Operads and Gamma-Homology of Commutative Rings

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Operads And Gamma-Homology Of Commutative Rings

Article in Mathematical Proceedings of the Cambridge Philosophical Society · June 1998 DOI: 10.1017/S0305004102005534 · Source: CiteSeer

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OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

Alan Robinson and Sarah Whitehouse

May

Introduction

In this pap er we construct and investigate the natural homology theory for

coherently homotopy commutative dg algebras usually known as E algebras We

call the theory homology for historical reasons see for instance

Since discrete commutative rings are E rings we obtain by sp ecialization a

new homology theory for commutative rings This sp ecial case is far from trivial

It has the following application in stable homotopy theory which was our original

motivation and which will b e treated in a sequel to this pap er The obstructions to

an E multiplicative structure on a sp ectrum lie under mild hyp otheses in the

cohomology of the corresp onding dual Steenro d algebra just as the obstructions

to an A structure lie in the Ho chschild cohomology of that algebra

The homology of a discrete commutative algebra B can b e understo o d as a

renement of Harrison homology which was originally dened as the homology of

the quotient of the Ho chschild complex by the sub complex generated by nontrivial

shue pro ducts It is b etter dened as the homology of a related complex which

one obtains by tensoring each term B with a certain integral representation V of

the symmetric group and passing to covariants Harrison theory works very

n n

well in characteristic zero but not otherwise A more satisfactory theory necessarily

involves the higher homology of the symmetric groups not only the covariants

H Our homology theory is constructed to do just that It is furthermore

completely dierent except in characteristic zero from AndreQuillen homology

which is related to a completely dierent class of problems Polynomial algebras

are acyclic for AndreQuillen theory by its construction but they are not generally

free E algebras and their homology is generally nonzero

There are two further signicant generalizations First there is a cyclic variant

of the homology of any E dg algebra This arises very naturally from our

construction in x The cyclic theory like standard cohomology is connected with

an obstructiontheoretic problem A full account will app ear elsewhere Second the

domain of denition can b e widened from the ab elian situation of dg algebras to the

case of sp ectra in stable homotopy theory so that one can dene the homology

of an E ring sp ectrum This is analogous to extending Ho chschild homology to

top ological Ho chschild homology which includes Mac Lane homology as a sp ecial

The second author was supp orted by a TMR grant from the Europ ean Union held at the

Lab oratoire dAnalyse Geometrie et Applications UMR au CNRS Universite ParisNord

The rst author was supp orted by the EU Homotopy Theory Network

Typ eset by A ST X

M E

ALAN ROBINSON AND SARAH WHITEHOUSE

case The generalization to sp ectra is not dicult but we have p ostp oned the

details to another sequel

We mention that E homology was invented indep endently by the rst author

and by Waldhausen during the s and outlined in various lectures including a

plenary lecture by the rst author to the Adams Memorial meeting in Some

details subsequently app eared in the second authors thesis and a preprint

Meanwhile Kriz and later Basterra were developing by metho ds very

dierent from ours an E cohomology theory for ring sp ectra which is extremely

likely to b e equivalent to ours

The pap er is organised as follows Section contains material on op erads In

Section we intro duce complexes over op erads and dene the realization of such

a complex A cyclic variant of the construction is also given Section

covers the most imp ortant case of realization namely the cotangent complex

of an E algebra This section also contains the denitions of homology

and cyclic homology and a transitivity theorem The pro of of a key

acyclicity lemma is deferred to App endix A In Section we further justify our

constructions by showing that their A analogues lead to Ho chschild and cyclic

homology of asso ciative algebras Section is devoted to the sp ecial case of

homology of discrete commutative algebras It is shown that for pairs of Q algebras

homology coincides with AndreQuillen homology and an example is given

to show the theories are dierent in general The nal section describ es a pro duct

in cohomology of a discrete commutative algebra

Operads cyclic operads and cofibrancy

We work in the category of chain complexes dg mo dules over a commutative

ground ring K We might equally well of course have chosen simplicial mo dules

Our principal denitions use Getzler and Kapranovs theory of cyclic op erads

but we require Markls nonunital version which is describ ed in

Op erads Let S denote the category of nite sets S and isomorphisms of

sets S the sub category of nonempty sets and S the category of based nite

sets and isomorphisms To avoid foundational diculties we assume without

further mention where necessary that these have b een replaced by equivalent small

sub categories Our constructions do not dep end up on the choice of sub category

One can for instance take just one set f ng in S for each n and similarly

f ng for each n in S so that b oth categories b ecome disjoint unions

of symmetric groups An op erad C has ob jects chain complexes C indexed by

all nite sets S isomorphisms C C induced by isomorphisms S T

S T

of sets and comp osition maps

C C C

t S T S t T

for all nite sets S and T and all elements t T where S t T is the deleted

sum S t T n ftg these data must satisfy standard conditions of functoriality and

asso ciativity of comp osition The induced isomorphisms give a left action of the

symmetric group of automorphisms of S on C One thinks of C as a parameter

S S S

space of op erations in the sense of universal algebra with inputs lab elled by S

and a single output the induced isomorphisms corresp ond to p ermutation of inputs

and the comp osition to substitution of the output of C for the input lab elled t in

t S

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

C The op erad C is said to b e E if for each S S the complex C is contractible

T S

and free It is obviously sucient to check this for S f ng for all

The standard example of an E op erad is D in which D is the nerve of the

category SS of isomorphisms of nite sets over S Comp osition in D is induced

by the deleted sum functor in S

Cyclic op erads A cyclic op erad can b e dened as an op erad with extra

structure a action on C which makes comp osition symmetric by

f ng

putting the output variable on the same fo oting as the n inputs see

Then it is clearly desirable to change the notation and denote by C what was

S t

previously denoted C Confusion can arise so we stress that from now on we shall

use the cyclic convention and include the output in the lab elling set

It seems b est to dene cyclic op erads directly A cyclic operad is a functor E

from the category S of nonempty nite sets to the category of chain complexes

together with comp osition op erations

E E E

st S T S t T

for all nite sets S T with at least two elements and all choices of s S t T

Here S t T denotes the deleted sum S n fsg t T n ftg and is required to

st st

b e a natural transformation of functors from S S to chain complexes having the

asso ciativity prop erty

st t u t u st

for s S t t T t t u U and the symmetry prop erty

st ts

where E E is induced by the isomorphism of sets and E

S t T T t S S

st ts

E E E interchanges factors and intro duces the usual sign

T T S

Some further notation will b e needed The comp osition

E E E

st S T V

is asso ciated with a partition of V into two subsets S n fsg and T n ftg Conversely

let V P t Q b e any partition of V into two nonempty sets We can dene the

asso ciated comp osition by writing P and Q for the disjoint unions P t fg and

Q t fg and taking

1 2

E E E

P Q

E cyclic op erads We call E an E cyclic operad if for all S S the

complex E is contractible and free It suces to check this for S f ng

S S

for all n The op erad D dened as in is an E cyclic op erad the comp osition

again b eing induced by the deleted sum functor

Cobrant op erads We adopt the notation intro duced in for adding new

p oints to a set S S and S are to denote S t fg S t fg and S t f g

resp ectively For each partition V S t T of V we have a comp osition map

1 2

E In a cobrant op erad provided S and T each have more E E

T S

than one element we want this map to b e the inclusion of a face of E so we

require it to b e an equivariant cobration and we require faces to intersect only

in faces of faces This leads to the following

ALAN ROBINSON AND SARAH WHITEHOUSE

Denition Let E denote the co equalizer of the maps

M M

2 1 2 12 1

E E E E E

S T P Q R

V S tT V P tQtR

where the sums are indexed by partitions of V into subsets of which S and T and

hence P and R have at least two elements each Asso ciativity of comp osition im

plies that induces a equivariant map E E which we call the inclusion

V V V

of the boundary The cyclic op erad E is cobrant if

for every V the inclusion of the b oundary is a equivariant cobration

there is a given augmentation E K when E has exactly two elements

invariant with resp ect to induced maps such that for every partition of

a set V W t fw g into a set and a singleton the mapping

1 1 1

E E E K E E

W fw g W W

where is the evident isomorphism W W t fw g V coincides with

mapping given by the comp osition

Cobrant noncyclic op erads are dened in a completely analogous way

The E tree op erad T We now construct a cobrant E cyclic op erad

The cyclic op erad D will not do the faces of D intersect in unacceptably large

sub complexes so that D D is not injective On the other hand we can

S S

form another cyclic op erad by taking E to b e the space of trees with ends

lab elled by the set S and to b e the op eration of grafting the end lab elled s to

the end lab elled t to pro duce a new edge of length This op erad has every E

contractible and it is cobrant but it is not an E op erad b ecause do es not

act freely on E In the realm of A op erads which are indexed by ordered nite

sets and have no action there is a corresp onding op erad in which the ob jects

are the complexes of cyclicallylab elled trees in the plane it is the analogue of the

top ological op erad of Stashe p olyhedra see

By combining the two constructions we obtain a cobrant E cyclic op erad the

tree operad T as follows We take T to b e the chain bicomplex asso ciated with

the bisimplicial set in which a k l bisimplex consists of a k simplex of the nerve

of the category SS

k 1

S S S

k k

together with an l simplex of the space T of trees lab elled by the set S and the

S k

simplicial op erators are dened in the obvious way The comp osition maps in

T are dened by using the deleted sum functor in the category S and the grafting

of trees as ab ove The op erad T inherits the E prop erty of D We show that it

also has the cobrancy of the op erad of trees To show that the inclusion of the

b oundary T is an equivariant cobration we have to verify that it is induced

by an injective map of bisimplicial sets and that the group of automorphisms of

V acts freely on its complement The freeness follows from the freeness on D V

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

For injectivity the essence is that a simplex lies in the face corresp onding to a

decomp osition V S t T if and only if it consists of trees in which there is an

internal edge of maximal length which separates the lab els S from the lab els T

and therefore two faces meet only where two sp ecied edges have maximal length

which is a face of a face or is empty as appropriate

Algebras modules and realization

Algebras and mo dules over an op erad Let C b e a cyclic op erad and K

the ground ring which is commutative with unit element

Denition An algebra over C is a chain complex of K mo dules A together with

A A for all nonempty sets S S which are structural maps C

natural in S and satisfy the usual condition

T S tT

S tT

0 01

A A C of equality of maps C

T S

By way of explanation we note that has b een adjoined to S and T as the

output variable for the op erad The element in S is a dummy lab el asso ciated

with the partition S t T as intro duced at the end of

When the smallest mo del is chosen for S which is the disjoint union of the

symmetric groups for n the naturality condition in the denition simply

means that is equivariant and so denes a map C A A where

n n n

acts on C on the left xing the output lab el and on A on the right

Denition An Amodule over C when A is a C algebra as ab ove is a chain

complex M together with structural maps C A M M which are

natural in S and satisfy the usual mo dule conditions

S T S tT

S tT

01 01

as maps C C A M M and

S T

T S tT

S tT

012 0

as maps C C A M M

S T

The ab ove algebras and mo dules are nonunital This defect will b e remedied in

the next section

Algebras and mo dules over an E op erad

From now on it is a standing assumption except where the reverse is stated

that all op erads are cyclic and E An algebra A over such an op erad C will

b e called an E algebra Since C is automatically augmented over the standard

commutative algebra op erad the ground ring K is an E algebra

For the purp oses of this pap er it suces to dene subalgebras and submo dules in

a naive way as chain sub complexes which are closed under the appropriate op erad

action If A is a subalgebra of B over C there is an inclusion homomorphism A B

and we call B an Aalgebra over C We shall usually work with K algebras where

K is the ground ring regarded as an algebra over C The unit element of K then

serves as a unit for the algebra When considering mo dules over a K algebra A we

require the induced K mo dule structure to b e the standard strict one

ALAN ROBINSON AND SARAH WHITEHOUSE

Cyclic and noncyclic complexes over an op erad

We now aim to construct the homotopical cotangent complex K B A M when

B is an E K algebra A a K subalgebra and M a B mo dule Although very

dierent in app earance and in construction this will play in our theory the role

analogous to that played in AndreQuillen theory by the cotangent complex of

Our cotangent complex will b e a ltered ob ject obtained by glueing together the

B M where V runs through the category S There is also ob jects C

a cyclic version Conceptually it resembles the realization of a simplicial ob ject

or the analogue describ ed in Because the realization sometimes has to b e

applied to sp ecies other than the standard B M it is worthwhile to formulate

a denition of the kind of general ob ject which can b e realized

Denition Let C b e a cyclic op erad A cyclic C complex is a cofunctor M

from the category S to the category of chain complexes together with the following

further data which sp ecify an action of C on M for each comp osition

C C C

st S T S t T

in C there is given a formal adjoint

C M M

S S t T T

to b e thought of as a cap pro duct corresp onding to the ab ove cup pro duct which

satises

the naturality condition

s t st

for all isomorphisms S s S s T t T t in S

the asso ciativity condition

32 1 4

C C M M

RtS tT

S R T

for all nite sets R S T

the asso ciativity condition

4 1 23

C C M M

RtS tT

T R S

for all nite sets R S T where interchanges factors

There are two asso ciativity conditions ab ove for the same reason as in the denition

of noncyclic op erad there are two typ es of iterated substitution to b e considered

Example Supp ose A is an algebra over the op erad C with structural maps

V S

C A A Then we can take M A and dene to b e

V S

S nfsg

S t T S nfsg T nftg T nftg T

C A C A A A A A

S S

giving a cyclic C complex

Now we need the noncyclic version which is slightly more complicated b ecause

the indexing sets have basep oints and there are as in the denition of op erads

corresp ondingly more cases to consider The basep oint may b e in any subset of a

partition We consistently write for the basep oint so that a typical based nite

set is S where S is an ob ject of S

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

Denition Let C b e a cyclic op erad A noncyclic C complex assigns to

each based nite set S a chain complex M dep ending cofunctorially up on S

and to each comp osition

0 01 0

C C C

S T S tT

in C a pair of formal adjoints

0 0 01

C M M

S S tT T

and

01 0

M C M

T S

S tT

satisfying the analogues of the naturality and asso ciativity conditions of

The asymmetry in this denition arises b ecause the comp osition corresp onds

to a partition of V S t T S t T into a subset which do es not contain the

basep oint and one which do es The rst of the two adjoints ab ove evaluates over S

where is a dummy lab el from the partition the second and takes values in M

evaluates over T and takes values in M where is a new dummy basep oint for

Example Just as the primary example of a cyclic C complex arises from

an algebra so the primary example of a noncyclic C complex arises from a

A M where A is a K algebra over mo dule In denition ab ove let M

C and M an Amo dule let b e where is the algebra structure and

b e where is the mo dule structure

The realization of a C complex

Let C b e a cobrant cyclic op erad and M a C complex We construct the

realization jMj by a pro cess resembling that for realizing a simplicial set We treat

the noncyclic case in detail b ecause it is more imp ortant for us then describ e the

dierences in the cyclic case which is imp ortant in cyclic homology There are

two steps in the construction

First we construct a complex jMj We take a direct sum over all V V t fg

in S our category of based sets having three or more elements

0 0

C M

V V

jV j

jMj is the quotient of this by the following identications

0 0

for each isomorphism S T in S and all x C all m M

S T

x m x m

for each partition V S t T of a set into two subsets the second con

taining the basep oint having at least two elements each we consider the

asso ciated comp osition as in

0 01 0 0

C C C C

S T S tT V

and we dene

0 01 0 0 0 01 01

C C M C M C M

S T S tT S S T T

ALAN ROBINSON AND SARAH WHITEHOUSE

by setting

0 01

where interchanges the factors C and C and intro duces the usual sign

S T

0 0

Then on the comp onent of the b oundary C M corresp onding to

V V

we make identications by requiring

x y m x y m

0 01 0

for all x C y C m M

S T V

The complex jMj thus dened is a quotient of

C M

n n

where n denotes the n element based set f ng This is b ecause

the identications imply that it suces to take one indexing set of each size

and pass to the quotient by the action of We can dene the skeletal ltration

of jMj by dening the k skeleton to b e the image of C M Just

n n

as in the standard simplicial realization construction the identications satisfy

compatibility conditions which guarantee that the k th ltration quotient of jMj is

isomorphic to C C M

k k k

We now describ e the second step in the construction which incorp orates the

b ottom ltration stage M Since C is contractible for an E op erad and is

trivial M is quasiisomorphic to the exp ected b ottom ltration stage C

Let V b e any based set in S having three or more elements Take any v V

and write T for V n fv g The partition V fv g t T has an asso ciated comp osition

v v

01 0

C C C

fv g T V

and action

01 0

M C M

T V

fv g

The standard isomorphisms T V n fv g t f g V and fv g f g taking

v to in each case convert this action into a map

0 0

C M M M

V V fg

If instead of v V we select V the other action map dened in

yields in identical fashion a map

0 0

C M M M

V V fg

Let This denes for each V in S having three or more elements

v V

0 0

a map C M M One can check immediately that the naturality and

V V

asso ciativity conditions in the denition of a C complex imply that these maps are

compatible with the identications used in the construction of jMj Therefore we

have a welldened map jMj M The nal realization jMj is dened to b e

the cobre of This completes the construction of the realization in the noncyclic case

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

Realization in the cyclic case

Now let M b e a cyclic C complex where C is a cobrant cyclic op erad We con

struct the cyclic realization jMj by mo difying the construction of as follows

We b egin with a sum indexed by all V in our category S of unbased nite sets

containing at least three elements

C M

V V

jV j

We alter the identications to take account of the extra symmetry available in that

there is now no basep oint they now read

for each isomorphism S T in S and all x C all m M

S T

x m x m

for each partition V S t T of a set into two subsets having at least two

elements each and we dene

2 1 2 2 1 1

C C M C M C M

S T S tT S S T T

by setting

2 1

where interchanges the factors C and C and intro duces the usual sign

S T

Then on the comp onent of the b oundary C M corresp onding to

V V

we make identications by requiring

x y m x y m

2 1

for all x C y C m M

S T

We have now completed the description of the rst stage which we denote jMj

of the cyclic realization

The identications ab ove mean in eect that jMj is a quotient of

C M

n n

n+1

where is the group of p ermutations of f ng

In analogy with we now exp ect to dene a map

jMj C M

cy 2

the cobre of which would b e jMj In actual fact a sign intervenes in the repre

sentation and we have to replace C by a dierent contractible free complex

The nerve of the category of isomorphisms of twoelement sets is a mo del for the

classifying space B and the nerve of the category of isomorphisms of ordered

twoelement sets is its universal cover E Let V b e any set in S having three or

ALAN ROBINSON AND SARAH WHITEHOUSE

more elements Take any v V and write T for V n fv g As in we have a

comp osition

C C C

fv g T

and action

C M M

fv g

Using the isomorphism T V we obtain from the adjoint a map which we

shall denote

C M M

V V

fv g

As f v g is an ordered twoelement set we can regard it as a chain of E We

dene

C M E M

V V

fv g

by setting

x f v g x

v V

This do es not yet resp ect the identications dening jMj But if we denote by

M the complex M with its structure twisted by the sign representation then

comp oses with the quotient map to give a welldened map

E M jMj

We nally dene the cyclic realization jMj to b e the cobre of this map

Remarks The sign in the last stage of the ab ove construction is needed

to ensure cancellation of the unwanted contributions from the two dummy lab els

in a partition as in identication ab ove

There is a natural map jMj jMj induced by the levelwise quotient

maps C M C M which are wellb ehaved with

n n n n

n n+1

resp ect to the identications in the construction

Uniqueness of E realization

We now prove that the homotopy typ e of the realization jMj or jMj do es not

dep end up on the cobrant cyclic op erad C used to construct it provided that C is

E The pro of uses the standard idea of comparison of resolutions

Lemma Let C and D be E cyclic operads with C cobrant Then there is a map

C D of cyclic operads and it is unique up to homotopy

Proof We construct equivariant maps C D commuting

n n n n

with all comp osition maps by using induction on n We note rst of all that

the unit axiom for E op erads means that will always commute

with comp ositions with C and D since the axiom reduces this to the naturality

prop erty Supp ose by inductive hyp othesis that we have equivariant for all

k n commuting with comp ositions as far as this makes sense We have to dene

The b oundary C is by a sum of copies of C C with i j

n n i j

and i j n amalgamated along C C C C The maps

i j i j

therefore induce a map C D equivariant with resp ect to

i j n n

the induced action of Since C is cobrant and D is contractible this map

n n

extends to a equivariant map C D which by construction retains

n n n

the compatibility with comp ositions Since the induction starts automatically with

n where the b oundary is empty the inductive pro of of existence is complete

The pro of of homotopy uniqueness is similar

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

Prop osition If M is a complex over one E cyclic operad then it is a

complex over every cobrant cyclic E operad and the homotopy type of the real

ization jMj or jMj in the cyclic case is independent of the cyclic cobrant E

operad used to construct it

Proof Let C and D b e cyclic E op erads with C cobrant By there is a map

of op erads unique up to homotopy from C to D If M is a D complex such a map

induces the structure unique up to homotopy of a C complex on M

Supp ose now that C D is a map of E op erads We show by induction on k

that is a homotopy equivalence of pairs C C D D This

k k k k k

is certainly true for k where the spaces are contractible and the b oundaries

are empty Supp ose it is true for k n The assembly of C and D from

n n

cobrations of lower spaces in the op erads as in the pro of of implies that

restricts to a homotopy equivalence C D But then C and

n n n n

D are contractible and the inclusions of the b oundaries are cobrations so

is a homotopy equivalence of pairs

When M has the C structure induced by the map there is a skeletonpreserving

induced map jj jMj jMj b etween the realizations constructed using the

C D

two dierent op erads On quotients of adjacent skeleta jj induces a map

C C M D D M

n n n n n n

n n

which is a homotopy equivalence b ecause has b een shown to b e a homotopy

equivalence of free complexes By induction and direct limit jj is a homotopy

equivalence Hence jMj is indep endent of the cobrant E op erad used A similar

pro of works in the cyclic case

The homology of the realization

Prop osition

Let M be a C complex where C is an E operad Then there is a homology

spectral sequence

E H E V M H jMj

q p p p pq

pq p

where V is the representation of on the homology of the tree space T

p p p

and acts diagonal ly on V M

p p p

When M is a cyclic C complex there is a corresponding homology spectral

sequence in the form

E H E V M H jMj

q p p pq cy

pq p

p+1

where V is the integral representation of on the homology of T

p p

Proof The sp ectral sequence obtained from the skeletal ltration of jMj or jM j

is indep endent of the particular cobrant E op erad used in the construction

Cho osing the E tree op erad of leads to the E terms given ab ove

The E term is in eect the hyp erhomology of the group or with

p p

co ecients in the complex V M or V M and the dierential d can

p p p

p p

b e expressed in terms of transfer in group hyp erhomology and the b oundary in M

ALAN ROBINSON AND SARAH WHITEHOUSE

Naturally there are the hyp erhomology sp ectral sequence

H V H M E

s p p t p

pst

and its analogue for the second case available as subsidiary sp ectral sequences for

calculating the E terms

It is known that the representation V is isomorphic over Z to HomLie Z

p p

where Lie is the Lie representation and Z the sign representation The mo dule

V is related to V and therefore to the Lie representations by a short exact sequence

p+1

V Ind V V

p p

The complex character of V is calculated in

The cotangent complex and the transitivity theorem

Intro duction We now apply the general theory of x to the case we are really

interested in the construction of the cotangent complex K B A M when A is

a subalgebra of the E dierential graded algebra B and M is any B mo dule

In the construction we shall use any cobrant E cyclic op erad C such as the

tree op erad T of the result is indep endent up to quasiisomorphism of the

choice It is no real loss of generality to assume that A B and M are at or

even pro jective over the ground ring K the structure of algebra or mo dule over

a cobrant op erad is homotopy invariant so that one can replace these ob jects by

pro jective resolutions and our realizations are homotopy invariant constructions

so the choice of pro jective resolution makes no dierence to the result Similarly

we may assume that A B is a cobration

We also prove a at basechange result showing that K B A M is essentially

indep endent of the ground ring

The notation of the ab ove intro duction will b e used throughout the section

Let K b e the following noncyclic C complex which was mentioned in For

each based nite set V V t fg in S we put

K A M

For every partition V S t T of V into nonempty sets we dene

0 0 10

C K K

S S tT T

by using the algebra structure map

S T T

C A A M A A M

and we dene

0 10

K K C

S T

S tT

by using the mo dule structure map

S T S

A C A M A M

It follows from the algebra and mo dule axioms that this is a C complex Its real

ization jK j we denote usually by K A M in order to stress that it dep ends in

an essential way up on the ground ring K

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

cotangent complex and homology groups Let A b e a subalgebra of

the E algebra B as in and M a B mo dule all these b eing assumed at over

K We dene the cotangent complex of B relative to A with co ecients M to

b e the quotient

K B A M K B M K A M

K K

Here we have assumed that A B is a cobration In a more general situation

the quotient can b e replaced by the cobre The homology of B over A is the

homology of this complex

H B A M H K B A M

the cohomology is corresp ondingly dened by

H B A M H RHom K B A B M

Remark For theoretical and practical reasons we have chosen to dene the

cotangent complex as a quotient This avoids the need to handle derived tensor

pro ducts over E algebras It also has the pleasant consequence that the imp or

tant transitivity theorem b elow b ecomes trivial to prove This advantage is

of course an illusion the counterbalancing disadvantage is that in order for our

denitions to b e useful we have to work to prove that K B A M and hence

H B A M are essentially indep endent of the ground ring K The pro of of this

at basechange theorem o ccupies most of the rest of x and the App endix

Transitivity theorem Let A B C be inclusions of E algebras and M

an E C module Then there is a cobration sequence of cotangent complexes

K B A M K C A M K C B M S K B A M

and therefore a long exact sequence of homology groups

H C B M H B A M H C A M H C B M

n n n n

and similarly for cohomology

Proof The denitions require A B and C to b e replaced by corresp onding pro jec

tive resolutions Then everything follows from the exact sequence connecting the

cobres of the three maps in the triple

K A M K B M K C M

K K K

We now b egin on the denitions and lemmas which we shall need for the other

main result of this section the at basechange theorem

A mo del for the derived tensor pro duct

We prop ose that the derived tensor pro duct of mo dules over an E algebra

should b e dened by the following construction In Prop osition b elow we shall

justify it in the case of at mo dules over a strictly commutative algebra which is

the only case we need in this pap er

ALAN ROBINSON AND SARAH WHITEHOUSE

Let C b e a cobrant cyclic E op erad We are going to make a realization like

those in and but with S or S replaced by S which is the category of

nite sets with r distinct basep oints and isomorphisms of sets which

preserve the basep oints in order There is formally no problem in extending the

denition given in to this case except that when S t T is a partition of the set

V V t f g in S the structural map

p q

C M M

pq S S tT T

must b e zero when S contains more than one of the basep oints as there is then

no natural way to structure T as a space with r basep oints Here p and q are

dummy lab els When r the homology of the realization is much simpler

than the homologies for which we obtained sp ectral sequences in b ecause the

analogues of V are now free mo dules That is why the following construction

p p

is valid

Let A b e a C algebra and let M M b e Amo dules Supp ose that A and all

the M are at over the commutative ground ring K We construct a C complex M

r V

on the category S by setting M M M A when V V t f g

V r r

is a set with r basep oints and when V S t T taking

q p

M M C

S tT T pq S

to b e

if S contains no basep oint

if S contains and no other

pq i

if S contains more than one basep oint

where is the structural map for the C algebra A and is that for the mo dule

L L L

M to b e the M We dene the left derived tensor product M

A A r A

realization jMj

The following prop osition is sucient justication for the purp oses of this pap er

of the ab ove denition The pro of actually go es rather further as most of it do es

not need R atness

Prop osition Let R be a commutative algebra over the ground ring K and

let M M where r be complexes of R modules in the sense of standard

homological algebra Suppose that R is at over K and al l the M are at over R

Then there is a quasiisomorphism

L L L

M M M M M M

R R r R R R r R

Proof For transparency we treat rst the case when r After we may

assume that C is the tree op erad of We shall also need the corresp onding

is the chain complex of trees which can b e A op erad in which the C

ord

emb edded in the plane with lab els n in cyclic order This op erad

has no p ermutations It is well known that C is a sub division of the Stashe

ord

op erad of asso ciahedra and that the homology of the nth complex mo dulo its

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

b oundary is a single copy of the ground ring If we construct the realization jMj

ord

of the complex

M M R M

1 2

with resp ect to C then the E term of the skeletal sp ectral sequence is the

ord

bar resolution and since everything is at over K we have E Tor M M

p p

Examination of the attaching maps in the structure of jMj shows that E E

ord

since R is strictly asso ciative Indeed we have assumed M is R at so this is quite

obvious as E for p and the homology sp ectral sequence converges to

M M

To complete the case r it now suces to prove that the S realization jMj

which we dened in is equivalent to the ordered realization jMj There is

ord

certainly a natural map jMj jMj induced by inclusion of op erads This map

ord

resp ects skeleta so induces a map of homology sp ectral sequences We have to

calculate the E term in the target sp ectral sequence Now the homology mo dulo

its b oundary of the complex of trees with lab els f n g is the tree

representation which restricts to the regular representation of and the

inclusion of the ordered trees induces a map which takes the homology generator to a

generator of this regular mo dule After taking covariants as the construction

of jMj requires we therefore have an isomorphism of E terms Thus the sp ectral

sequences are isomorphic and so jMj jMj is a quasiisomorphism Combining

ord

this with the rst result of the pro of shows that jMj is quasiisomorphic to M

M The result is now proved for r

The pro of for r follows exactly the same lines The only dierence is the

inclusion of a counting argument to match the numb ers of generators in the free

mo dules involved in the two E terms for these no longer have rank one We omit

the details

The following acyclicity lemma is central to the results of the present section

Lemma Let K be a commutative ring and M a K module Then the complex

K K M is acyclic

Contemplating conguration spaces makes one think that should b e true

but the only pro of we know is combinatorial and lengthy This pro of is given

in App endix A The rst consequence of the Lemma is that one can calculate

homology relative to the ground ring without normalizing by quotienting by

K K M

Prop osition Let A be an E algebra over the ground ring K and M an

Amodule Then

H AK M H K A M

Proof Since quotienting by the acyclic complex K K M is a quasiisomorphism

we have K AK M K A M

Naturally one would like to b e able to describ e H B A M in an equally

simple way for any E pair of algebras A B The tensor p owers of A would have

to b e replaced by derived tensor p owers of B over A We have little doubt that this

could b e done but the resulting elegant statement might not justify the technical

mischief with derived p owers which would b e needed to prove the result and later

to apply it The following theorem is a go o d substitute

ALAN ROBINSON AND SARAH WHITEHOUSE

Theorem Flat basechange for K B A M Let K be a commutative

ring and R a at commutative K algebra Then

For every E algebra A and every Amodule M which are at over the

ground ring R there is a quasiisomorphism

K A M K A M K R M

R K K

If A is a subalgebra of the E algebra B and M is a B module al l these

being R at then the quasiisomorphism type of K B A M is the same

whether the ground ring be taken to be K or R

Proof Supp ose L is a commutative ring such that K L R In the ap

plication L will actually b e either K or R As in no generality is lost by

assuming that R is a strictly commutative subalgebra of the E algebra A We

have dened K A M as the realization of a certain C complex K This means

that jK j is dened rst as a certain quotient of C A M then

K A M jK j is constructed as the cobre of a map jK j A M all tensor

pro ducts b eing over L

We construct a ltration of jK j and A M and therefore of K A M by

dening the pth ltration stage F K A M to b e the image of the submo dule

in which at most p of the tensor factors from A lie outside R This resp ects all

necessary identications and is thus a valid denition of a ltration in which

p p

K R M F F F F F K A M

L L

p p

Let us consider the quotient F F Under the action of every tensor a

a m with p factors outside R is equivalent to an element in which only

a a are outside R and mo dulo lower ltrations this element is unique up to

the action of Provided that R is Lat and p it follows from

p np

p p

that F F is quasiisomorphic to E AR AR AR

p R R R

where there are p factors AR Now this is quite indep endent of L So if we take

the natural ltered map b etween the two ltered complexes

K A M K A M

K R

asso ciated with the two choices L K and L M we know that it induces

p p

equivalences of ltration quotients F F for all p Therefore the map

K A M F K A M K A M F K A M

K K R R

is a quasiisomorphism But F K A M K R M and F K A M

K K R

K R M which is acyclic by so this relation is precisely of the statement

When A is a subalgebra of the E algebra B and M is a B mo dule all these

b eing R at we have a diagram

K R M K R M

K K

K A M K B M K B A M

K K

K A M K B M K B A M

R R

in which two columns are cobrations by ab ove and two rows are cobrations

by denition The diagram implies that the vertical map on the right b etween the

two mo dels for K B A M is a quasiisomorphism

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

Cyclic homology and cohomology Let A b e an algebra over the co

brant cyclic E op erad C with K as ground ring We then have the cyclic C

complex jMj describ ed in which has M A with structural maps induced

by the multiplication in A We denote the cyclic realization of jMj by K A

The cyclic homology and cyclic cohomology are dened in terms of the cyclic

realization K A

cy cy

H A H K A

H A H Hom K A K

The A analogue Hochschild and cyclic homology

The ab ove theory is sp ecically for E structures and is new We now construct

the precise analogue for A homotopyasso ciative structures and show that this

just leads to a new description of the familiar Ho chschild homology and cyclic

homology of asso ciative algebras

We replace S with the category S of cyclical lyordered nite sets and order

is a nite preserving isomorphisms The automorphism group of an ob ject of S

cyclic group If is chosen as basep oint in an ob ject S of S its complement S is

totally ordered and the group of automorphisms preserving the basep oint is trivial

We redene op erads and cyclic op erads for the new case replacing the category

S in and by S The comp osition op erations have the form

A A A

st S T S t T

where S t T has the unique cyclic ordering obtained by concatenating the total

orderings on S n fsg and T n ftg We say a cyclic op erad A is A if A is contractible

for each S and the cyclic group C acts freely on A Cobrancy is dened as

S S

b efore Next we intro duce algebras over a cyclic A op erad A and mo dules over

these algebras by analogy with The simplest examples are asso ciative rings and

bimo dules resp ectively Similarly cyclic and noncyclic Acomplexes are dened by

precise analogy with and The archetyp es are M A in the cyclic

case and M A M in the noncyclic case where A is an asso ciative or A

algebra and M an Abimo dule The realizations jMj and jMj are dened just as

in and the category S b eing replaced everywhere by S and the symmetric

group in by the cyclic group C

n n

Homology of the A realization

We have the following analogue of It is very much simpler than the E ver

sion b ecause the represention V is replaced the homology of the space of cyclically

ordered ptrees which is free of rank one

Prop osition

Let M be a Acomplex where A is an A operad Then there is a homology

spectral sequence

E H M H jMj

q p pq

When M is a cyclic Acomplex the spectral sequence has the form

H EC M H jMj E

q p C p pq cy

p+1

where M indicates that the C module structure of M is twisted

p p p

by the sign representation

ALAN ROBINSON AND SARAH WHITEHOUSE

Proof Just as in the E case of the sp ectral sequence obtained from the

skeletal ltration of jMj or jM j is indep endent of the particular cobrant cyclic

A op erad used in the construction We may therefore cho ose the A tree operad

T which is constructed just as in but with the category S replaced by the

category S of cyclicallyordered sets This leads to the E terms given ab ove

Corollary

Let M be the Acomplex with M A M where A is an associa

tive K algebra and M an Abimodule Then the homology of jMj is the

Hochschild homology of A with dimension shifted by one

H jMj HH A M for r

r r

Let N be the cyclic Acomplex with N A Then the homology of jN j

S cy

is the cyclic homology of A with a dimension shift

H jN j HC A for r

r cy r

Proof Since M is discrete the E term of the sp ectral sequence of

collapses to the edge E A M Analysis of the identications in jMj

p p

shows that d A M A M is the Ho chschild b oundary Thus

E is simply the standard Ho chschild complex shifted down and truncated

Using a mo del where S has one set of each size we have

cy cy

C C C T T

n n

is the space of planar ntrees and C C is the bar construction on where T

the cyclic group which p ermutes the lab els f ng of these trees Therefore

jN j is a bicomplex which has m k st group

cy cy n

C C C T T A

m n k

n n

cy cy

where T is the b oundary of T the fullygrown trees We lter by n Since the

n n

cy cy cy

complex T is a Stashe n cell C T T has only one homology group

n n n

generated by the homology class c of the cycle denoted c in Thus each

n n

ltration quotient is a bicomplex for which the second standard sp ectral sequence

column homology rst collapses We conclude that the sp ectral sequence asso ci

ated to our ltration has E H C A where the action of the

m n

cyclic group on the tensor pro duct includes the usual sign

On the other hand the cyclic homology of A is given by Tsygans bicom

plex This has A in the m nth p osition and the horizontal dierentials

are alternately T and N the morphisms in the standard p ero dic resolution of

the cyclic group C Filtration by n gives rise to a sp ectral sequence with

E H C A

m n

There is an equivalence from the p erio dic resolution to the bar resolution which

takes the generator to N jT j jT jN jT in even degrees and to T jN j jT jN jT

in o dd degrees We use it to construct a chain map from Tsygans bicomplex

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

with the row n deleted to the bicomplex representing jN j Explicitly we

dene

n cy cy n

A C C C T T A

mn m n n

n n

by setting

N jT j jN jT c a for m even

T jN j jN jT c a for m o dd

The map commutes with horizontal dierentials since we b egan with a map of

C complexes To prove that it commutes with vertical dierentials one needs

a calculation like that which proves that Tsygans diagram is a bicomplex and

the fact that the vertical Ho chschild dierential arising from the identications in

jN j carries c to c Finally is a map of ltered bicomplexes which has bide

cy n n

gree and which induces an isomorphism on the E terms of the asso ciated

sp ectral sequences Hence induces isomorphisms HC A H jN j

r r cy

Explicit complexes in the strictly commutative case

Let B b e a strictly commutative algebra which is at over a commutative ring

A and let M b e a B mo dule By and we may take A as the ground ring

in calculating K B A M and H B A M Accordingly we denote simply

by The cotangent complex K B A M is quasiisomorphic to K B M by

When constructed using the tree op erad T this is a bicomplex

1 1

C B A M C S C T T B M

pq q p

S S

Here denotes a generic ob ject of S and denotes the same ob ject minus its

basep oint The vertical dierential d of the bicomplex is the dierential of the

twosided bar construction on the category S The horizontal dierential d is the

dierential in the chain complex C T except that chains in the b oundary T are

identied with lower skeleta by relation When n the relative chain

complex C T T has to b e interpreted conventionally as A in degree

n n

We can make this smaller and more explicit by replacing S with the mo del

in which there is just one ob ject f k g for each k Then one has to

make many choices ab out how to identify an arbitrary quotient set of f k g

with some f l g See for example the lab elling convention describ ed in the

App endix Any coherent system of choices gives a complex

C B A M C C T T B M

pq q k p k k

k k

which is quasiisomorphic to though the precise horizontal dierential d

dep ends up on the choices Once more the vertical dierential is that of the two

sided bar construction on the symmetric groups There is a dual version for

cohomology when B is pro jective

Since we are working in the discrete case the subsidiary sp ectral sequence of

collapses to an edge and we have the following sp ectral sequence

E H V B M H B A M

q p p pq

where V is the mo dule given by the reduced homology of the treespace T

p p p

When B is pro jective there is a dual sp ectral sequence in cohomology

q p pq

E H V HomB M H B A M

p p

ALAN ROBINSON AND SARAH WHITEHOUSE

Theorem The edge q of the spectral sequence above is precisely the

complex used in dening the Harrison cohomology Harr B A M of B with

a shift in degree Therefore there are natural transformations

H B A M Harr B A M H B A M Harr B A M

p p

when B is at resp projective which are isomorphisms when B contains a eld

of characteristic zero

Proof We give the details for homology The edge of the sp ectral sequence

has terms

p p

H V B M V B M E

p p p

Now we describ e the structure of the mo dule V H T further details

p p p p

can b e found in The tree space T has the homotopy typ e of a wedge

of p spheres of dimension p A set of indep endent homology generators

is given by f c j g where c is the cycle consisting of cyclically lab elled

p p p

trees in the plane Let s where is the sign of and the sum is

ipi

over i p ishues in In it is shown that s c for i p

p ipi p

and that these relations completely determine the mo dule structure of V It

p p

follows that V B is isomorphic to B mo dulo the submo dule of shue

decomp osables

It remains to identify the dierential d E E It is straightforward to

p p

check that d c x x m c bx x m where b denotes

p p p p

the usual Ho chschild b oundary map The edge E is therefore the quotient of

the Ho chschild complex by the shue decomp osables which is precisely Harrisons

Harr B A M complex Hence E

The edge map of the sp ectral sequence gives a natural transformation

H B A M Harr B A M

p p

When B contains a eld of characteristic zero the higher homology of the symmetric

groups is zero so the sp ectral sequence collapses to the edge and the ab ove is an

isomorphism

Prop osition

H B A M M H B A M Der B M

B A

H B A M Exalcom B M

Here Exalcom B M the mo dule of innitesimal Aalgebra extensions of B

by M is as dened in x

Proof In the bicomplex C d C B M The image of

the horizontal dierential d C C is spanned by the usual relations for

dierentials of pro ducts It follows that H B A M is the mo dule of Kahler

dierentials M Similarly the zeroth cohomology group is Der B M

B A

Supp ose rst that B is Apro jective In the sp ectral sequence we have

E Harr B A M and so H B A M Harr B A M This E

is the mo dule of Asplit innitesimal commutative Aalgebra extensions of B by

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

M Since B is pro jective this coincides with Exalcom B M the mo dule of all

innitesimal Aalgebra extensions of B by M

In the general case when B is not Apro jective we have to use a simplicial

resolution and elementary prop erties of AndreQuillen cohomology It is elementary

that cohomology extends to simplicial rings with co ecients in a simplicial

mo dule the cotangent complex of a simplicial ring is a simplicial dg mo dule

and one simply takes the asso ciated total complex The cobrancy of the op erad

ensures that this is a homotopy invariant of the simplicial ring This said we may

replace the algebra B by an AndreQuillen resolution P consisting of p olynomial

algebras over A Filtering the cotangent complex by the simplicial degree gives

a sp ectral sequence

q pq

E H P A M H B A M

On the edge we have E H P A M Der P M by so by def

p A p

inition the AndreQuillen cohomology D B A M is just E In particular

E Exalcom B M But E by the rst case since P is pro jective and

A p

p olynomial The sp ectral sequence now gives the result

Corollary When B contains a eld of characteristic zero

p p

H B A M D B A M H B A M D B A M

p p

where D is AndreQuil len homology

Proof Again we give the details for homology If B is at over A and contains a

eld of characteristic zero then Harrison homology coincides with AndreQuillen

homology so the result is given by If B is not at we replace it by a

simplicial Andre resolution by p olynomial algebras P As in the pro of of

this is the preferred metho d for strictly commutative rings We again obtain a

sp ectral sequence E H P A M H B A M Since each P is

q p pq i

at gives H P A M M and all higher homology groups are

i P

P A

zero by Thus the sp ectral sequence collapses to the edge where E is exactly

an AndreQuillen resolution of B giving the result The case of cohomology is

similar except that at is everywhere replaced by pro jective

In general homology is dierent from AndreQuillen homology and from Har

rison homology The following example shows this and reveals a nontrivial dier

ential in the sp ectral sequence of

Example First take B A F Then is a nonb ounding

Harrison cycle by the calculation in x Thus Harr F F F and

by our element exists in E Since or the transitivity theorem

implies that H F F F and similarly for Andre homology this cycle

must map by the only available dierential d to a nonzero element of E The

only such element is where generates H F for the mo dule V is

trivial Thus H Harr

Now let us take B to b e the p olynomial algebra F X A F M B X

F A brief calculation with shues shows that E Harr F X F F con

tains no nonzero element of degree two in X Therefore X X E is an

innite cycle which is not in the image of d and therefore is not a b oundary So

H F X F F and H is not Andres H

ALAN ROBINSON AND SARAH WHITEHOUSE

Theorem

Let B and C be Aalgebras with B at over A and let M be a B C module

Then the complex K B C C M is quasiisomorphic to K B A M

so that

H B C C M H B A M

Let B and C be at Amodules and M a B C module Then there is a

quasiisomorphism

K B C A M K B A M K C A M

and therefore H B C A M H B A M H C A M

If B is an etale Aalgebra then H B A M H B A M for

every B module M

Proof Since B is at over the discrete commutative ring A the cotangent

complex K B A M is equivalent to K B M Also B C is at over C and

A A

K B C M may b e replaced by K B C M But standard identities with

A C A

the tensor pro duct show that K B C M K B M b ecause these are

C A A

realizations of isomorphic complexes

We have an exact triangle corresp onding to the triple A C B C

K C A M K B C A M K B C C M

A A

Using the quasiisomorphism of this can b e split by the map

K B A M K B C A M

The arguments of Andre x for the homology of a separable eld

extension generalize to show that this can b e deduced from and the long

exact sequence of a triple as was observed by Quillen x

A product

In this section we prove the following theorem giving a graded anticommutative

pro duct in the cohomology of a commutative algebra This pro duct is not asso

ciative We b elieve it is a graded Lie pro duct but we have not yet veried all the

details of the Jacobi identity

Theorem There is a graded anticommutative product in cohomology

l m lm

H B A B H B A B H B A B

We b egin by explaining the idea of the construction which mimics the Lie bracket

in Ho chschild cohomology We recall that this is dened as a graded commutator

of circle pro ducts where the circle pro duct f g is an alternating sum over i of

substitution of g into f in the ith place As in x realization using the tree op erad

gives rise to the following bicomplex for cohomology of a discrete commutative

algebra B in which as b efore denotes the complement of the basep oint in the

set of the category S

1 1

C S C B A B Hom C T T B B

q p

S S

1 1

C S C T HomB B Hom

q p

S S

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

l m

For co chains f C g C the bracket f g is dened using a dierence

f g g f where f g C is the sum over all p ossible ways of

inserting g into f as indicated schematically by the diagram

@ g

However it is necessary to use a diagonal approximation in the construction

Lack of strict commutativity complicates matters and forces us to add a correction

term to our bracket

Now we give the details of the pro of of Theorem The rst ingredient is the

following coop erad structure which is closely related to a coop erad discussed by

Ginzburg and Kapranov x

Lemma The chain complexes fC T U S g form a cooperad

e e e

0 01 0

for U V t W C T C T Proof We dene C T

V W

W V U

An internal edge in a U tree t divides the tree into two parts If t has an internal

edge such that one of these parts is lab elled by V and the other by W then t

V W

is given by cutting t at this internal edge to pro duce a tree lab elled by V and

a tree lab elled by W If t has no such internal edge we set t The

V W

new lab elling sets V W are b est thought of as quotient sets of U obtained by

identifying all elements of W V resp ectively It is easy to see that the s are

V W

chain maps satisfying the required coasso ciativity condition

Secondly we need a diagonal approximation on the chains on the category S

Recall that such a diagonal approximation exists and that for the bar resolution

it may b e chosen to b e strictly coasso ciative and co commutative up to homotopy

The homotopy H say is itself commutative up to homotopy Now

we combine the diagonal approximation with taking induced isomorphisms on

quotient sets For each partition U V t W we have a chain map

V W

C S C S C S

j j j j j j

i i k i i k k

where the s start at U the s at V and the s at W We denote by

such maps constructed with the homotopy H in place of and so on

Finally we have the structure maps of the endomorphism op erad of B

S T S t T

HomB B HomB B HomB B

for each element s of S

l m lm

Now the map C C C is given by

Y X

f g f g

V W

0 1 0 0

U S U V tW

ALAN ROBINSON AND SARAH WHITEHOUSE

where is a suitably signed switch of factors and is the image under of

Now consider the graded commutator

hf g i f g g f

We check how this b ehaves with resp ect to the dierentials Since the s are

V W

chain maps we have d f g d f g f d g for the vertical dierential

d The horizontal dierential d consists of the internal b oundary in the tree

spaces plus extra terms d D say Again f g f g f g

since the s are chain maps Analysis of the identications in the cotangent

V W

complex shows that D can b e expressed in terms of the circle pro duct Since

B is strictly commutative we may consider the pro duct co chain C that

is b b where is the star tree lab elled by S Then

sS s s S

D f f f A calculation shows

D hf g i hD f g i hf D g i E f g

where E f g C is an error term which results from the diagonal approx

imation not b eing strictly commutative It can b e describ ed as follows

e e e

0 012 0

C T C T C T For U X t Y t Z we dene

p X Y Z p

Y X U

C T as follows Supp ose a U tree t has exactly two internal edges meet

ing at the ro ot the part ab ove one b eing a subtree lab elled by Y and the part ab ove

t is given by splitting the the other b eing a subtree lab elled by Z Then

X Y Z

tree t in the evident manner at these internal edges into a star tree lab elled X

and two subtrees lab elled Y and Z If the tree t cannot b e split as indicated we

set t

X Y Z

We have

Y X

E f g f g

X Y Z

Y Z

0 1 0 0

U S U X tY tZ

where the p ermutation is simply the necessary reordering of factors with

appropriate sign Now dene f g C by the same formula as for E f g

but using the homotopy H to replace by Then by construction we

have d f g d f g f d g E f g and by considering the maps

it is not hard to check that d f g d f g f d g

Finally dene a bracket by

f g hf g i f g

From the discussion ab ove this map is wellb ehaved with resp ect to the dier

entials and so induces a map in cohomology

Using the fact that H is commutative up to a homotopy H say we may con

struct a homotopy b etween f g and g f So the bracket is graded

anticommutative in cohomology This completes the pro of of Theorem

Remarks The bracket describ ed ab ove is compatible with the Lie pro duct in

Harrison cohomology x For co cycles it is simply the usual bracket of

derivations

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

If n is o dd or the characteristic of B is then the circle pro duct g g g passes

n n

to cohomology giving an op eration H B A B H B A B

If the same constructions are carried out in the A situation of x the error

term E f g is always zero and one recovers the Lie pro duct of Gerstenhab er on

Ho chschild cohomology x

ALAN ROBINSON AND SARAH WHITEHOUSE

Appendix A Acyclicity of K A M

Contraction of a certain complex without permutations

We construct then contract a certain chain complex related to K A M It is

obtained by glueing together the chains on the various tree spaces T for n

For simplicity we may as well take M to b e the ground ring A The construction

of our complex K requires a lab elling convention for trees which is detailed

b elow The contraction requires an ordering convention for the edges of a tree

Both these conventions are somewhat arbitrary at this stage but they have to b e

compatible with each other

Ordering convention Let t T b e an ntree It therefore has a ro ot lab elled

and leaves lab elled n Let b e the arc shortest path in t from the leaf i to

the ro ot Then t We intro duce a total ordering on the set of edges of t as

follows If x y are edges then x precedes y written x y if either x and y are in

some common arc with y nearer the ro ot or minfi j x g minfj j y g

i i j

This do es dene a total ordering in which an internal edge o ccurs at the rst

moment after all edges ab ove it have b een counted When no internal edge is

available the next leaf in descending order is taken So the leaf n is p erversely

rst The ro ot is last

The trees tx and tnx An internal edge x in an ntree t divides the tree into two

The p ortion including the ro ot and the edge x itself is a subtree called tnx The

other part containing some leaves and x itself but not the ro ot is called the sub

tree over x and is written tx It is much b etter to regard tx as the identication

space obtained by crushing the subtree tnx to a single edge and tnx as obtained

by identifying tx to an edge If x is a leaf or the ro ot of t the symb ols tx and

tnx are interpreted as either the whole of t or the tree consisting of a single leaf as

appropriate Now we have to decide how to lab el these quotient trees

Lab elling convention A quotient tree such as tx is naturally lab elled by subsets

forming a partition of the set f ng b ecause a new leaf or ro ot inherits all

the lab els on the subtree it came from We replace these subsets by r

lab elling the subsets in increasing order of their minimal elements

The p oint of the lab elling convention is that the conventional ordering intro duced

ab ove is compatible with identifying a subtree to a single edge provided one regards

a subtree as enumerated when all its edges have b een enumerated For instance a

subtree containing the ro ot is always lab elled and comes last in the conventional

ordering

Now we are ready to start dening our chain complex To b egin with we use

reduced cubical chains b ecause T is naturally a cubical complex

Denition Let K b e K where the complexes K are dened inductively

n n

as follows

K is the chain complex C T of the onep oint tree space T

as a for n supp ose that we have already dened the complex K

quotient of C T Then the complex K is obtained by attaching

along the sub complex C T of fullygrown trees The C T to K

n n

takes the generator corresp onding attaching map C T K

n n

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

nr r

to an ntree t with fullygrown edge x to the class tx tnx

in K where r is the numb er of leaves in tx

The previous identications in K ensure that the attaching map is welldened

and indep endent of the choice of the edge x and evidently K is by construction

a quotient of C T The cub es tx and tnx are of course lab elled by the

convention ab ove and oriented by the ordering convention It should b e noted that

whenever t has more than one internal edge at least one of the cub es tx and tnx

is a degenerate face

Sub dividing K We shall show that there is a natural geometricallyinspired

contraction of the complex K It is not easy to describ e in terms of the cubical

chains b ecause geometrically the image of T is deformed through T in a way

n n

which is not cellular but diagonal on the cub es

Therefore we replace each cubical complex T by its natural simplicial sub divi

sion in which each r cub e is replaced by r r simplices An ntree b elongs to one or

other of these dep ending up on which internal edges are longer than which others

Diagonal simplices in T contain trees having certain edges of equal length Every

cubical chain is a chain of the simplicial sub division so we have enlarged C T

and we make identications among these exactly as b efore to obtain a chain com

plex K quasiisomorphic to and containing K But we continue to use cub es

as blo cks of simplices sums of generators in K

Informal description of the contraction The contraction of K closely fol

lows this geometrical idea A lab elled ntree t passes through N stages t t t

during the homotopy where t t and N is the total numb er of edges of t In the

tree t there are two identical copies of each of the rst i edges in the conventional

order and one copy of the others As identical edges must have the same length

t represents a diagonal cub e in some T having the same dimension as t The

i nj

homotopy connecting t and t is represented by a tree like t but with one new

i i i i

edge b elow the two copies of the ith edge connecting the most recentlydoubled

edge to the undoubled part This is a cub e of dimension one higher Shrinking

one undoubled edge or two identical edges to a p oint is a cubical face op erator

therefore has t and t as faces Finally t is the sum of two copies of t A

i i i N

more formal description follows

The double of a tree Let t b e an ntree The double Y t is the ntree obtained

by taking two identical copies t and t of the tree t and grafting them by the ro ots

onto the two leaves of the unique tree in T Pairs of identical edges have the

same length We lab el the result as follows The two leaves formerly lab elled i are

marked i and i in t and t resp ectively Then all lab els are multiplied by two

to give integers

The construction We actually dene t and t by induction on i We set

i i i

t t If t has b een dened and x is the ith edge of t in the conventional

i i

ordering we dene t to b e the result of grafting the double Y tx by its ro ot

i i

onto the leaf x of t nx We dene t by shrinking the grafted internal edge

i i i i

formerly the ro ot of the double Y tx of t to a p oint It follows from the

i i

inductive denition that t contains two copies of edges x x and one copy of

i i

the highernumb ered edges We note that t and t have b een dened cub e by

i i

cub e or a blo ck of generators of K at a time n

ALAN ROBINSON AND SARAH WHITEHOUSE

We have to lab el t and t As for the doubling construction we give the two

i i

and i without changing copies of the leaf formerly designated i the lab els i

the lab els on the undoubled leaves Then we replace the lab els in bijective order

preserving fashion with the integers s for some s

Example

t (t)

1 2 1 2

@ @

4 4

@ @

5 6 5

8 12

@ @

a @ @

7 10

@ @

@ @ 13

6 8

3 3 9

0 0

Denition If t is a cub e corresp onding to a treeshap e with a total of N edges

t we dene t

We claim that this denes a contracting homotopy of K by sp ecifying it on

the generating simplices a cubical blo ck at a time To prove this we must verify

that where K K factors through the chain complex of

a p oint So we have to investigate how commutes with resp ect to face relations

This includes verifying that resp ects the identications used to dene K

As we are still working with cubical blo cks in K even though some of them may

b e diagonal cub es with certain co ordinates equal it is the cubical face op erators

we have to check Let x b e the ith edge of a tree t corresp onding to a certain

cub e also denoted t in K If x is an internal edge there is a face op erator

i i

corresp onding to shrinking the length of x and of all edges forced to have the same

length to zero When the length of x stretches to we have the opp osite face

i i

of the cub e which by construction of K is identied with tx tnx which is

i i

the zero chain when x lies b etween two internal edges of t

By checking the geometrical details we can now verify a whole slew of cubical

identities such as to give one instance

t tx tx tnx

i j i i j f i

when x is an edge of tx which implies i j and where f denotes the numb er

i j

of edges of tx The enumeration of faces is more complex here than in the case

of the usual simplicial or cubical identities b ecause of the branching of trees But

some of the formulae simply assert that a certain face is degenerate and is therefore

a zero chain For instance the ab ove formula gives a nonzero right hand side only

in two cases rst when x is the ro ot of t and i j second when x has nothing

j i

but leaves ab ove it

In calculating these identities it is essential to rememb er that aects identical

edges simultaneously and not separately and likewise for i

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

The cubical identities in full Let x and x b e edges of t and let f b e the

i j

numb er of edges of tx We denote the numb er of free edges leaves plus ro ot of

a tree s by l s Then

If i j and x is an internal edge or the ro ot of tx

i j

l tnx l tnx

j i j i

t tx tx tnx

i j i i j f i

If i j and x is an internal edge of tnx

i j

l tnx

j i

t t x tnx

i j j i j f i

If i j x is an internal edge of t and x is the q th edge of tx

i j i

l tx l tnx

j i j i

t t nx tx

i j j i q i

For all i such that x is a leaf of t

t t tx

i i i i

If i j x is an internal edge of t and x is the pth edge of tnx

i j i

l tnx l tx

j i j i

t t x tnx

i j j i p i

If i j and x is an internal edge of t

t t

i j j i

For all i

t t t

i i i i i

If i j and x is an internal edge of t

t t

i j j i

If i j

i j j i

The rst ve identities together with the lab elling convention imply that

is compatible with the identications used in dening K and is therefore well

dened The fourth identity gives according to the dimension of the cub e t

t if dim t

where is the unique tree The rst identity gives when x is the last edge

ro ot of t

t if dim t

N N

t if dim t

From the cubical identities it follows that is a chain homotopy from

where is the identity map and is a p oint map as ab ove to twice the identity

map There is additional checking to b e done on chains is given by

n where denotes the star tree with n leaves Therefore is

nullhomotopic by the chain homotopy and K is contractible

ALAN ROBINSON AND SARAH WHITEHOUSE

Acyclicity of K A M

The chain complex K A M is constructed as the cobre of a map from a

partial realization jMj to M It very easily follows that K A M is acyclic if

jMj is contractible so this contractibility is what we have to prove It is sucient

to treat the case when the co ecient mo dule M is A

We describ e jMj Just as K in the previous section was constructed by glueing

together the tree spaces T according to certain lab elling conventions so jMj is

obtained by glueing together the spaces C of a cobrant cyclic E op erad C

n n

for which we shall use the tree op erad In the new context it can b e seen that the

somewhat arbitrary lab elling convention is actually quite immaterial a dierent

choice leads by conjugation in symmetric groups to homotopic glueing maps and

so to a quasiisomorphic result But a choice has to b e made

Thus jMj is an extended version of K incorp orating the actions of the sym

metric groups One tries to contract it by applying brewise the contraction of

K This amounts to constructing a coherent system of higher homotopies among

the contractions obtained by twisting the original contraction by all elements of

the symmetric group One exp ects to b e able to do this since if and are two

contractions of a complex then is a homotopy of homotopies from to

Construction of jMj We construct jMj using the cobrant tree op erad T of

Since the symmetric group acts trivially on the nth tensor p ower of A over

itself the realization is built by glueing together the complexes T The free

n n

chain complex T has generators

n n

j j j t

in dimension k dim t where k and t is a simplex or cub e

k n

of the tree space T The b oundary is given by

j j j t j j t

k k

j j j j t

j j k

j j t

k k

j j j t

where t is the b oundary in T and t is dened using the p ermutation action of

n k

on the lab els of T

n n

The identications which form jMj from the chain complexes T mirror

n n

those used to form K from the T In the latter case we recall when an internal

edge x of t has length the chain t of K is identied with tx tnx which are

j j j

trees lab elled by our convention This lab elling convention is suciently functorial

to allow us to identify when x has length the chain j j t with

j k

j j tx j j tnx

k j k j

where and are the induced p ermutations of conventional lab elling sets for tx

i i j

and tnx For instance if have already b een dened then is uniquely

j k i i

OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

determined by the stipulation that tx b e the conventional lab elling

i i k j

of t x By these means we can dene cubical face op erators in

i i k j j

jMj just as in K

In a totally analogous way we can extend the denition of the op erators to

jMj setting

j j t j j t

j k k j

where j j is the induced string of p ermutations of the conventional lab elling

set of t We dene to b e the alternating sum but we can not

j j

exp ect this to b e a contraction b ecause of the form of the b oundary op erator in

jMj Nor is it true that b ecause the action of do es not preserve

i i n

the conventional ordering which is essentially used in the denition of

In the following denition and all that follows we use the notation to denote

any p ermutation induced by on a set of tree lab els derived by our conventions

The context always implies exactly what the trees in question are so it is not

necessary to burden the notation with any heavy details

Denition We dene an op erator on the chains of jMj by setting

j j t j j t

j j j k k

Theorem The chain complex K A M is acyclic

Proof We have remarked ab ove that it is sucient to prove that jMj is con

tractible and that we may take M to b e A We simply claim that the homotopy

dened ab ove is a contraction of jMj

To see this one rep eatedly uses the relation in K to calculate

that when t is a tree with at least one internal edge the relation

t t

j j k j j k

k j

j j r j r k

k j

j k

k j

j k

holds in jMj and a minor variant when t is a startree Then straightforward

calculation with the formulae dening and gives

in dimension

where is a p oint map The theorem is therefore proved

ALAN ROBINSON AND SARAH WHITEHOUSE

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OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS

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Email address carmathswarwickac uk sarahmathunivparis fr

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