HOH This property is already valid on the chain-level:

T: E(gl(A)) o E(g1(A))

->

E(gl(A)) e E(gl(A))

is an automorphism of the graded k-algebra

E(gl(A)) 0 E(gl(A));

furthermore,

A: E(91(A))

+

1303“”) 0 E(91(A))

is a homomorphism of graded k-algebras,and for all

A(x) = x 9 1 + 1 o x = TA(x)

x E g1(A).

Remark 2.2.13

The primitive part of

(1)

Preliminaries.

Let

k

be a field of characteristic

graded Hopf-algebra over Consider

I(H) = Ker€ =

H*(gl(A)).

0, and let

H

be any connected

k. Q Hn’ and look at the two exact sequences n21

1(a) own 3 1(a) + mm + o

o + pm) + 1m) 5 1(a) sun) which define

P(H)

and

0(3).

indecompgsable elements of the primitive elements of

The elements of

Q(H)

H, whereas the elements of

are called the

P(H)

are called

H.

More explicitely, we have

Q(H) = H 1 o Hz/HIOH 1 0H3/H10H2 + H 2 0H 1 6 P(H) = {x E H: A(x) = x 8 1 + 1 8 x}. There is a natural homomorphism

whenever

H

P(H)

+ 0(a),

is commutative and cocommutative

(2)

The graded Lie algebra structure of

Let

H

(cf.

[MM,4.18,p.234]).

P(H).

be a graded Hopf-algebra.

Define the graded commutator

( 1)

which is an isomorphism

hq p

for

hp 6 Hp,

[,]: H 6 H + H

q E H q , p,q 2 0.

by

[hp,hq] = hphq -

151 (H,[,])

becomes a graded Lie algebra in the following sense:

[hp, h q]

= - ( - 1) Pq [hq. h P]

(11)

( 1) Pr [hprlhtrll +

( _ 1) QP [hq’[hr’hp]] +

-

(1)

( _ 1) rq [hr.[hp.hq]] = 0

(for homogeneous elements of the indicated degrees)

Claim.

(P(H).[,])

is a graded Lie subalgebra of

We h av e show th a t

Now,

A

(xpyq)

A

=

[XPIYq] E P ( H )P+q

(xp) A (yq)

(xp

=

e 1 + 1 e

xpyq

0 1

xpyq

fo r +

+

1

x9

0

e

(H,[,]).

KP 6 P ( H )p’ Yq E P ( H )q .

xp)(yq

yq

+

(

8 1

-1 Pg

)

+

1

yq

0

0

Yq)

“p

=

.

Thus A([xp,yq]) = [xP,yq] @ 1 + 1 8 [xp,yq]. Conseguence.

Let

of characteristic

A

be a unital associative algebra over a field

0.

Consider the graded

cocommutative) Hopf-algebra

H = H*(gl(A)).

is an abelian graded Lie algebra

Remark 2.2.14

from its primitive part Let

[,]

Reconstruction of a graded

cocommutative) Hopf-algebra

(a)

(with

H

k

(connected, commutative and Its primitive part

P(H)

E 0).

(connected, commutative and

over a field

k

of characteristic

0

P(H).

V =

e V be any (positively) graded k-vector space. K K21 of V over k, with the following Consider the tensor algebra T(V) grading: T°(V) = k

Tn(V) = Vn e (V o V)n e (V 6 V 8 V)n e ...

where

Z VK K1+K2+. . .+Km=n 1

Put

(VOm)n =

I

= T(V)/I, where

A(V)

8 VK

yq E Vq,

grading of nonical map

p,q 2

1.

T(V)), A(V) T(V) + A(V)

Furthermore, k 0 V + A(V) (I

. m

is the two-sided ideal in

is generated by all graded commutators XP 6 VP,

8...® VK 2

Since

I

n 2 1

xp 8 yq -

T(V)

which

(-1)qq 8 xp,

is homogeneous

(for the total

inherits a quotient grading such that the cais a homomorphism of graded k-algebras. is a monomorphism of graded k-vector spaces

is generated by quadratic elements, for the usual grading of

A(V) sense 3

is the free graded commutative k-algebra on

V

T(V)).

in the following

152 For every commutative graded k-algebra graded k-vector spaces graded k-algebras

f: V + B

T: A(V) + B

B

and every homomorphism of

there is a unique homomorphism of such that

V -——————+ A(V)

;\\N

(//§ B

is commutative. The structure of

A(V)

is easy to describe.

Consider first two particular cases: (i)

V

2v

= 0

for all

v 2 1.

Then

A(V) = E(V), the exterior algebra of

(11)

V2v+1

Then

A(V) = S(V), the symmetric algebra of

= O

for all

V

over

k.

v 2 0. V

over

k.

(with, in

both cases, a total grading coming from the interior grading of analogy with the grading of Since

T(V)

A(V 0 W) = A(V) 0 A(W)

V, in

above).

(graded tensor product), we obtain imme-

diately the general case:

Decompose

v=vov_

withV=

+

Then

+

e

v,v_=

K even

K

o

v.

K odd

K

A(V) = S(V+) 0 E(V_)

Furthermore,

s(v+) =

0

S(V ), E(V_) =

K even

K

9

E(VK)

K odd

(all tensor products are graded tensor products of graded k-algebras (direct limits!)). Note that (b)

An(VK) = O

Let now

L

be a

for

n 1 0 mod K.

(positively) graded abelian Lie algebra over a

field

k

of characteristic

space

V

without extra-structure, its universal enveloping algebra

U(L)

Since

(in the graded sense) equals

cussion of U(L)

0.

U(L)

A(V)

in the general case).

in our particular setting,

too,

L

is merely a graded k-vector

(cf.

[MM]

for a detailed dis-

We shall nevertheless write

in order to accentuate the context

of ideas. U(L)

is a connected commutative and cocommutative

(graded) Hopf-algebra.

This is clear, once you have observed the following: The diagonal

A: L + L x L, which isalumwmorphism of graded (abelian)

Lie algebras, prolongs to the comultiplication

A: U(L) + U(Lx L) =

U(L) 0 U(L), which is a homomorphism of graded k-algebras.

The twisting

153 morphism

T: U(L) o U(L) + U(L) 0 U(L)

algebra

U(L) 0 U(L).

Thus the equality

fied on

L, which generates

x 9 1 + 1 e x = T°A(x) (c)

U(L)

for all

is an automorphism of the graded A = ToA

need only be veri-

as an algebra over

k.

But

A(x) =

x 6 L.

The foregoing observations, together with 2.2.13

(2), yield imme-

diately: For every connected commutative and cocommutative H

(graded) Hopf-algebra

there isa natural homomorphism of graded algebras

which is induced by the inclusion phism of Hopf-algebras). Theorem:

P(H) c H (o

w: U(P(H)) + H

is actually a homomor—

The main result is the following

In the situation above

(char k = 0!)

we have:

w:U(P(H))

+ H

is an isomorphism Proof.

This is a particular case of [MM, 4.18,p.234] combined with

[MM, 5.18,p.244].

Note that the standard (i.e. trivially graded) ver-

sion of this theorem can be found in [BL, p.15].

The important fact is the possibility to reconstruct ject over

H

as a free ob-

P(H).

In order to tie finally everything together, we have to show the follow— ing Proposition 2.2.15

k

a

field of characteristic

0, A

ciative k-algebra, H = H*(gl(A)) the homology of

a unital asso-

gl(A), with its struc-

ture as a connected commutative and cocommutative (graded) Hopi-algebra (2.2.12(4)).

Then

P(H) = Prim H*(g1(A))

Proof. via

(in the sense of 2.2.9)

Recall first the (adjoint) action of

g = glw(k)

on

E = E(goA)

ezg » DerkE, where

n 6(x)((x10a1)A...A(xn0an)) = i:1(x10a1)A...A[x,xi]0aiA.../\(xnean) Considering

the

action of

g

on

E 6 E

given by

x + 6(x)91+186(x),

it is easy to verify that

Ae(x) = (9(x) O 1 + 1 9 6(x))A (i.e. the comultiplication

Thus

93

is a coideal of

A

for all

x e g

is a g-homomorphism)

E:

A(gE)cE09E+9EOE and

E(g 0 Mg

becomes a quotient coalgebra of

E = E(g O A).

154 But, revising the arguments about the well-definedness tivity) of the multiplication on that everything

works already well on

2.2.12(3) and (4)) E(g 9 A) tive

H,(gl(A))

(graded) Hopf—algebra over

u = u1

we have:

E(g 9 A)9.

Thus (look at

is a connected commutative and cocommutak.

Furthermore, the differential of For

(and associa—

(Lemma 2.2.111), we see

A...A up, v = v1

d(uov) = (du)ov +

E(g e A)g

A...A vq

is a (graded) derivation:

(think of equivalence classes)

(—1)pu0dv

This is easily verified:

W( )A...A(p)A( )A...A( ) u10

110

00

00

O

0

0

0

O

1.11 O

and

[(

O

O

)1 = 0 vj

0

The explicit formula for (E(9 0 A)

,d)

v1

vq

O

),( 0

O

1 s i s p, d

1 s j S q

yields immediately the assertion.

is a differential graded Hopf-algebra.

Thus

In order to prove

the assertion of our proposition, we have to show

(1)

P(E(g o A)g)

PE(g 0 A’g

(in the sense of 2.2.6(3))

(2)

P(H*(gl(A)))

H,(P(E(g O A)g))

Look first at (1). Recall our reduced isomorphism (2.2.5(3))

(kIYnl o A“) e (sgn)

I En(g e A)g

Yn

c e (a1,...,an) o 1 (where

e

E: mod gEn(9 o A)

A a a a _ 1 n Ea — E1c(1) A...A Enc(n))

which identifies

“‘[Un] o A“) o

(sgn)

with :a(9 a my

(2.2.7)

Yn

Let us calculate

“a “a A(Eo) = p+g=nAP’q(Ea)

the following formula:

“a =

Ap,q(Eo)

. an

u an

(§)E(°) Ep.cp ° Ep.ao'

w h are

“a Ap,q(Ea)

i 3 given by

155 the sum running over all on

p E Yn

which are increasing on

[1,p]

and

[p+1,n]; we have exlicitely

“an

=

"Eap

=

'Eo,op

a

a

0(1)

Mp)

Ep(1),op 0

if and only if or

“a

_

Ap,q(Eo) - 0

for all

(p,q)

q > 0.

But this is equivalent to saying that there is no nontrivial partition (I.J)

of

{1....,n} which is invariant for the action of

{1,...,n},which means precisely that

a

on

a 6 Un, the conjugacy class of

t = (1,2,...,n). Now we have to show (2) By(1) and the argument in the proof of 2.2.10 (we have actually a projection onto the complex of primitive elements in

E(g 8 A)g)

we get:

H*(P(E(g o A)g) c PH*(gl(A)). But primitive homology classes have primitive representatives. we get equality, as claimed. This finishes the last section of our lectures.

Thus

156 Comments on chapter II.

The skeleton character of cyclic cohomology for the construction of

noncommutative de Rham homology (considered as a direct limit which inverts the suspension operator

s: cf.

theorem 1.1.18, for cyclic homology.

[Co]) has his counterpart in

This result of M. Karoubi's is

one step of his program of find the appropriate range for his character maps from Quillen's K-groups to de Rham cohomology (lying in reduced cyclic homology; of.

[Kb]).

Our second section deals with cyclic ho-

mology of commutative algebras in characteristic zero.

The ideas and

results are due to J.L. Loday and D. Quillen [L.Qu] and culminate in theorem 1.2.18.

It was not difficult to replace all original spectral

sequence arguments by simpler mixed complex patterns. As to the second part, which attacks the “additive K-theory" aspect of cyclic homology, I have first collected all the matrial of invariant

theory which enters implicitely or explicitely in the proofs of the last section.

There I have tried to strip down the correspondence cy-

clic homology - Lie algebra homology to its bare essentials.

2.2.10 is due to J.L. Loday and D. Quillen.

Theorem

The very last section on

the Hopf-algebra aspect of the result is sometimes sketchy, since a little bit non-thematic.

157 References to chapter II.

[An,L]

André, M.:

Méthode simpliciale en algebre homologique et alge-

bre commutative. [An,B]

Andre, M.:

[Be]

Behrens, E.A.:

LNM 32. Heidelberg: Springer 1967

Homologie des algebras commutatives. Berlin, Hei-

delberg, New York: Springer 1974 Ring Theory.

New York, London: Academic Press

1972

[BL]

Bourbaki, N.: Bléments de Mathématique. de Lie II, III.

[Bou]

Bourbaki, N.:

Elements de Mathématique.

Paris... [C.E.],[Co]

Groupes et Algébres

Hermann, Paris 1972. Algebre. Ch.1o.

: Masson 1980

as in chapter I

[Fo]

Fogarty, J.:

Invariant Theory.

[Gr]

Green, J.A.:

Polynomial Representations of

New York: Benjamin 1965 GLn.

LNM 830.

Berlin, Heidelberg, New York: Springer 1980 [H.St.] [Hu]

as in chapter I Humphreys.

J.:

Linear Algebraic Groups.

Berlin,

New York:

Springer 1975 [Kb]

Karoubi, M.:

Homologie cyclique des groupes et des algebras.

Homologie cyclique et K-théorie algebrique, I et II. Homologie cyclique et régulateurs en K-théorie alge-

brique.

C. R. Acad. Sci. Paris, série I,vol. 297 (1963)

381-384 at 447-450 at 513-516 at 557-560 [L.Qu.] [Ma]

as in chapter I Matsumura,

H.:

Commutative algebra,

2nd ed.

London...:

Benjamin/Cummings: 1980 [ML]

as in chapter I

[M.M.]

Milnor, J.W., Moore, J.C.: Ann. Math. 81

On the Structure of Hopf Algebras.

(1965), 211-264.

158 Further References.

The reader will find a lot of references in [Co],but also in [Ks].

He

should also consult [Ca]

Cartier, P.:

Homologie cyclique:

rapport sur les travaux

récents de Connes, Karoubi, Loday, Quillen... Sém. Bourbaki, exp. 621, Février 1984 The natural first extension of the material exposed in these lectures would be [Ka], already cited, and, I think, [Gw]

Goodwillie, Th.G.: loopspace.

Cyclic homology, derivations and the free Topology, 24(2)

(1985), 187-215

For those who are interested in noncommutative differential calculus (see 11.1.1) and in the

Zz—graded version of cyclic cohomology, I rec-

ommed [Kt]

Kastler, D.:

Cyclic Cohomology within thedifferential Envelope.

Preprint. CPT Marseille Luminy.

159

LIST OF SYMBOLS AND NOTATIONS

1 ED. q

H0" (A) 60 c; (A) 62 H; (A) 62

3

Elm: 3H" 7 1

Tot (M)

13

'FPTot (M), ”FPTot (M) I

a

Ep,q'

II

C(A)

a

Ema

19

c (A)

20

14

13

64

(BM,d) ct (A)

65

n, (A) 86 n (A) 90

CMA)

22

An (A) 94

HMA)

22

nix/k 9"

(BM,d)

24

HBR (A)

Ho. (M) C (A)

24 33

117

[1,3]

121

S (n)

122

H. (g)

130

(E (EM!) gl (A)

Fun (A) ficn (A)

ct (A)

45 45

58

130

133

Prim H. (9| (A) )

P (H)

150

144

160

INDEX

approximation theorem 10

Homology of Lie algebras 130

convergence (of a spectral sequence) 7

Hopf-algebra

cyclic

cohomology 60 homology 20 de Rham cohomology 95,117

complex 94 homology 96 differential envelope 90

graded 145 commutative and cocommutative 149 K'éhler-de Rham complex 95

limit term (of a spectral sequence) 7 mixed complex 23 cochain complex 63 Morita-equivalent 74

d.g. algebra 25

normalized Hochschild complex 42

double complex 12

primitive elements 141, 144, 150

enveloping algebra 17,127

reduced

exact couple 1 derived 3 filtration 1 bounded 7 first and second 13

Hochschild cohomology 59

complex 18 homology 18

cyclic homology 45 Hochschild homology 45 Schur algebra 122

shuffle product 105, 107, 110 spectral sequence 6 strongly homotopy A-map 30 total complex 12, 13