XX THE SEGAL CONJECTURE

explain in the next section we can analyze the singular terms in in terms of

these sub quotient theories

Assume that k is einvariant for every prop er sub quotient J of Theorem

J

and let Y E P G

i If G is not elementary Ab elian then k Y EG

G

r r r

ii If G Zp then k Y EG is the direct sum of p copies of

G

r

k S

GG

arning the nonequivariant theory k is usually quite dierent from the W

GG

underlying nonequivariant theory k k

e

As we shall explain in Section we can use Adams sp ectral sequences to analyze

the free terms in

Theorem Assume that k is split and k is b ounded b elow and let Y

G

E P

Y EG i If G is not elementary Ab elian then k

G

r

k is nite Y EG is the ii If G Zp and H dimensional then k

G

r r r

direct sum of p copies of k S

The hyp othesis that H k b e nite dimensional in ii is extremely restrictive

although it is satised trivially when k is the sphere sp ectrum The hyp othesis

are is actually necessary We shall see in Section that the theories B

G

G

einvariant for nite groups They satisfy all other hyp otheses of our theorems

are dierent In such cases the calculation of k Y EG but here k and k

G

GG

falls out from the einvariance which must b e proven dierently and

reduction is now the case of the following immediate inductive Carlssons

G

consequence of the rst parts of Theorems and

Theorem Supp ose that G is not elementary Ab elian Assume

i k is einvariant for all elementary Ab elian sub quotients J

J

ii k is split and k is b ounded b elow for all nonelementary Ab elian sub

J

KK

quotients J H K

Then k is einvariant for all sub quotients J including J G

J

to cohomotopy and the pro of of the Segal conjecture it only remains Returning

r

to prove that the map in is an isomorphism when G Zp We assume

that the result has b een proven for q r Comparing Theorems and

we see that the map in is a map b etween free mo dules on the same

APPROXIMATIONS OF SINGULAR SUBSPACES OF GSPACES

numb er of generators It suces to show that is a bijection on generators which

means that it is an isomorphism in degree r Here is a map b etween free

mo dules on the same numb er of generators over the padic integers Z so that it

p

will b e an isomorphism if it is a monomorphism when reduced mo d p

EG H F where H F is the Eilenb ergMacLane G To prove this let k F

p G

p

sp ectrum asso ciated to the constant Mackey functor at F that we obtain from

p

This theory like any other theory represented by a function sp ectrum IX

G

F EG is einvariant Since H F F we have a unit map S H F

p G

p p

and we comp ose with H F S k There is an k to obtain

G G G

p

induced map S S k and a little calculation shows that it sends the

GG GG

unit in S to an element that is nonzero mo d p We can also check that the

are all einvariant By sub quotient theories k the naturality of we have the

J

commutative diagram

r

r

Y EG

Y EG

G

G

 

r

r

k Y EG

Y EG k

G

G

Y The left map is The b ottom map is an isomorphism since k

G

r r r

the sum of p copies of S k and is therefore a

GG

monomorphism mo d p Thus the top map is a monomorphism mo d p and this

concludes the pro of

J H Gunawardena and H Miller The Segal conjecture for elementary Ab elian J F Adams

pgroupsI Top ology

G Carlsson Equivariant stable homotopy and Segals Burnside ring conjecture Annals Math

J Caruso J P May and S B Priddy The Segal conjecture for elementary Ab elian pgroupsI I

Top ology

Approximations of singular subspaces of Gspaces

Let SX denote the singular set of a Gspace X namely the set of p oints with

nontrivial isotropy group The starting p oint of the pro of of Theorem is the

space level observation that the inclusions

SX X and S EG

XX THE SEGAL CONJECTURE

induce bijections

S X EG X X EG X S X X

G G G

We may represent theories on nite GCW complexes via colimits of space level

homotopy classes of maps The precise formula is not so imp ortant What is

imp ortant is that when calculating k X EG we get a colimit of terms of the

G

general form SW Z We can replace S here by other functors T on spaces that

G

satisfy appropriate axioms and still get a cohomology theory in X called k X T

G

transformations T T induce Such functors are called S functors Natural

maps of theories contravariantly We have a notion of a cobration of S functors

and cobrations give rise to long exact sequences In sum we have something like

a cohomology theory on S functors T

that approximates the singular functor S We construct a ltered S functor A

Let A A G b e the partially ordered set of nontrivial elementary Ab elian

subgroups of G thought of as a Gcategory with a map A B when B A

e the classifying space B A is Gcontractible with G acting by conjugation If G

In fact if C is a central subgroup of order p then the diagram A AC C

displays the values on an ob ject A of three Gequivariant functors on A together

with two equivariant natural transformations b etween them these induce a G

homotopy from the identity to the constant Gmap at the vertex C

we construct a top ological We can parametrize A by p oints of SX Precisely

A

Gcategory A X whose ob jects are pairs A x such that x X there is a mor

if B A and y x and G acts by g A x g Ag g x phism A x B y

Pro jection on the X co ordinate gives a functor A X SX where SX is a cate

gory in the trivial way and B A X BSX SX is a Ghomotopy equivalence

of B A X is Gcontractible Let AX B A X B A We The subspace B A

still have a Ghomotopy equivalence AX SX but now A is an S functor and

our equivalences give a map of S functors F or any space Y we have

k Y A k Y S k Y EG

G G G

The functor A arises from geometric realizations of simplicial spaces and carries

the simplicial ltration F A here F A and F A A where r rank G

q r

Insp ection of denitions shows that the successive sub quotients satisfy

q A

F AF AX G X

q q

H

AN INVERSE LIMIT OF ADAMS SPECTRAL SEQUENCES

Here runs over the Gconjugacy classes of strictly ascending chains A A

q

of nontrivial elementary Ab elian subgroups of G H is the isotropy group of

namely fg jg A g A i q g and A A For each normal subgroup

i i q

K of a subgroup H of G there is an S functor C K H whose value on X is

K

G X and as S functors

H

q

F AF A C A H

q q

By direct insp ection of denitions we nd that for any space Y

K

k Y k Y C K H

H K G

This is why the xed p oint functors enter into the picture

To prove Theorem we restrict attention to Y E P If G is not elementary

K

Ab elian then Y is contractible and the sub quotients H K are prop er for all

r

pairs K H that app ear in If G Zp and q r this is still true

r

All these terms vanish by hyp othesis If G Zp we are left with the case

q r Here A H G for all chains there are pp chains

G

and Y S Using Theorem follows

G Carlsson Equivariant stable homotopy and Segals Burnside ring conjecture Annals Math

J Caruso J P May and S B Priddy The Segal conjecture for elementary Ab elian pgroupsI I

Top ology

An inverse limit of Adams sp ectral sequences

We turn to the pro of of Theorem Its hyp othesis that k is split allows

G

us to reduce the problem to a nonequivariant one and the hyp othesis that the

underlying nonequivariant sp ectrum k is b ounded b elow ensures the convergence

of the relevant Adams sp ectral sequences We prove Theorem by use of a

particularly convenient mo del Y for E P namely the union of the Gspheres

nV

S where V is the reduced regular complex representation of G It is a mo del

G H

since V fg and V for H P

V

In general for any representation V there is a Thom sp ectrum BG Here we

V BG may think of V as the negative of the representation bundle EG

G

regarded as a map V BG BO Z If V is suitably oriented for example

V

if V is complex there is a Thom isomorphism showing that H BG is a free

H BGmo dule on one generator of degree n where n is the real dimension

v

of V We take cohomology with mo d p co ecients For V W there is a

W V V W

map f BG BG such that f H BG H BG carries

XX THE SEGAL CONJECTURE

to W V Here V H BG is the Euler class of V which is the

v w

Euler class of its representation bundle For a split Gsp ectrum k we have an

G

isomorphism

V G V

k BG k S EG

W V

For V W the map f k BG k BG corresp onds under the

V W

isomorphisms to the map induced by e S S The pap er of mine cited at

the end gives details on all of this With our mo del Y for E P we now see that

q

nV G

lim k BG Y EG k Y EG k

q

G q

Rememb er that we are working padically we complete sp ectra at p without

change of notation The inverse limit E of Adams sp ectral sequences of an inverse

r

sequence fX g of b ounded b elow sp ectra of nite typ e over the padic integers Z

n p

converges from

E Ext colim H X F

A n p

nV

to lim X With X k BG this gives an inverse limit of Adams sp ectral

n n

sequences that converges from

nV

colim H BG F E Ext H k

p A

to k Y EG The colimit is taken with resp ect to the maps

G

nV nV

V H BG H BG

H

Since V fg V restricts to zero in H BH for all H P A theorem of

implies that V must b e nilp otent if G is not elementary Ab elian and Quillen

this implies that E This proves part i of Theorem

r

r p

Now assume that G Zp Let L BG Then V H

nV

colim H BG H BGL

It is easy to write L down explicitly and the heart of part ii is the following

purely algebraic calculation of Adams Guna wardena and Miller which gives the

E term of our sp ectral sequence

H BGL F and regard S t as a trivial A Theorem Let S t

A p

r r

mo dule Then S t is concentrated in degree r and has dimension p The

quotient homomorphism H BGL S t induces an isomorphism

Ext K S t F Ext K H BGL F

A p A p

for any nite dimensional Amo dule K

FURTHER GENERALIZATIONS MAPS BETWEEN CLASSIFYING SPACES

The notation S t stands for Steinb erg GLr F acts naturally on everything

p

in sight and S t is the classical Steinb erg representation

r r r

Let W b e the wedge of p copies of S It follows by convergence that

nV

there is a compatible system of maps W BG that induces an isomorphism

nV

k Y EG k W k W lim k BG

G

This gives Theorem ii It also implies the following remarkable corollary

which has had many applications

Cor ollary The wedge of spheres W is equivalent to the homotopy limit

V nV

of the Thom sp ectra BG In particular with G Z S is BG

equivalent to the sp ectrum holim R P

i

J F Adams J H Gunawardena and H Miller The Segal conjecture for elementary Ab elian

pgroupsI Top ology

J Caruso J P May and S B Priddy The Segal conjecture for elementary Ab elian pgroupsI I

Top ology

J P May Equivariant constructions of nonequivariant sp ectra Algebraic Top ology and Alge

braic K theory Princeton Univ Press

ov Cohomology of nite groups and elementary Ab elian subgroups D Quillen and B Venk

Top ology

Further generalizations maps b etween classifying spaces

Even b efore the Segal conjecture was proven Lewis McClure and I showed

that it would have the following implication Let G and b e nite groups and

e that let AG b e the Grothendieck group of free nite G sets Observ

b e the augmentation ideal of AG AG is an AGmo dule and let I

Theorem There is a canonical isomorphism

BG B AG

I I

The map AG BG B can b e describ ed explicitly in

terms of transfer maps and classifying maps and the pap er of mine cited at the

end gives more ab out the relationship b etween the algebra on the left and the

top ology on the right A free G set T determines a principal bundle

EG T EG T

G G

XX THE SEGAL CONJECTURE

which is classied by a map T EG T B It also determines a not

G

necessarily connected nite cover

EG T EG fg B G

G G

T BG EG T Both and which has a stable transfer map

G

are additive in T and is the unique homomorphism such that

T T T

In principle this reduces to pure algebra the problem of computing stable maps

b etween the classifying spaces of nite groups Many authors have studied the

relevant algebra Nishida Martino and Priddy Harris and Kuhn Benson and

Feshback and Webb among others and have obtained a rather go o d under

standing of such maps We shall not go into these calculations Rather we shall

place the result in a larger context and describ e some substantial generalizations

Recall that we interpreted the consequences of the Sullivan conjecture for maps

b etween classifying spaces as statements ab out equivariant classifying spaces Anal

ogously Theorem is a consequence of a result ab out the susp ension Gsp ectra

of equivariant classifying spaces

Theorem The cohomology theory B is einvariant There

G I

G

fore the map EG induces an isomorphism

BG B EG B S B

G G I

G G

In degree zero this is The isomorphism on the right comes from XVI

Theorem The description of the map of that result is obtained by describing

the map of Theorem in nonequivariant terms using the splitting theorem for

G

B of VI I the splitting theorem for the homotopy groups of susp ension

G

sp ectra of XIX and some diagram chasing

We next p oint out a related consequence of the generalization of the Segal

conjecture to families In it we let b e a normal subgroup of a nite group

Theorem The pro jection E induces an isomorphism

B E A

I

F

G

in the This is just the degree zero part of Theorem for the family F

group the last isomorphism is a consequence of XVI With the Burnside

ring replaced by the representation ring a precisely analogous result holds in

K theory but in that context the result generalizes to an arbitrary extension of

FURTHER GENERALIZATIONS MAPS BETWEEN CLASSIFYING SPACES

compact Lie groups Of course these may b e viewed as calculations of equivariant

characteristic classes It is natural to ask if Theorems and admit a common

generalization or b etter if the completion theorems of which they are sp ecial cases

admit a common generalization

A result along these lines was proven by Snaith Zelewski and myself Here for

the rst time in our discussion we let compact Lie groups enter into the picture

We consider nite groups G and J and a compact Lie group Let AG J

b e the Grothendieck group of principal G J bundles over nite G J sets

This is an AG J mo dule and we can complete it at the ideal I F J As in

G

F J is the family of subgroups H of G J such that H J e VI Ix

G

Theorem There is a canonical isomorphism

B J B AG J

I G G G I

F F J J

G G

Again the map AG J B J B is given on principal

G G G

G J bundles as comp osites of equivariant classifying maps and equivariant

transfer maps Although the derivation is not quite immediate this result is a

consequence of an invariance result exactly analogous to the version of the Segal

conjecture given in Theorem

Theorem Let b e a normal subgroup of a compact Lie group with

nite quotient group G Let S b e a multiplicatively closed subset of AG and

B is let I b e an ideal in AG Then the cohomology theory S 

I

G

H invariant where

H fSupp P jP S and P I g

The statement is identical with that of Theorem except that we have substi

tuted B for S as the second variable of our bitheory We could generalize

a bit further by substituting E X for any nite CW complex X What

The elementary pgroup case of the pro of of the other Gspaces can b e substituted

Segal conjecture makes it clear that one cannot substitute an arbitrary Gspace

In fact very little more than what we have already stated is known

Theorem sp ecializes to give the analog of Theorem

Theorem Let F b e a family in G where G The map E F

induces an isomorphism

 S B  E F B

I

F

G G

XX THE SEGAL CONJECTURE

We can restate this in Mackey functor form as in Theorem and then deduce

a conceptual formulation generalizing Theorem

Theorem For every family F in G the map

F K I F B F E F B

is an equivalence of Gsp ectra

This extends the calculational consequences to the RO Ggraded represented

theories Exactly as in Sections all of these theorems reduce to the following

sp ecial case

Let b e a normal subgroup of a compact Lie group such Theorem

that the quotient group G is a nite pgroup Then the theory  B

p

G

is einvariant

The pro of is a b o otstrap argument starting from the Segal conjecture When

is nite the result can b e deduced from the generalized splitting theorem of

XIX and the case of the Segal conjecture for that deals with the family of

subgroups of that are contained in When is a nite extension of a torus

the result is then deduced by approximating by an expanding sequence of nite

groups this part of the argument entails rather rather elab orate duality and colimit

arguments together with several uses of the generalized Adams isomorphisms

XVI Finally the general case is deduced by a transfer argument

As is discussed in my pap er with Snaith and Zelewski and more extensively in

the survey of Lee and Minami these results connect up with and expands what is

known ab out the Segal conjecture for compact Lie groups

D J Benson and M Feshbach Stable splittings of classifying spaces of nite groups Top ology

Stable decomp ositions of classifying spaces of nite Ab elian pgroups J Harris and N Kuhn

Math Pro c Camb Phil So c

CN Lee and N Minami Segals Burnside ring conjecture for compact Lie groups in Algebraic

top ology and its applications MSRI Publications SpringerVerlag

L G Lewis J P May and J E McClure Classifying Gspaces and the Segal conjecture

Canadian Math So c Conf Pro c Vol Part

J Martino and Stewart Priddy The complete stable splitting for the classifying space of a nite

group Top ology

J P May Stable maps b etween classifying spaces Cont Math Vol

J P May V P Snaith and P Zelewski A further generalization of the Segal conjecture Quart

J Math Oxford

G Nishida Stable homotopy typ e of classifying spaces of nite groups Algebraic and Top olog

to the memory of T Miyata ical Theories

CHAPTER XXI

Tate cohomology Generalized

by J P C Greenlees and J P May

In this chapter we will describ e some joint work on the generalization of the

Tate cohomology of a nite group G with co ecients in a Gmo dule V to the Tate

cohomology of a compact Lie group G with co ecients in a Gsp ectrum k There

G

has b een a great deal of more recent work in this area with many calculations and

indicate some of the main directions applications We shall briey

J P C Greenlees Representing Tate cohomology of Gspaces Pro c Edinburgh Math So c

J P C Greenlees and J P May Generalized Tate cohomology Memoirs Amer Math So c

No

Denitions and basic prop erties

Tate cohomology has long played a prominent role in nite group theory and

its applications For a nite group G and a Gmo dule V the Tate cohomology

H V is obtained as follows One starts with a free resolution

G

P P Z

of Z by nitely generated free ZGmo dules dualizes it to obtain a resolution

Z P P

renames P P and splices the two sequences together to obtain a Zgraded

i

i

exact complex P of nitely generated free ZGmo dules with a factorization

The complex P is called a complete resolution of Z P Z P of d

and H V is dened to b e the cohomology of the co chain complex Hom P V

G

G

G

There results a norm exact sequence that relates H V H V and H V

G G

XXI GENERALIZED TATE COHOMOLOGY

In connection with Smith theory Swan generalized this algebraic theory to a

cohomology theory H X V on Gspaces X using HomP C X V Swan

G

to ok X to b e a Gsimplicial complex but singular chains could b e used When

and X is a CWcomplex with a cellular action by G there is G S or G S

a closely analogous theory that is obtained by replacing P by Zu u where u

has degree or Here HomP C X V has dierential

dp x p dx pu i x

where i C S or i C S is the fundamental class For S this is p erio dic

cyclic cohomology theory

We shall give a very simple denition of a common generalization of these vari

ants of Tate theory In fact as part of a general norm cobration sequence we

shall asso ciate a Tate Gsp ectrum tk to any Gsp ectrum k where G is any

G G

compact Lie group The construction is closely related to the stable homotopy

limit problem and to nonequivariant stable homotopy theory

We have the cob er sequence

EG S E G

and the pro jection EG S induces the canonical map of Gsp ectra

k F S k F EG k

G G G

Taking the smash pro duct of the cob ering with the map we obtain

the following map of cob erings of Gsp ectra

k EG

k

k EG G

G

G

id id

F EG k EG F EG k G

F EG k G E G

G

We have seen most of the ingredients of this diagram in our discussion of the Segal

conjecture We intro duce abbreviated notations for these sp ectra Dene

f k k EG

G G

We call f k the free Gsp ectrum asso ciated to k It represents the appropri

G G

ate generalized version of the Borel homology theory H EG X Precisely if

G

DEFINITIONS AND BASIC PROPERTIES

k is split with underlying nonequivariant sp ectrum k then by XVI

G

Ad G

k EG f k X X

G G

We refer to the homology theories represented by Gsp ectra of the form f k as

G

Borel homology theories We refer to the cohomology theories represented by the

f k simply as f cohomology theories Dene

G

f k F EG k EG

G G

It is clear that the map Id f k f k is always an equivalence so that

G G

the Gsp ectra f k and f k can b e used interchangeably We usually drop the

G G

notation f preferring to just use f Dene

f k k E G

G G

We call f k the singular Gsp ectrum asso ciated to k

G G

Dene

c k F EG k

G G

We call ck the geometric completion of k The problem of determining the

G G

b ehavior of k ck on Gxed p oint sp ectra is the stable homotopy limit

G G

problem We have already discussed this problem in several cases and we have

seen that it is b est viewed as the equivariant problem of comparing the geometric

completion ck with the algebraic completion k of k at the augmentation

G G I G

ideal of the Burnside ring or of some other ring more closely related to k As

G

one would exp ect ck represents the appropriate generalized version of Borel

G

cohomology H EG X Precisely if k is a split Gsp ectrum with underlying

G G

nonequivariant sp ectrum k then by XVI

k EG X ck X

G G

We therefore refer to the cohomology theories represented by Gsp ectra ck as

G

Borel cohomology theories We refer to the homology theories represented by the

ck as chomology theories

G

Finally dene

tk F EG k EG f ck

G G G

We call tk the Tate Gsp ectrum asso ciated to k It is the singular part of the

G G

geometric completion of k Our primary fo cus will b e on the theories represented

G

by the tk These are our generalized Tate homology and cohomology theories G

XXI GENERALIZED TATE COHOMOLOGY

With this cast of characters and with the abbreviation of id to the diagram

can b e rewritten in the form

f k

k

f k

G

G

G

f k ck tk

G G G

The b ottom row is the promised norm cobration sequence The theories rep

resented by the sp ectra on this row are all einvariant

The denition implies that if X is a free Gsp ectrum then

tk X and tk X

G G

Similarly if X is a nonequivariantly contractible Gsp ectrum then

ck X and f k X

G G

By denition Tate homology is a sp ecial case of chomology

tk X ck EG X

G n G n

The two vanishing statements imply that Tate cohomology is a sp ecial case of

f cohomology

n n

f k EG X tk X

G G

In fact on the sp ectrum level the vanishing statements imply the remarkable

equivalence

tk F EG k EG F E G EG k F E G f k

G G G G

It is a consequence of the denition that tk is a ring Gsp ectrum if k is a

G G

G

ring Gsp ectrum and then tk is a ring sp ectrum

G

Much of the force of our denitional framework comes from the fact that

is a diagram of genuine and conveniently explicit Gsp ectra indexed on representa

tions so that all of the Zgraded cohomology theories in sight are RO Ggradable

The RO Ggrading is essential to the pro ofs of many of the results discussed b e

low Nevertheless it is interesting to give a naive reinterpretation of the xed

p oint cobration sequence asso ciated to the norm sequence

With our denitions the Tate homology of X is

G

tk X tk X

G G

ORDINARY THEORIES ATIYAHHIRZEBRUCH SPECTRAL SEQUENCES

Since any k is eequivalent to j for a naive Gsp ectrum j and Tate theory is

G G G

einvariant we may as well assume that k i j Provided that X is a nite

G G

G

GCW complex the sp ectrum tk X is then equivalent to the cob er of an

G

appropriate transfer map

AdG AdG

j X j EG X G

G hG G

hG G

j X F EG j X

G G

A description like this was rst written down by Adem Cohen and Dwyer When

G is nite X S and j is a nonequivariant sp ectrum k given trivial action by

G

G this reduces to

k BG F BG k

The interpretation of Tate theory as the third term in a long sequence whose other

terms are Borel k homology and Borel k cohomology is then transparent

A Adem R L Cohen and W G Dwyer Generalized Tate homology homotopy xed p oints

and the transfer Contemp orary Math Volume

J D S Jones Cyclic homology and equivariant homology Inv Math

R G Swan A new metho d in xed p oint theory Comm Math Helv

Ordinary theories AtiyahHirzebruch sp ectral sequences

Let M b e a Mackey functor and V b e the Gmo dule M Ge The norm

sequence of HM dep ends only on V if M and M are Mackey functors for which

M Ge as Gmo dules then the norm cobration sequences of M Ge

HM and HM are equivalent We therefore write

G

H X V tHM X and H X V tHM X

G

For nite groups G this recovers the TateSwan cohomology groups as the nota

tion anticipates We sketch the pro of The simple ob jects to the eyes of ordinary

cohomology are cells and the calculation dep ends on an analogue of the skeletal

ltration of a CW complex that mimics the construction of a complete resolution

The idea is to splice the skeletal ltration of EG with its SpanierWhitehead

dual More precisely we dene an integer graded ltration on EG or rather on

XXI GENERALIZED TATE COHOMOLOGY

its susp ension sp ectrum by letting

i

i

C EG for i EG S

i

F EG

S for i

i

D EG for i

i

The ith sub quotient of this ltration is a nite wedge of sp ectra S G and the

E term of the sp ectral sequence that is obtained by applying ordinary nonequiv

ariant integral homology is a complete resolution of Z Therefore if one takes the

smash pro duct of this ltration with the skeletal ltration of X and applies an

equivariant cohomology theory k one obtains the AtiyahHirzebruchTate

G

sp ectral sequence

pq pq p

q

tk X E H X k

G G

q

Here k is the underlying nonequivariant sp ectrum of k and k k regarded

G q

as a Gmo dule To see that the target is Tate cohomology as claimed note that

the cohomological description of the Tate sp ectrum gives

X EG X k EG tk

G G

There are comp ensating shifts of grading in the identications of the E terms and

of the target so that the grading works out as indicated in

When k HM the sp ectral sequence collapses at the E term by the dimen

G

sion axiom and this proves that tHM X is the TateSwan cohomology of X

G

In general we have a whole plane sp ectral sequence but it converges strongly

to tk X provided that there are not to o many nonzero higher dierentials

G

When k is a ring sp ectrum it is a sp ectral sequence of dierential algebras

G

With a little care ab out the splice p oint and the mo del of EG used we can

apply part of this construction to compact Lie groups G of dimension d In

this case there is a gap in the appropriate ltration of EG

i

i

EG S C EG for i

i

F EG

S for d i

i

D EG for i d

d

The gap is dictated by the fact that the SpanierWhitehead dual of G is G S

In the case of Eilenb ergMacLane sp ectra this gives an explicit chain level

G G

calculation of the co ecient groups H V H S V in terms of the ordinary

ORDINARY THEORIES ATIYAHHIRZEBRUCH SPECTRAL SEQUENCES

unreduced homology and cohomology groups of the classifying space BG

n

H BG V if n

n n

H V tHM

if d n

G

H BG V if n d

nd

However we would really like a chain complex for calculating the ordinary Tate

cohomology of GCW complexes X and for groups of p ositive dimension it is not

obvious how to make one At present we only have such descriptions for G S

and G S In these cases we can exploit the obvious cell structure on G and the

and S H for EG to put a cunning GCW structure on standard mo dels S C

EG X and to derive an appropriate ltration of EG X when G acts cellularly

on X In the case of S the resulting chain complex is a cellular version of Jones

complex for cyclic cohomology and this proves that tH Z X is the p erio dic

1

S

cyclic cohomology H X as dened by Jones in terms of the singular complex

1

S

of X There is a precisely analogous identication in the case of S In general

the problem of giving EG X an appropriate ltration app ears to b e intractable

although a few other small groups are under investigation

Despite this diculty we still have sp ectral sequences of the form for

q

general compact Lie groups G where k k is now regarded as a G

q

mo dule However in the absence of a go o d ltration of EG X we construct

the sp ectral sequences by using a Postnikov ltration of k In this generality the

G

X V used to describ e the E terms are not familiar ones ordinary Tate groups H

G

and systematic techniques for their calculation do not app ear in the literature One

approach to their calculation is to use the skeletal ltration of X together with

and change of groups More systematic approaches involve the construction

of sp ectral sequences that converge to H X V and there are several sensible

G

candidates This is an area that needs further investigation and we shall say no

more ab out it here

We have similar and compatible sp ectral sequences for Borel and f cohomology

and in these cases to o the E terms dep end only on the graded Gmo dule k

as one would exp ect from the einvariance of the b ottom row of Diagram

This very weak dep endence on k makes the b ottom row much more calculationally

G

accessible than the top row

XXI GENERALIZED TATE COHOMOLOGY

Cohomotopy p erio dicity and ro ot invariants

For nite groups G the Segal conjecture directly implies the determination of

the Tate sp ectrum asso ciated to the sphere sp ectrum S Indeed we have

G

tS F EG S EG S EG EG

G G

I I

For instance if G is a pgroup then

tS EG

G

p

and we may calculate from the splitting theorem XIX that after completion

M

G H

tS X EW H X

G G

W H

G

H

With X S the summand for H G is S and it follows that for each

G the AtiyahHirzebruchTate sp ectral sequence denes a ro ot invariant on the

stable stems Its values are cosets in the Tate cohomology group H G S

Essentially the ro ot invariant assigns to an element S all elements of E

of the appropriate ltration that pro ject to the image of in the E term of the

sp ectral sequence

These invariants have not b een much investigated b eyond the classical case of

G C the cyclic group of order p In this case our construction agrees with

p

earlier constructions of the ro ot invariant Indeed this is a consequence of the

observations that if G C and k i k is the Gsp ectrum asso ciated to a

G

nonequivariant sp ectrum k then

G

tk holimRP k

G

i

and if G C for an o dd prime p and k i k then

p G

G

k tk holimL

G

i

where L is the lens space analog of RP Taking k S there results a sp ectral

i i

sequence that agrees with our AtiyahHirzebruchTate sp ectral sequence and was

used in the classical denition of the ro ot invariant

Similarly if G is the circle group and k i k then

G

G

tk holimC P k

G

i

These are all sp ecial cases of a phenomenon that o ccurs whenever G acts freely on

the unit sphere of a representation V and this phenomenon is the source of p erio dic

nV

b ehavior in Tate theory The p oint is that the union of the S is then a mo del

THE GENERALIZATION TO FAMILIES

for EG and we can use this mo del to evaluate the right side as a homotopy limit

in the equivalence This immediately gives These equivalences

allow us to apply nonequivariant calculations of Davis Mahowald and others of

sp ectra on the right sides to study equivariant theories We will say a little more

ab out this in Section It also gives new insight into the nonequivariant theories

G

In particular if k is a ring sp ectrum then tk is a ring sp ectrum Lo oking

G

nonequivariantly at the right sides this is far from clear

D M Davis and M Mahowald The sp ectrum P bo Pro c Cambridge Phil So c

1

D M Davis D C Johnson J Klipp enstein M Mahowald and S Wegmann The sp ectrum

P B P hi Trans American Math So c

1

The generalization to families

The theory describ ed ab ove is only part of the story it admits a generalization

in which the universal free Gspace EG is replaced by the universal F space E F

for any family F of subgroups of G The denitions ab ove deal with the case

F feg and there is a precisely analogous sequence of denitions for any other

family We have the cob ering

E F S E F

and the pro jection E F S induces a Gmap

k F S k F E F k

G G G

Taking the smash pro duct of the cob ering with the map we obtain

the following map of cob erings of Gsp ectra

k E F

k

G k E

G

G

id id

F E F k G E F F E F k G

F E F k G E F

Dene the F free Gsp ectrum asso ciated to k to b e

G

f k k E F

F G G

We refer to the homology theories represented by Gsp ectra f k as F Borel

F G

homology theories Dene

f k F E F k E F

G G F

XXI GENERALIZED TATE COHOMOLOGY

Again Id f k f k is an equivalence hence we usually use the

F G G

F

notation f Dene the F singular Gsp ectrum asso ciated to k to b e

F G

f k k E F

G G

F

Dene the geometric F completion of k G to b e

c k F E F k

F G G

We refer to the cohomology theories represented by Gsp ectra c k as F

F G

Borel cohomology theories The map k c k of is the ob ject

G F G

of study of such results as the generalized AtiyahSegal completion theorem and

the generalized Segal conjecture of AdamsHaeb erlyJackowskiMay As in these

results one version of the F homotopy limit problem is the equivariant problem

of comparing the geometric F completion c k with the algebraic completion

F G

k of k at the ideal I F of the Burnside ring or at an analogous ideal in a

G I G

F

ring more closely related to k Observe that we usually do not have analogs of

G

and for general families F the Adams isomorphism XVI and the

discussion around it are relevant at this p oint

Dene

c k t k F E F k E F f

F G F G G

F

We call t k the F Tate Gsp ectrum asso ciated to k These Gsp ectra rep

F G G

resent F Tate homology and cohomology theories With this cast and with the

abbreviation of id to the diagram can b e rewritten in the form

f k

k f k

F G

G G

F

f k c k t k

G F G F G

F

We call the b ottom row the F norm cobration sequence The theories repre

sented by the sp ectra on this row are all F invariant

The diagram leads to a remarkable and illuminating relationship b etween the

Tate theories and the F homotopy limit problem Recall that I F AG is the

intersection of the kernels of the restrictions AG AH for H F

Theorem The sp ectra c k are I F complete The sp ectra f k G

F G F

and t k are I F complete if k is b ounded b elow

F G G

THE GENERALIZATION TO FAMILIES

We promised in XXx to relate the questions of when

k F K I F k F E F k c k

G I F G G F G

and

k E F k K I F

G G

are equivalences The answer is rather surprising

Theorem Let k b e a ring Gsp ectrum where G is nite Then is an

G

equivalence if and only if is an equivalence and t k is rational

F G

The pro of is due to the rst author and will b e discussed in XXIVx We shall

turn to relevant examples in the next section

When G is nite and k is an Eilenb ergMacLane Gsp ectrum HM the F Tate

G

Gsp ectrum t HM represents the generalization to homology and cohomology

F

theories on Gspaces and Gsp ectra of certain AmitsurDressTate cohomology

theories H M that gure prominently in induction theory We again obtain

F

generalized AtiyahHirzebruchTate sp ectral sequences in the context of families

These vastly extend the web of symmetry relations relating equivariant theory

with the stable homotopy groups of spheres In particular for a nite pgroup G

if we use the family P of all prop er subgroups of G we obtain a sp ectral sequence

P G

whose E term is H and which converges to We have moved the

p

groups BWH from the target to ingredients in the calculation of E In this

sp ectral sequence the ro ot invariant of an element lies in degree at least

q

q jGj The ro ot invariant measures where elements are detected in E of the

sp ectral sequence and the dep endence on the order of G indicates an increasing

dep endence of lower degree homotopy groups of spheres on higher degree homotopy

groups of classifying spaces

More generally if G is any nite group we use the family P to obtain two re

lated sp ectral sequences b oth of which converge to the completion of the nonequiv

ariant stable homotopy groups of spheres at nP where nP is the pro duct of

those primes p such that ZpZis a quotient of G For example if G is a nonab elian

group of order pq p q then nP p and the sp ectral sequences provide a

mechanism for the prime q to aect stable homotopy groups at the prime p One

of the sp ectral sequences is the AtiyahHirzebruchTate sp ectral sequence whose

G P

The other comes from a E term is the AmitsurDressTate homology H

ltration of EG in terms of the regular representation of G These sp ectral se

quences lead to new equivariant ro ot invariants and the basic BredonJonesMiller

XXI GENERALIZED TATE COHOMOLOGY

ro ot invariant theorem generalizes to the sp ectral sequence constructed by use of

the regular representation

A W M Dress Contributions to the theory of induced representations Springer Lecture Notes

in Mathematics Vol

J P C Greenlees Tate cohomology in commutative algebra J Pure and Applied Algebra

J D S Jones Ro ot invariants cuprpro ducts and the KahnPriddy theorem Bull London

Math So c

H R Miller On Joness KahnPriddy theorem Springer Lecture Notes in Mathematics Vol

Equivariant K theory

Our most interesting calculation shows that for any nite group G tK is a

G

rational Gsp ectrum namely

i tK K J Q

G

where J is the Mackey functor of completed augmentation ideals of representation

rings and i ranges over the integers In this case the relevant AtiyahHirzebruch

Tate sp ectral sequence is rather amazing Its E term is torsion b eing annihilated

by multiplication by the order of G If G is cyclic then E E and the sp ectral

sequence certainly converges strongly In general the E term dep ends solely

on the classical Tate cohomology of G and not at all on its representation ring

whereas tK dep ends solely on the representation ring and not at all on the

G

Tate cohomology Needless to say the pro of of is not based on use of the

sp ectral sequence

In fact and the generalization is easier to prove than the sp ecial case t K

F G

turns out to b e rational for every family F Again there results an explicit

calculation of t K as a wedge of Eilenb ergMacLane sp ectra Let J F b e the

F G

intersection of the kernels of the restrictions RG RH for H F It is

clear by character theory that

J F fjg if the group generated by g is in F g

and we dene a rationally complementary ideal J F by

J F fjg if the group generated by g is not in F g

Then generalizes to

t K K RJ F Q i

F G J F

EQUIVARIANT K THEORY

where RJ F denotes the Mackey functor whose value at GH is the com

J F

pletion at the ideal J F j of the quotient RH J F jH This is consistent

H

with since when F feg J F jH is a copy of Z generated by the regular

representation of H and JH maps isomorphically onto RH Z It follows in all

cases that the completions t K are contractible

F G I F

The following folklore result is proven in our pap er on completions at ideals of the

G

Burnside ring On passage to the unit S K induces the homomorphism

G G

AG RG that sends a nite set X to the p ermutation representation C X

We regard RGmo dules as AGmo dules by pullback

Theorem The completion of an RGmo dule M at the ideal J F of RG

is isomorphic to the completion of M at the ideal I F of the Burnside ring AG

In fact the pro of shows that the ideals I F RG and J F of RG have the

same radical Therefore the generalized completion theorem of AdamsHaeb erly

JackowskiMay discussed in XIV implies that

K F E F K

G I F G

is an equivalence By and Theorem this in turn implies that

k E F k K I F

G G

is an equivalence In fact the latter result was proven by the rst author b efore

the implication was known we shall explain his argument and discuss the algebra

b ehind it in Chapter XXIV

As a corollary of the calculation of tK we obtain a surprisingly explicit

G

calculation of the nonequivariant K homology of the classifying space BG

Q Z J G Z and K BG K BG

J

G

In fact and b oth follow easily once we know that tK is rational

G

Given that we have the exact sequence

G

EG Q tK EG Q K K

G G

which turns out to b e short exact The AtiyahSegal theorem shows that

RG K EG Q Q

J G

XXI GENERALIZED TATE COHOMOLOGY

where is the Bott element Rationally the K homology of EG is a summand

G G

Q It is not hard to identify the maps of K and in fact K EG Q

and conclude that

tK fRGZg Q

G J

Since as explained in XIXx all rational Gsp ectra split this gives the exact

equivariant homotopy typ e claimed in Now we can deduce by analy

sis of the inte gral norm sequence using the AtiyahSegal completion theorem to

EG identify K

G

We must still say something ab out why tK and all other t K are rational

G F G

An inductive scheme reduces the pro of to showing that t K E P is rational

F G

where P is the family of prop er subgroups of G If V is the reduced regular

V

complex representation of V then S is a mo del for E P It follows that for any

G

X is the lo calization of K mo dule sp ectrum M and any sp ectrum X M E P

G

G

X away from the Euler class which is the total exterior p ower M V RG

Since V is in J P it restricts to zero in all prop er subgroups Since the pro duct

over the cyclic subgroups C of G of the restrictions RG RC is an injection

V and the conclusion holds trivially unless G is cyclic In that case the

AtiyahHirzebruchTate sp ectral sequence for t K X gives that primes that

P G

do not divide the order n of G act invertibly since n annihilates the E term An

easy calculational argument in representation rings handles the remaining primes

The evident analogs of all of these statements for real K theory are also valid

In the case of connective K theory we do not have the same degree of p erio dicity

to help and the calculations are harder Results of Davis and Mahowald give the

following result

Theorem If G C for a prime p then

p

Y

n

H J tk u

G

nZ

and similarly for connective real K theory

This result led us to the overoptimistic conjecture that its conclusion would

generalize to arbitrary nite groups However Bayen and Bruner have shown that

the conjecture fails for b oth real and complex connective K theory

Finally we must p oint out that the restriction to nite groups in the discussion

ab ove is essential even for G S something more complicated happ ens since

G

in that case tK is a homotopy inverse limit of wedges of even susp ensions of G

FURTHER CALCULATIONS AND APPLICATIONS

G

K and each even degree homotopy group of tK is isomorphic to Z

G

where is the canonical irreducible onedimensional representation of G In

particular tK is certainly not rational Similarly still taking G S each

G

G

even degree homotopy group of tk is isomorphic to Z In this case we

G

can identify the homotopy typ e of the xed p oint sp ectrum

Y

1

S n

1 1

tk u k u

S S

nZ

J F Adams JP Haeb erly S Jackowski and J P May A generalization of the AtiyahSegal

completion theorem Top ology

D Bayen and R R Bruner Real connective K theory and the quaternion group Preprint

J P C Greenlees K homology of universal spaces and lo cal cohomology of the representation

ring Top ology

J P C Greenlees and J P May Completions of Gsp ectra at ideals of the Burnside ring

Adams Memorial Symp osium on Algebraic Top ology Vol London Math So c Lecture Notes

Vol

Further calculations and applications

Philosophically one of the main dierences b etween the calculation of the Tate

K theory for nite groups and for the circle group is that the Krull dimension of

RG is one in the case of nite groups and two in the case of the circle group Quite

generally the complexity of the calculations increases with the Krull dimension of

the co ecient ring It is relevant that the Krull dimension of RG for a compact

connected Lie group G is one greater than its rank

For nite groups most calculations that have b een carried out to date con

cern ring Gsp ectra k like those that represent K theory that are so related to

G

cob ordism as to have Thom isomorphisms of the general form

V j G j G V

X k k X

V

for all complex representations V Let eV S S b e the inclusion Ap

V V

k S we obtain an element of plying eV to the element k S

G G

V G

k S k S The Thom isomorphism yields an isomorphism b etween this

G V

G

group and the integer co ecient group k and there results an Euler class

jV j

G

V k As in our indication of the rationality of tK lo calizations and

G

jV j

other algebraic constructions in terms of such Euler classes can often lead to ex plicit calculations

XXI GENERALIZED TATE COHOMOLOGY

This works particularly well in cases such as pgroups where G acts freely on

a pro duct of unit spheres S V S V for some representations V V

n n

This implies that the smash pro duct S V S V is a mo del for

n

EG and there results a ltration of EG that has sub quotients given by wedges

of smash pro ducts of spheres This gives rise to a dierent sp ectral sequence for

the computation of tk X When X S the E term can b e identied as

G

G

Cech cohomology H k BG of the k mo dule k BG with resp ect to the

0

J G

the ideal J V V k The relevant algebraic denitions will b e

n

G

given in Chapter XXIV These groups dep end only on the radical of J and when

k is No etherian it turns out that J has the same radical as the augmentation

G

k ideal J Kerk

G

The interesting mathematics b egins with the calculation of the E term where

the nature of the Euler classes for the particular theory b ecomes imp ortant In

fact this sp ectral sequence collapses unusually often b ecause the complexity is

controlled by the Krull dimension of the co ecients In cases where one can

one can often also deduce the homotopy typ e of calculate the co ecients tk

G

G G

the xed p oint sp ectrum tk b ecause tk is a mo dule sp ectrum over k

G G

However the p erio dic and connective cases have rather dierent avors In the

p erio dic case the algebra of the co ecients has a eldlike app earance and is

more often enough to determine the homotopy typ e of the xed p oint sp ectrum

G

tk In the connective case the algebra of the co ecients in the answer has

G

the app earance of a complete lo cal ring and some sort of Adams sp ectral sequence

argument seems to b e necessary to deduce the top ology from the algebra In very

exceptional circumstances such as the use of rationality in the case of K one

G

can go on to deduce the equivariant homotopy typ e of tk

G

In the discussion that follows we consider equivariant forms k of some familiar

G

nonequivariant theories k We may take k to b e i k but any split Gsp ectrum

G

with underlying nonequivariant sp ectrum k could b e used instead Technically

it is often b est to use F EG i k This has the advantage that its co ecients

can often b e calculated and it can b e thought of as a geometric completion of

any other candidate and an algebraic completion of any candidate for which a

completion theorem holds

The most visible feature of the calculations to date is that the Tate construction

tends to decrease chromatic p erio dicity We saw this in the case of K where the

G

p erio dicity reduced from one to zero This app ears in esp ecially simple form in a

theorem of Greenlees and Sadofsky if K n is the nth Morava K theory sp ectrum

FURTHER CALCULATIONS AND APPLICATIONS

whose co ecient ring is the graded eld

n

v deg v p K n F v

n n p

n

then

tK n

G

In fact this is a quite easy consequence of Ravenels result that K n BG is

nitely generated over K n Another example of this nature is a calculation of

Fa jstrup which shows that if the sp ectrum KR that represents K theory with

reality is regarded as a C sp ectrum then the asso ciated Tate sp ectrum is trivial

These calculations illustrate another phenomenon that app ears to b e general it

seems that the Tate construction reduces the Krull dimension of p erio dic theories

More precisely the Krull dimension of tk is usually less than that of k In the

G

G G

is nite case of Morava K theory one deduces from Ravenels result that K n

G

over K n and thus has dimension The contractibility of tK n can then b e

G

thought of as a degenerate form of dimension reduction More convincingly work

is of Greenlees and Sadofsky shows that for many p erio dic theories for which k

G

is nite dimensional over a eld The higher dimensional one dimensional tk

G

G

case is under consideration by Greenlees and Strickland

This reduction of Krull dimension is reected in the E term of the sp ectral

sequence cited ab ove When k is v p erio dic for some n one typically rst proves

n

that some v i n is invertible on tk and then uses the lo calisation of the

i G

norm sequence

h h i i

G

EG k EG v v k tk

G

i i G G

to assist calculations For example consider the sp ectra E n with co ecient rings

E n Z v v v v

p n

n

v

1

E p K we deduce from Since there is a cob er sequence E p

that v is invertible on tE p More generally v is invertible on a suitable

G n

completion of tE n

G

The intuition that the Tate construction lowers Krull dimension is reected in

the following conjecture ab out the sp ectra BP h ni with co ecient rings

BP hni Z v v v

n

p

XXI GENERALIZED TATE COHOMOLOGY

Conjecture DavisJohnsonKlippensteinMahowaldWegmann

Y

C n

p

BP hn i tBP hni

C

p

p

nZ

The cited authors proved the case n the case n was due to Davis and

Mahowald Since BP hni has Krull dimension n the depth of the conjecture

increases with n

We end by p ointing the reader to what is by far the most striking application of

okstedt Hesselholt generalized Tate cohomology In a series of pap ers Madsen B

and Tsalidis have used the case of S and its subgroups to carry out fundamentally

imp ortant calculations of the top ological cyclic homology and thus of the algebraic

K theory of numb er rings It would take us to o far aeld to say much ab out this

Madsen has given two excellent surveys In another direction Hesselholt and

Madsen have calculated the co ecient groups of the S tate sp ectrum asso ciated

to the p erio dic J theory sp ectrum at an o dd prime The calculation is consistent

with the following conjecture

Conjecture Hesselhol tMadsen

Y

1

n n S

K K K K tJ

G

nZ nZ

where K is the Adams summand of pcomplete K theory with homotopy groups

concentrated in degrees mo d p

okstedt and I Madsen Top ological cyclic homology of the integers Asterisque M B

M Bokstedt and I Madsen Algebraic K theory of lo cal numb er elds the unramied case

Aarhus University Preprint No

D M Davis and M Mahowald The sp ectrum P bo Pro c Cambridge Phil So c

1

D M Davis D C Johnson J Klipp enstein M Mahowald and S Wegmann The sp ectrum

P BP hi Trans American Math So c

1

L Fa jstrup Tate cohomology of p erio dic Real Ktheory is trivial Pro c American Math So c

to app ear

J P C Greenlees and H Sadofsky The Tate sp ectrum of v p erio dic complex oriented theories

n

Math Zeitschrift To app ear

J P C Greenlees and H Sadofsky The Tate sp ectrum of theories with one dimensional co e

cient ring Preprint

J P C Greenlees and N P Strickland Varieties for chromatic group cohomology rings In

preparation

1

Tate sp ectrum for J Bol So c Math Mex L Hesselholt and I Madsen The S

FURTHER CALCULATIONS AND APPLICATIONS

LHesselholt and I Madsen Top ological cyclic homology of nite elds and their dual numb ers

Preprints

I Madsen The cyclotomic trace in algebraic Ktheory Pro ceedings of the ECM Paris

Progress in Mathematics Vol Birkhauser

I Madsen Algebraic K theory and traces Aarhus University Preprint No

D C Ravenel Morava K theories and nite groups Cont Mathematics

S Tsalidis Top ological Ho chschild homology and the homotopy limit problem Preprint

S Tsalidis On the top ological cyclic homology of the integers Preprint

XXI GENERALIZED TATE COHOMOLOGY

CHAPTER XXI I

ve new algebra Bra

The category of S mo dules

Let us return to the intro ductory overview of the stable homotopy category given

in XI Ix As said there Elmendorf Kriz Mandell and I have gone b eyond the

foundations of Chapter XI I to the construction of a new category of sp ectra the

category of S mo dules that has a smash pro duct that is symmetric monoidal

asso ciative commutative and unital up to coherent natural isomorphisms on the

p ointset level The complete treatment is given in EKMM and an exp osition

has b een given in EKMM The latter emphasizes the logical development of the

foundations Here instead we will fo cus more on the structure and applications

of the theory Working nonequivariantly in this chapter we will describ e the

new categories of rings mo dules and algebras and summarize some of their more

imp ortant applications All of the basic theory generalizes to the equivariant

context and working equivariantly we will return to the foundations and outline

the construction of the category of S mo dules in the next chapter We b egin work

here by summarizing its prop erties

with additional An S mo dule is a sp ectrum indexed on some xed universe U

structure and a map of S mo dules is a map of sp ectra that preserves the additional

structure The sphere sp ectrum S and more generally any susp ension sp ectrum

X has a canonical structure of S mo dule The category of S mo dules is denoted

It is symmetric monoidal with unit ob ject S under a suitable smash pro duct M

S

which is denoted and it also has a function S mo dule functor which is denoted

S

F The exp ected adjunction holds

S

M M F N P P M M N

S S S S

XXI I BRAVE NEW ALGEBRA

Moreover for based spaces X and Y there is a natural isomorphism of S mo dules

X Y X Y

S

When regarded as a functor from spaces to S mo dules rather than as a functor

left adjoint to the zeroth space functor from spaces to sp ectra is not

rather we have an adjunction

X M S M T M X M

S S

Here the space of maps M S M is not even equivalent to M As observed by

S

Hastings and Lewis this is intrinsic to the mathematics since M is symmetric

S

monoidal M S S is a commutative top ological monoid and it therefore cannot

S

S b e equivalent to the space QS

For an S mo dule M and a based space X the smash pro duct M X is an

S mo dule and

M X M X

S

Cylinders cones and susp ensions of S mo dules are dened by smashing with

I I and S A homotopy b etween maps f g M N of S mo dules is a

map M I N that restricts to f and g on the ends of the cylinder The

ty

X M is the appropriate function sp ectrum F X M is not an S mo dule F

S

substitute and must b e used when dening co cylinder path and lo op S mo dules

The category M is co complete has all colimits its colimits b eing created in

S

That is the colimit in S of a diagram of S mo dules is an S mo dule that is S

the colimit of the given diagram in M It is also complete has all limits The

S

limit in S of a diagram of S mo dules is not quite an S mo dule but it takes values

of L sp ectra that lies intermediate b etween sp ectra and S in a category S L

mo dules Limits in S L are created in S and the forgetful functor M S L

S

has a right adjoint that creates the limits in M We shall explain this scaolding

S

in XXI I Ix For pragmatic purp oses what matters is that limits exist and have

the same weak homotopy typ es as if they were created in S

There is a free S mo dule functor F S M It is not quite free in the

S S

usual sense since its right adjoint U M S is not quite the evident forgetful

S S

functor This technicality reects the fact that the forgetful functor M S L

S

is a left rather than a right adjoint Again for pragmatic purp oses what matters

is that U is naturally weakly equivalent to the evident forgetful functor

S

We dene sphere S mo dules by

n n

S F S

S S

THE CATEGORY OF S MODULES

We dene the homotopy groups of an S mo dule to b e the homotopy groups of the

underlying sp ectrum and nd by the adjunction cited in the previous paragraph

that they can b e computed as

n

M hM S M

n S

S

From here we develop the theory of cell and CW S mo dules precisely as we

n

develop ed the theory of cell and CW sp ectra taking the spheres S as the domains

S

n

of attaching maps of cells CS We construct the derived category of S mo dules

S

by adjoining formal inverses to the weak equivalences and nd that denoted D

S

D is equivalent to the homotopy category of CW S mo dules The following

S

fundamental theorem then shows that no homotopical information is lost if we

replace the stable homotopy category hS by the derived category D

S

Theorem The following conclusions hold

i The free functor F S M carries CW sp ectra to CW S mo dules

S S

ii The forgetful functor M S carries S mo dules of the homotopy typ es

S

of CW S mo dules to sp ectra of the homotopy typ es of CW sp ectra

iii Every CW S mo dule M is homotopy equivalent as an S mo dule to F E

S

for some CW sp ectrum E

The free functor and forgetful functors establish an adjoint equivalence b etween

the stable homotopy category hS and the derived category D This equivalence

S

of categories preserves smash pro ducts and function ob jects Thus

D F E M hS E M

S S

D F E F E F hS E E

S S S S

F E E F E F E

S S S S

and

F F E E F F E F E

S S S S

We can describ e the equivalence in the language of closed mo del categories

in the sense of Quillen but we shall say little ab out this Both S and M are

S

mo del categories whose weak equivalences are the maps that induce isomorphisms

of homotopy groups The q cobrations or Quillen cobrations are the retracts

of inclusions of relative cell complexes that is cell sp ectra or cell S mo dules

The q brations in S are the Serre brations namely the maps that satisfy the

covering homotopy prop erty with resp ect to maps dened on the cone sp ectra

XXI I BRAVE NEW ALGEBRA

n

CS where q and n The q brations in M are the maps M N

S

q

of S mo dules whose induced maps U M U N are Serre brations of sp ectra

S S

EKMM A Elmendorf I Kriz M A Mandell and J P May Rings mo dules and algebras in

stable homotopy theory Preprint

0

A Elmendorf I Kriz M A Mandell and J P May Mo dern foundations for stable EKMM

homotopy theory In Handb o ok of Algebraic Top ology edited by IM James North Holland

pp

H Hastings Stabilizing tensor pro ducts Pro c Amer Math So c

L G Lewis Jr Is there a convenient category of sp ectra J Pure and Applied Algebra

ring spaces and J P May with contributions by F Quinn N Ray and J Tornehave E

1

E ring sp ectra Springer Lecture Notes in Mathematics Volume

1

Categories of Rmo dules

Let us think ab out S mo dules algebraically There is a p erhaps silly analogy

that I nd illuminating Algebraically it is of course a triviality that Ab elian

groups are essentially the same things as Zmo dules Nevertheless these notions

are conceptually dierent Thinking of brave new algebra in stable homotopy

theory as analogous to classical algebra I like to think of sp ectra as analogues of

Ab elian groups and S mo dules as analogues of Zmo dules While it required some

thought and work to gure out how to pass from sp ectra to S mo dules now that

we have done so we can follow our noses and mimic algebraic denitions word for

word in the category of S mo dules thinking of as analogous to and F as

S Z S

analogous to Hom

Z

We think of rings as Zalgebras and we dene an S algebra R by requiring a

unit S R and pro duct R R R such that the evident unit and asso

S

ciativity diagrams commute We say that R is a commutative S algebra if the

evident commutativity diagram also commutes We dene a left Rmo dule simi

larly requiring a map R M M such that the evident unit and asso ciativity

S

diagrams commute

N For a right Rmo dule M and left Rmo dule N we dene an S mo dule M

R

by the co equalizer diagram

Id

S

M N

M N M R N

R

S S S

Id

S

and are the given actions of R on M and N Similarly for left Rmo dules where

M and N we dene an S mo dule F M N by an appropriate equalizer diagram

R

We then have adjunctions exactly like those relating and Hom in algebra

R R

CATEGORIES OF RMODULES

If R is commutative then M N and F M N are Rmo dules the category

R R

M of Rmo dules is symmetric monoidal with unit R and we have the exp ected

R

adjunction relating and F We can go on to dene R R bimo dules and to

R R

derive a host of formal relations involving smash pro ducts and function mo dules

over varying rings all of which are exactly like their algebraic counterparts

For a left Rmo dule M and a based space X M X M X and

S

F X M are left Rmo dules If K is an S mo dule then M K is a left and

S S

F M K is a right Rmo dule We have theories of cob er and b er sequences

S

of Rmo dules exactly as for sp ectra We dene the free Rmo dule generated by a

sp ectrum X to b e

F X R F X

R S S

Again the right adjoint U of this functor is naturally weakly equivalent to the

R

forgetful functor from Rmo dules to sp ectra We dene sphere Rmo dules by

n n n

S F S R S

R S

R S

and nd that

n

M hM S M

n R

R

There is also a natural weak equivalence of Rmo dules F S R

R

We develop the theory of cell and CW Rmo dules exactly as we develop ed the

n

theory of cell and CW sp ectra using the spheres S as the domains of attaching

R

maps However the CW theory is only of interest when R is connective R

n

for n since otherwise the cellular approximation theorem fails We construct

the derived category D from the category M of Rmo dules by adjoining formal

R R

inverses to the weak equivalences and nd that D is equivalent to the homotopy

R

category of cell Rmo dules

Browns representability theorem holds in the category D a contravariant

R

setvalued functor k on D is representable in the form k M D M N if and

R R

only if k converts wedges to pro ducts and converts homotopy pushouts to weak

pullbacks However as recently observed by Neeman in an algebraic context

Adams variant for functors dened on nite cell Rmo dules only holds under a

countability hyp othesis on R

The category M is a mo del category The weak equivalences and q brations

R

are the maps of Rmo dules that are weak equivalences and q brations when re

garded as maps of S mo dules The q cobrations are the retracts of relative cell

Rmo dules It is also a tensored and cotensored top ological category That is its

XXI I BRAVE NEW ALGEBRA

Hom sets are based top ological spaces comp osition is continuous and we have

adjunction homeomorphisms

M M F X N T X M M N M M X N

R S R R

Recently Hovey Palmieri and Strickland have axiomatized the formal prop

erties that a category ought to have in order to b e called a stable homotopy

category The idea is to abstract those prop erties that are indep endent of any

underlying p ointset level foundations and see what can b e derived from that

starting p oint Our derived categories D provide a wealth of examples

R

M Hovey J H Palmieri and N P Strickland Axiomatic stable homotopy theory Preprint

A Neeman On a theorem of Brown and Adams Preprint

The algebraic theory of Rmo dules

The categories D are b oth to ols for the the study of classical algebraic top ol

R

ogy and interesting new sub jects of study in their own right In particular they

subsume much of classical algebra The Eilenb ergMacLane sp ectrum HR asso

ciated to a commutative discrete ring R is a commutative S algebra and the

Eilenb ergMacLane sp ectrum HM asso ciated to an Rmo dule is an HRmo dule

Moreover the derived category D is equivalent to the algebraic derived cate

HR

gory D of chain complexes over R and the equivalence converts derived smash

R

pro ducts and function mo dules in top ology to derived tensor pro ducts and Hom

functors in algebra In algebra the homotopy groups of derived tensor pro duct

and Hom functors compute Tor and Ext and we have natural isomorphisms

R

M N HM HN T or

n HR

n

for a right Rmo dule M and left Rmo dule N and

n

E xt M N F H M H N

n HR

R

for left Rmo dules M and N where HM is taken to b e a CW HRmo dule

Now return to the convention that R is an S algebra By the equivalence of hS

and D we see that homology and cohomology theories on sp ectra are subsumed

S

as homotopy groups of smash pro ducts and function mo dules over S Precisely

for a CW S mo dule M and an S mo dule N

M N M N

n S n

and

n

F M N N M

n S

THE ALGEBRAIC THEORY OF RMODULES

These facts suggest that we should think of the homotopy groups of smash

pro duct and function Rmo dules ambiguously as generalizations of b oth Tor and

Ext groups and homology and cohomology groups Thus for a right cell Rmo dule

M and a left Rmo dule N we dene

R R

T or M N M N M N

n R

n n

and for a left cell Rmo dule M and a left Rmo dule N we dene

n n

E xt M N F M N N M

n R

R R

We assume that M is a cell mo dule to ensure that these are welldened derived

category invariants

These functors enjoy many prop erties familiar from b oth the algebraic and top o

logical settings For example assuming that R is commutative we have a natural

n

asso ciative and unital system of pairings of R mo dules R R

n



L N L M Ext Ext M N Ext

R

R R R

Similarly setting D M F M R a formal argument in duality theory implies

R R

a natural isomorphism

n

R

Ext T or D M N M N

R

n R

for nite cell Rmo dules M and arbitrary Rmo dules N Thought of homologically

this isomorphism can b e interpreted as SpanierWhitehead duality for a nite cell

Rmo dule M and any Rmo dule N

R n

D M N M N

R

n R

There are sp ectral sequences for the computation of these invariants As usual

n

for a sp ectrum E we write E E E

n n

Theorem For right and left Rmo dules M and N there is a sp ectral se

quence

R R



E Tor M N Tor M N

pq pq pq

For left Rmo dules M and N there is a sp ectral sequence

pq pq pq

M N M N Ext E Ext



R R

If R is commutative these are sp ectral sequences of dierential R mo dules and

oneda pairings on the E the second admits pairings converging from the evident Y

terms to the natural pairings on the limit terms

XXI I BRAVE NEW ALGEBRA

Setting M F X in these two sp ectral sequences we obtain universal co e

R

cient sp ectral sequences

coefficient For an Rmo dule N and any sp ec Theorem Universal

trum X there are sp ectral sequences of the form

R



Tor R X N N X

and

R X N N X Ext



R

Replacing R and N by Eilenb ergMac Lane sp ectra HR and HN for a discrete

ring R and Rmo dule N we obtain the classical universal co ecient theorems

Replacing N by F Y and by F F Y R in the two universal co ecient sp ectral

R R R

sequences we obtain Kunneth sp ectral sequences

unneth For any sp ectra X and Y there are sp ectral se Theorem K

quences of the form

R



Tor R X R Y R X Y

and

R X R Y R X Y Ext



R

Under varying hyp otheses the Kunneth theorem in homology generalizes to an

Eilenb ergMo ore typ e sp ectral sequence Here is one example

Theorem Let E and R b e commutative S algebras and M and N b e R

mo dules Then there is a sp ectral sequence of dierential E Rmo dules of the

form

R E



E M E N E M N Tor

pq R

pq

The homotopical theory of Rmo dules

Thinking of the derived category of Rmo dules as an analog of the stable ho

motopy category we have the notion of an Rring sp ectrum which is just like the

classical notion of a ring sp ectrum in the stable homotopy category

THE HOMOTOPICAL THEORY OF RMODULES

Definition An Rring sp ectrum A is an Rmo dule A with unit R

A A A in D such that the following left and right unit A and pro duct

R R

diagram commutes in D

R

id id

o o

R A A R A A

R R R

L

L r

L

r

L

r

L

r

L r

L r

L

r

L

r

L

r

L r

xr

A

A is asso ciative or commutative if the appropriate diagram commutes in D If A

R

is asso ciative then an Amo dule sp ectrum M is an Rmo dule M with an action

A M M such that the evident unit and asso ciativity diagrams commute

R

in D

R

If A and B are Rring sp ectra then so is A B If A and B are Lemma

R

asso ciative or commutative then so is A B

R

When R S S ring sp ectra and their mo dule sp ectra are equivalent to classical

ring sp ectra and their mo dule sp ectra By neglect of structure an Rring sp ectrum

A is an S ring sp ectrum and thus a ring sp ectrum in the classical sense its unit is

the comp osite of the unit of R and the unit of A and its pro duct is the comp osite

of the pro duct of A and the canonical map

A A A A A A

S R

If A is commutative or asso ciative as an Rring sp ectrum then it is commutative

and asso ciative as an S ring sp ectrum and thus as a classical ring sp ectrum The

Rring sp ectra and their mo dule sp ectra play a role in the study of D analogous

R

to the role played by ring and mo dule sp ectra in classical stable homotopy theory

Moreover the new theory of Rring and mo dule sp ectra provides a p owerful con

structive to ol for the study of the classical notions The p oint is that in D we

R

have all of the internal structure such as cob er sequences that we have in the

stable homotopy category

This can make it easy to construct Rring sp ectra and mo dules in cases when

a direct pro of that they are merely classical ring sp ectra and mo dules is far more

dicult if it can b e done at all We assume that R is a commutative S algebra

and illustrate by indicating how to construct M I M and M Y for an Rmo dule

M where I is the ideal generated by a sequence fx g of elements of R and Y is

i

a countable multiplicatively closed set of elements of R We shall also state some

XXI I BRAVE NEW ALGEBRA

results ab out when these mo dules have Rring structures and when such structures

are commutative or asso ciative

We have isomorphisms

n

M hM S M

n R

R

n n

The susp ension M is equivalent to S M and for x R the comp osite

R n

R

map of Rmo dules

id x

n

S M

R M

M

R

R

R

n

is a mo dule theoretic version of the map x M M

Definition Dene M xM to b e the cob er of the map and let

M M xM b e the canonical map Inductively for a nite sequence

fx x g of elements of R dene

n

M x x M N x N where N M x x M

n n n

For a sequence X fx g dene M X M tel M x x M where the

i n

telescop e is taken with resp ect to the successive canonical maps

Clearly we have a long exact sequence

x



M M xM M M

q q q n q n

If x is regular for M xm implies m then induces an isomorphism

of R mo dules

M xM M x M

If fx x g is a regular sequence for M in the sense that x is regular for

n i

M x x M for i n then

i

M x x M M x x M

n n

and similarly for a p ossibly innite regular sequence X fx g The following

i

result implies that M X M is indep endent of the ordering of the elements of the

set X We write RX instead of RX R

Lemma For a set X of elements of R there is a natural weak equivalence

RX M M X M

R

In particular for a nite set X fx x g

n

Rx x Rx Rx

n R R n

THE HOMOTOPICAL THEORY OF RMODULES

If I denotes the ideal generated by X then it is reasonable to dene

M I M M X M

However this notation must b e used with caution since if we fail to restrict

attention to regular sequences X the homotopy typ e of M X M will dep end on

the set X and not just on the ideal it generates For example quite dierent

mo dules are obtained if we rep eat a generator x of I in our construction

i

To construct lo calizations let fy g b e any sequence of elements of Y that is

i

we may conal in the sense that every y Y divides some y If y R

i i n

i

n

i

represent y by an Rmap S S which we also denote by y Let q

i i

R R

and inductively q q n Then the Rmap

i i i

n

i

M M S y id S

R R i

R R

q

i 1

represents multiplication by y Smashing over R with S we obtain a sequence

i

R

of Rmaps

q

q

i1

i

S M S M

R R

R R

Definition Dene the lo calization of M at Y denoted M Y to b e the

M in D we may S telescop e of the sequence of maps Since M

R R

R

M of the telescop e as a natural map regard the inclusion of the initial stage S

R

R

M M Y

Since homotopy groups commute with lo calization we see immediately that

induces an isomorphism of R mo dules

M Y M Y

As in Lemma the lo calization of M is the smash pro duct of M with the

lo calization of R

Lemma For a multiplicatively closed set Y of elements of R there is a

natural equivalence

RY M M Y

R

Moreover RY is indep endent of the ordering of the elements of Y For sets X

and Y RX Y is equivalent to the comp osite lo calization RX Y

The b ehavior of lo calizations with resp ect to Rring structures is now immediate

XXI I BRAVE NEW ALGEBRA

Pr oposition Let Y b e a multiplicatively closed set of elements of R If

A is an Rring sp ectrum then so is AY If A is asso ciative or commutative

then so is AY

Proof It suces to observe that RY is an asso ciative and commutative

Rring sp ectrum with unit and pro duct the equivalence

RY RY RY Y RY

R

This do esnt work for quotients since RX X is not equivalent to RX How

ever we can analyze the problem by analyzing the deviation and by Lemma

we may as well work one element at a time We have a necessary condition for Rx

to b e an Rring sp ectrum that is familiar from classical stable homotopy theory

Lemma Let A b e an Rring sp ectrum If AxA admits a structure of

Rring sp ectrum such that A AxA is a map of Rring sp ectra then

x AxA AxA is null homotopic as a map of Rmo dules

Thus for example the Mo ore sp ectrum S is not an S ring sp ectrum since

the map S S is not null homotopic We have the following sucient

condition for when Rx do es have an Rring sp ectrum structure

where Rx and Rx Then Theorem Let x R

m m m

Rx admits a structure of Rring sp ectrum with unit R Rx Therefore

for every Rring sp ectrum A and every sequence X of elements of R such that

Rx and Rx if x X has degree m AX A admits

m m

a structure of Rring sp ectrum such that A AX A is a map of Rring

sp ectra

For an Rring sp ectrum A and an element x as in the theorem we give AxA

Rx A the pro duct induced by one of our constructed pro ducts on Rx and

R

the given pro duct on A We refer to any such pro duct as a canonical pro duct

on AxA We also have sucient conditions for when the canonical pro duct is

unique and when a canonical pro duct is commutative or asso ciative

Theorem Let x R where Rx and Rx

m m m

Let A b e an Rring sp ectrum and assume that AxA Then there

m

is a unique canonical pro duct on AxA If A is commutative then AxA is

commutative If A is asso ciative and AxA then AxA is asso ciative

m

This leads to the following conclusion

THE HOMOTOPICAL THEORY OF RMODULES

Theorem Assume that R if i is o dd Let X b e a sequence of non

i

zero divisors in R such that RX is concentrated in degrees congruent to zero

mo d Then RX has a unique canonical structure of Rring sp ectrum and it is

commutative and asso ciative

This is particularly valuable when applied with R MU The classical Thom

sp ectra arise in nature as E ring sp ectra and give rise to equivalent commutative

S algebras In fact insp ection of the presp ectrum level denition of Thom sp ectra

in terms of Grassmannians rst led to the theory of E ring sp ectra and therefore

of S algebras Of course

MU Zx jdeg x i

i i

Thus the results ab ove have the following immediate corollary

Theorem Let X b e a regular sequence in MU let I b e the ideal gen

erated by X and let Y b e any sequence in MU Then there is an MU ring

sp ectrum M U X Y and a natural map of MU ring sp ectra the unit map

MU M U X Y

such that

MU M U X Y

realizes the natural homomorphism of MU algebras

MU MU I Y

If MU I is concentrated in degrees congruent to zero mo d then there is a

unique canonical pro duct on M U X Y and this pro duct is commutative and

asso ciative

In comparison with earlier constructions of this sort based on the BaasSullivan

theory of manifolds with singularities or on Landweb ers exact functor theorem

where it applies we have obtained a simpler pro of of a substantially stronger

result since an MU ring sp ectrum is a much richer structure than just a ring

sp ectrum and commutativity and asso ciativity in the MU ring sp ectrum sense

are much more stringent conditions than mere commutativity and asso ciativity of the underlying ring sp ectrum

XXI I BRAVE NEW ALGEBRA

Categories of Ralgebras

In the previous section we considered Rring sp ectra which are homotopical

versions of Ralgebras We also have a p ointwise denition of Ralgebras that is

just like the denition of S algebras That is Ralgebras and commutative R

algebras A are dened via unit and pro duct maps R A and A A A

R

such that the appropriate diagrams commute in the symmetric monoidal category

M All of the standard formal prop erties of algebras in classical algebra carry

R

over directly to these brave new algebras For example a commutative Ralgebra

A is the same thing as a commutative S algebra together with a map of S algebras

R A the unit map and the smash pro duct A A of commutative Ralgebras

R

A and A is their copro duct in the category of commutative Ralgebras

Some of the most subtantive work in EKMM concerns the understanding of the

A of Ralgebras and commutative Ralgebras The crucial categories A and C

R R

p oint is to b e able to compute the homotopical b ehavior of formal constructions in

these categories Technically what is involved is the homotopical understanding

of the forgetful functors from A and CA to M Although not in itself enough

R R R

to answer these questions the context of enriched mo del categories is essential

to give a framework in which they can b e addressed We shall indicate some of

the main features here but this material is addressed to the relatively sophisti

cated reader who has some familiarity with enriched category and mo del category

theory It provides the essential technical underpinning for the applications to

Bouseld lo calization and top ological Ho chschild homology that are summarized

in the following two sections

Both A and CA are tensored and cotensored top ological categories In fact

R R

they are top ologically complete and co complete which means that they have not

only the usual limits and colimits but also indexed limits and colimits Limits are

created in the category of Rmo dules but colimits are less obvious constructions

In the absence of basep oints in their Hom sets these categories are enriched over

of unbased spaces The cotensors in b oth cases are the function the category U

S algebras F X A with the Ralgebra structure induced from the diagonal

S

on X and the pro duct on A The tensors are less familiar They are denoted

A X These are dierent constructions in the two cases but X and A

A A C

R R

we write A X when the context is understo o d We have adjunctions

A A F X B U X A A B A A X B

R S R R

and similarly in the commutative case Some idea of the structure and meaning of

CATEGORIES OF RALGEBRAS

tensors is given by the following result For Ralgebras A and B and a space X we

say that a map f A X B of Rmo dules is a p ointwise map of Ralgebras

if each comp osite f i A B is a map of Ralgebras where for x X

x

i A A X is the map induced by the evident inclusion fxg X

x

Proposition For Ralgebras A and spaces X there is a natural map of

Rmo dules

A X A X

such that a p ointwise map f A X B of Ralgebras uniquely determines

a map f A X B of Ralgebras such that f f The same statement

holds for commutative Ralgebras

More substantial results tell how to compute tensors when X is the geometric

realization of a simplicial set or simplicial space These results are at the heart of

the development and understanding of mo del category structures on the categories

A and CA In b oth categories the weak equivalences and q brations are the

R R

maps of Ralgebras that are weak equivalences or q brations of underlying R

mo dules It follows that the q cobrations are the maps of Ralgebras that satisfy

the left lifting prop erty with resp ect to the acyclic q brations The LLP is

recalled in VIx However the q cobrations admit a more explicit description

as retracts of relative cell Ralgebras or cell commutative Ralgebras Such

cell algebras are constructed by using free algebras generated by sphere sp ectra as

the domains of attaching maps and mimicking the construction of cell Rmo dules

using copro ducts pushouts and colimits in the relevant category of Ralgebras

The question of understanding the homotopical b ehavior of the forgetful functors

from A and CA to M now takes the form of understanding the homotopical

R R R

b ehavior of q cobrant algebras retracts of cell algebras with resp ect to these

forgetful functors However the formal prop erties of mo del categories have nothing

to say ab out this homotopical question

In what follows let R b e a xed q cobrant commutative Ralgebra Since R

is the initial ob ject of A and of CA it is q cobrant b oth as an Ralgebra

R R

and as a commutative Ralgebra However it is not q cobrant as an Rmo dule

Therefore the most that one could hop e of the underlying Rmo dule of a q cobrant

Ralgebra is the conclusion of the following result

Theorem If A is a q cobrant Ralgebra then A is a retract of a cell R

mo dule relative to R That is the unit R A is a q cobration of Rmo dules

XXI I BRAVE NEW ALGEBRA

The conclusion fails in the deep er commutive case The essential reason is that

the free commutative Ralgebra generated by an Rmo dule M is the wedge of the

j

and passage to orbits obscures the homotopy typ e of symmetric p owers M

j

the underlying Rmo dule The following technically imp ortant result at least gives

the homotopy typ e of the underlying sp ectrum

Theorem Let R b e a q cobrant commutative S algebra If M is a cell

Rmo dule then the pro jection

j j

E M M

j j

j

is a homotopy equivalence of sp ectra

The following theorem provides a workable substitute for Theorem It shows

that the derived smash pro duct is represented by the p ointset level smash pro duct

of Rmo dules one that in particular includes the underlying on a large class E

R

Rmo dules of all q cobrant Ralgebras and commutative Ralgebras

Theorem There is a collection E of Rmo dules of the underlying ho

R

motopy typ es of CW sp ectra that is closed under wedges pushouts colimits of

countable sequences of cobrations homotopy equivalences and nite smash pro d

ucts over R and that contains all q cobrant Rmo dules and the underlying R

mo dules of all q cobrant Ralgebras and all q cobrant commutative Ralgebras

N M are weak Moreover if M M are Rmo dules in E and

i i i n R

equivalences where the N are cell Rmo dules then

i

N N M M

R R n R R n R R n

is a weak equivalence Therefore the cell Rmo dule N N represents

R R n

M M in the derived category D

R R n R

W G Dwyer and J Spalinski Homotopy theories and mo del categories In A handb o ok of

algebraic top ology edited by I M James NorthHolland pp

G M Kelly Basic concepts of enriched category theory London Math So c Lecture Note

Series Vol Cambridge University Press

D G Quillen Homotopical algebra Springer Lecture Notes in Mathematics Volume

lo calizations of Rmo dules and algebras Bouseld

Bouseld lo calization is a basic to ol in the study of classical stable homotopy

theory and the construction generalizes readily to the context of brave new alge

bra In fact using our mo del category structures this context leads to a smo other

BOUSFIELD LOCALIZATIONS OF RMODULES AND ALGEBRAS

treatment than can b e found in the classical literature More imp ortant as we shall

sketch any brave new algebraic structure is preserved by Bouseld lo calization

Let R b e an S algebra and E b e a cell Rmo dule A map f M N of

Rmo dules is said to b e an E equivalence if

id f E M E N

R R R

is a weak equivalence An Rmo dule W is said to b e E acyclic if E W and

R

a map f is an E equivalence if and only if its cob er is E acyclic We say that

an Rmo dule L is E lo cal if f D N L D M L is an isomorphism for

R R

any E equivalence f or equivalently if D W L for any E acyclic Rmo dule

R

W Since this is a derived category criterion it suces to test it when W is a

cell Rmo dule A lo calization of M at E is a map M M such that is

E

an E equivalence and M is E lo cal The formal prop erties of such lo calizations

E

discussed by Bouseld carry over verbatim to the present context There is a mo del

structure on M that implies the existence of E lo calizations of Rmo dules

R

Theorem The category M admits a new structure as a top ological mo del

R

category in which the weak equivalences are the E equivalences and the cobra

tions are the q cobrations in the standard mo del structure that is the retracts

of the inclusions of relative cell Rmo dules

We call the brations in the new mo del structure E brations They are deter

mined formally as maps that satisfy the right lifting prop erty with resp ect to the

E acyclic q cobrations namely the q cobrations that are E equivalences The

RLP is recalled in VIx One can characterize the E brations more explicitly

but the following result gives all the relevant information Say that an Rmo dule

L is E brant if the trivial map L is an E bration

Theorem An Rmo dule is E brant if and only if it is E lo cal Any R

mo dule M admits a lo calization M M at E

E

In fact one of the standard prop erties of a mo del category shows that we can

factor the trivial map M as the comp osite of an E acyclic q cobration

M M and an E bration M so that the rst statement implies

E E

the second The following complement shows that the lo calization of an Rmo dule

at a sp ectrum not necessarily an Rmo dule can b e constructed as a map of

Rmo dules

XXI I BRAVE NEW ALGEBRA

Proposition Let K b e a CWsp ectrum and let E b e the Rmo dule F K

R

Regarded as a map of sp ectra a lo calization M M of an Rmo dule M at

E

E is a lo calization of M at K

The result generalizes to show that for an Ralgebra A the lo calization of an

Amo dule at an Rmo dule E can b e constructed as a map of Amo dules

Proposition Let A b e a q cobrant Ralgebra let E b e a cell Rmo dule

and let F b e the Amo dule A E Regarded as a map of Rmo dules a lo calization

R

M M of an Amo dule M at F is a lo calization of M at E

F

Restrict R to b e a q cobrant commutative S algebra in the rest of this section

We then have the following fundamental theorem ab out lo calizations of Ralgebras

Theorem For a cell Ralgebra A the lo calization A A can b e

E

constructed as the inclusion of a sub complex in a cell Ralgebra A Moreover if

E

f A B is a map of Ralgebras into an E lo cal Ralgebra B then f lifts to a

map of Ralgebras f A B such that f f if f is an E equivalence then

E

f is a weak equivalence The same statements hold for commutative Ralgebras

The idea is to replace the category M by either the category A or the cat

R R

egory CA in the development just sketched That is we attempt to construct

R

new mo del category structures on A and CA in such a fashion that a factor

R R

ization of the trivial map A as the comp osite of an E acyclic q cobration

and a q bration in the appropriate category of Ralgebras gives a lo calization of

the underlying Rmo dule of A The argument do esnt quite work to give a mo del

structure b ecause the mo dule level argument uses vitally that a pushout of an

E acyclic q cobration of Rmo dules is an E equivalence There is no reason to

b elieve that this holds for q cobrations of Ralgebras However we can use Theo

rems to prove that it do es hold for pushouts of inclusions of sub complexes

in cell Ralgebras along maps to cell Ralgebras This gives enough information

to prove the theorem

The theorem implies in particular that we can construct the lo calization of R at

E as the unit R R of a q cobrant commutative Ralgebra This leads to a

E

new p ersp ective on lo calizations in classical stable homotopy theory To see this

to the stable homotopy category D E we compare the derived category D

R R

E

asso ciated to the mo del structure on M that is determined by E Thus D E

R R

is obtained from D by inverting the E equivalences and is equivalent to the full R

BOUSFIELD LOCALIZATIONS OF RMODULES AND ALGEBRAS

sub category of D whose ob jects are the E lo cal Rmo dules Observe that for a

R

cell Rmo dule M we have the canonical E equivalence

R M R M id M

R E R

The following observation is the same as in the classical case

Lemma If M is a nite cell Rmo dule then R M is E lo cal and there

E R

fore is the lo calization of M at E

We say that lo calization at E is smashing if for all cell Rmo dules M R M

E R

is E lo cal and therefore is the lo calization of M at E The following observation

is due to Wolb ert

t If lo calization at E is smashing then the cat Proposition Wolber

egories D E and D are equivalent

R R

E

These categories are closely related even when lo calization at E is not smash

ing as the following elab oration of Wolb erts result shows Rememb er that R is

assumed to b e commutative

Theorem The following three categories are equivalent

i The category D E of E lo cal Rmo dules

R

whose ob jects are the R mo dules E of D ii The full sub category D

E R R

E E

that are E lo cal as Rmo dules

iii The category D F of F lo cal R mo dules where F R E

R E E R

E

This implies that the question of whether or not lo calization at E is smashing

is a question ab out the category of R mo dules and it leads to the following

E

factorization of the lo calization functor In the case R S this shows that the

commutative S algebras S and their categories of mo dules are intrinsic to the

E

classical theory of Bouseld lo calization

Theorem Let F R E The E lo calization functor

E R

D D E

E R R

is equivalent to the comp osite of the extension of scalars functor

R D D

E R R R

E

and the F lo calization functor

D D F

F R R

E E

XXI I BRAVE NEW ALGEBRA

Corollary Lo calization at E is smashing if and only if all R mo dules

E

are E lo cal as Rmo dules so that

D D F D E

R R R

E E

We illustrate the constructive p ower of Theorem by showing that the alge

braic lo calizations of R considered in Section actually take R to commutative

Ralgebras on the p oint set level and not just on the homotopical level as given

by Prop osition Thus let Y b e a countable multiplicatively closed set of ele

ments of R Using Lemma we see that lo calization of Rmo dules at RY

is smashing and is given by the canonical maps

R M RY M id M

R R R

Theorem The lo calization R RY can b e constructed as the unit

of a cell Ralgebra

By multiplicative innite lo op space theory and our mo del category structure on

the category of S algebras the sp ectra k o and k u that represent real and complex

connective K theory can b e taken to b e q cobrant commutative S algebras The

sp ectra that represent p erio dic K theory can b e reconstructed up to homotopy by

k o or k u That is inverting the Bott element

U O

and KU k u KO k o

U O

We are entitled to the following result as a sp ecial case of the previous one

Theorem The sp ectra KO and KU can b e constructed as commutative

k o and k ualgebras

In particular KO and KU are commutative S algebras but it seems very hard

to prove this directly Wolb ert has studied the algebraic structure of the derived

categories of mo dules over the connective and p erio dic versions of the real and

complex K theory S algebras

For nite groups G Theorem applies with the same pro of Remark

to construct the p erio dic sp ectra KO and KU of equivariant K theory as com

G G

mutative k o and k u algebras As we shall discuss in Chapter XXIV this leads

G G

to an elegant pro of of the AtiyahSegal completion theorem in equivariant K

cohomology and of its analogue for equivariant K homology

TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY

A K Bouseld The lo calization of sp ectra with resp ect to homology Top ology

J P May Multiplicative innite lo op space theory J Pure and Applied Algebra

J Wolb ert Toward an algebraic classication of mo dule sp ectra Preprint University of

Chicago Part of PhD thesis in preparation

Top ological Ho chschild homology and cohomology

As another application of brave new algebra we describ e the top ological Ho ch

schild homology of Ralgebras with co ecients in bimo dules We assume familiar

ity with the classical Ho chschild homology of algebras as in Cartan and Eilenb erg

for example The study of this topic and of top ological cyclic homology which

takes top ological Ho chschild homology as its starting p oint and involves equivari

ant considerations is under active investigation by many p eople We shall just

give a brief intro duction

We assume given a q cobrant commutative S algebra R and a q cobrant R

algebra A If A is commutative we require it to b e q cobrant as a commutative

Ralgebra We dene the enveloping Ralgebra of A by

e op

A A A

R

op

where A is dened by twisting the pro duct on A as in algebra If A is commu

e E

A A and the pro duct A A is a map of Ralgebras We tative then A

R

also assume given an A Abimo dule M it can b e viewed as either a left or a

e

right A mo dule

Definition Working in derived categories dene top ological Ho chschild

homology and cohomology with values in D by

R

R

e e

A M THH A M M A and THH A M F

A R A

e

If A is commutative then these functors take values in the derived category D

A

On passage to homotopy groups dene

e

A R

A M A M Ext A M Tor M A and THH THH

e

A R

When M A we delete it from the notations

Since we are working in derived categories we are implicitly taking M to b e

e R

a cell A mo dule in the denition of THH A M and approximating A by a

e

weakly equivalent cell A mo dule in the denition of THH A M

R

R

Proposition If A is a commutative Ralgebra then THH A is isomor

e

e

to a commutative A algebra phic in D A

XXI I BRAVE NEW ALGEBRA

R

The mo dule structures on THH A M have the following implication

Proposition If either R or A is the Eilenb ergMac Lane sp ectrum of a

R

commutative ring then THH A M is a pro duct of Eilenb ergMac Lane sp ectra

We have sp ectral sequences that relate algebraic and top ological Ho chschild

homology For a commutative graded ring R a graded R algebra A that is at

as an R mo dule and a graded A A bimo dule M we dene

e

pq pq

A

R





e A M A M Ext A M Tor M A and HH HH





R pq pq

A

where p is the homological degree and q is the internal degree This algebraic

denition would not b e correct in the absence of the atness hyp othesis When

R



A M A we delete it from the notation If A is commutative then HH

op op

is a graded A algebra Observe that A A

In view of Theorem the sp ectral sequence of Theorem sp ecializes to

give the following sp ectral sequences relating algebraic and top ological Ho chschild

homology

Theorem There are sp ectral sequences of the form

R e op



Tor E A A A

pq

pq pq

e

A R



Tor E A M M A THH

pq pq pq

and

pq pq pq

A M A M THH E Ext



e

R

A

If A is a at R mo dule so that the rst sp ectral sequence collapses then the

initial terms of the second and third sp ectral sequences are resp ectively

R



HH A M A M and HH



R

This is of negligible use in the absolute case R S where the atness hyp oth

esis on A is unrealistic However in the relative case it implies that algebraic

Ho chschild homology and cohomology are sp ecial cases of top ological Ho chschild

homology and cohomology

Theorem Let R b e a discrete ungraded commutative ring let A b e an

Rat Ralgebra and let M b e an A Abimo dule Then

HR R

HA HM THH A M HH

and

THH HA HM HH A M

HR R

TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY

HR R

THH HA as Aalgebras If A is commutative then HH A

We concentrate on homology henceforward In the absolute case R S it is

S

natural to approach THH A M by rst determining the ordinary homology of

S

THH A M using the case E H F of the following sp ectral sequence and

p

then using the Adams sp ectral sequence A sp ectral sequence like the following

one was rst obtained by Bokstedt Under atness hyp otheses there are variants

in which E need only b e a commutative ring sp ectrum eg Theorem b elow

Theorem Let E b e a commutative S algebra There are sp ectral sequence

of dierential E Rmo dules of the forms

E R op e



E Tor E A E A E A

pq

pq pq

and

e

E A

R



E M E A E THH A M Tor E

pq

pq pq

There is an alternative description of top ological Ho chschild homology in terms

of the brave new algebra version of the standard complex for the computation of

p

Ho chshild homology Write A for the pfold p ower of A and let

R

A A A and R A

R

b e the pro duct and unit of A Let

A M M and M A M

R r R

b e the left and right action of A on M We have cyclic p ermutation isomorphisms

p p

M A A A M A

R R R R

The top ological analogue of passage from a simplicial k mo dule to a chain com

plex of k mo dules is passage from a simplicial sp ectrum E to its sp ectrum level

geometric realization jE j this construction is studied in EKMM

R

Definition Dene a simplicial Rmo dule thh A M as follows Its R

p

mo dule of psimplices is M A Its face and degeneracy op erators are

R

p

id if i

r

i pi

d

id id id if i p

i

p

id if i p

i pi

and s id id id Dene

i

R R

thh A M jthh A M j

XXI I BRAVE NEW ALGEBRA

R R

When M A we delete it from the notation writing thh A and jthh A j

R

Proposition Let A b e a commutative Ralgebra Then thh A is a

R R

commutative Aalgebra and thh A M is a thh Amo dule

As in algebra the starting p oint for a comparison of denitions is the relative

R

twosided bar construction B M A N It is dened for a commutative S algebra

R an Ralgebra A and right and left Amo dules M and N Its Rmo dule of p

p

simplices is M A N There is a natural map

R

R

B A A N N

of Amo dules that is a homotopy equivalence of Rmo dules More generally there

is a natural map of Rmo dules

R

B M A N M N

A

that is a weak equivalence of Rmo dules when M is a cell Amo dule The relevance

of the bar construction to thh is shown by the following observation which is the

same as in algebra We write

R R

B A B A A A

e R R

A A B A is an A Abimo dule on the simplicial level B

Proposition For A Abimo dules M there is a natural isomorphism

R R

e

B A M thh A M

A

e

Therefore for cell A mo dules M the natural map

id

R R R

e e

A THH A M M B A M thh A M

A A

e

is a weak equivalences of Rmo dules or of A mo dules if A is commutative

e

While we assumed that M is a cell A mo dule in our derived category level

denition of THH we are mainly interested in the case M A of our p ointset

e e

level construction thh and A is not of the A homotopy typ e of a cell A mo dule

except in trivial cases However Theorem leads to the following result

e

Theorem Let M A b e a weak equivalence of A mo dules where

e

M is a cell A mo dule Then the map

R R R R

thh id thh A M thh A A thh A

e

is a weak equivalence of Rmo dules or of A mo dules if A is commutative There

R R

fore THH A M is weakly equivalent to thh A

TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY

R R

thh A Corollary In the derived category D THH A

R

Use of the standard simplicial ltration of the standard complex gives us the

promised variant of the sp ectral sequence of Theorem For simplicity we

restrict attention to the absolute case R S

Theorem Let E b e a commutative ring sp ectrum A b e an S algebra

e

and M b e a cell A mo dule If E A is E at there is a sp ectral sequence of the

form

S E



E A E M E thh A M E HH

pq

pq pq

If A is commutative and M A this is a sp ectral sequence of dierential E A

algebras the pro duct on E b eing the standard pro duct on Ho chschild homology

McClure Schwanzl and Vogt observed that when A is commutative as we

assume in the rest of the section there is an attractive conceptual reinterpreta

R

tion of the denition of thh A Recall that the category CA of commutative

R

Ralgebras is tensored over the category of unbased spaces By writing out the

whose realization is the circle and comparing faces and standard simplicial set S

degeneracies it is easy to check that there is an identication of simplicial com

mutative Ralgebras

R

thh A A S

Passing to geometric realization and identifying S with the unit complex numb ers

we obtain the following consequence

Schw Vogt For commutative Ralgebras anzl Theorem McClure

A there is a natural isomorphism of commutative Ralgebras

R

A S thh A

R

The pro duct of thh A is induced by the co diagonal S S S The unit

R

A thh A is induced by the inclusion fg S

R

The adjunction that denes tensors implies that the functor thh A pre

serves colimits in A something that is not at all obvious from the original def

inition The theorem and the adjunction imply much further structure on

R

thh A

XXI I BRAVE NEW ALGEBRA

Corollary The pinch map S S S and trivial map S

induce a homotopy coasso ciative and counital copro duct and counit

R R R R

thh A thh A thh A and thh A A

A

R

that make thh A a homotopical Hopf Aalgebra

The pro duct on S gives rise to a map

A S S A S A S S

it ir t r r

e Corollary For an integer r dene S S by e

r

The induce p ower op erations

r R R

thh A thh A

These are maps of Ralgebras such that

r s r s

id

and the following diagrams commute

R R

thh A S thh A

r s

s r +

R R

thh A S thh A

Consider naive S sp ectra and let S act trivially on R and A Via the adjunc

R

tion the map gives rise to an action of S on thh A

R

Corollary thh A is a naive commutative S Ralgebra If B is a

naive commutative S Ralgebra and f A B is a map of commutative R

R

algebras then there is a unique map f thh A B of naive commutative

S Ralgebras such that f f

Finally the description of tensors in Prop osition leads to the following result

Corollary There is a natural S equivariant map of Rmo dules

R

thh A A S

such that if B is a commutative Ralgebra and f A S B is a map of

Rmo dules that is a p ointwise map of Ralgebras then f uniquely determines a

R

map of Ralgebras f thh A B such that f f

TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY

M Bokstedt Top ological Ho chschild homology Preprint

M Bokstedt The top ological Ho chschild homology of Z and Zp Preprint

H Cartan and S Eilenb erg Homological Algebra Princeton Univ Press

1

H H R R S for E ring sp ectra J Pure and J E McClure R Schwanzl and R Vogt T

1

Applied Algebra To app ear

XXI I BRAVE NEW ALGEBRA

CHAPTER XXI I I

ariant foundations Brave new equiv

by A D Elmendorf L G Lewis Jr and J P May

Twisted halfsmash pro ducts

We here give a quick sketch of the basic constructions b ehind the work of the

last chapter Although the basic source EKMM is written nonequivariantly it

applies verbatim to the equivariant context in which we shall work in this chapter

We shall take the opp ortunity to describ e some unpublished p ersp ectives on the

role of equivariance in the new theory

The essential starting p oint is the twisted halfsmash pro duct construction from

LMS Although we have come this far without mentioning this construction it is

in fact central to equivariant stable homotopy theory Before describing it we shall

motivate it in terms of the main theme of this chapter which is the construction of

the category of Lsp ectra As we shall see this is the main step in the construction

of the category of S mo dules

Fix a compact Lie group G and a Guniverse U and consider the category GS U

j

of Gsp ectra indexed on U Write U for the direct sum of j copies of U Recall

that we have an external smash pro duct GS U GS U GS U and an

internal smash pro duct f GS U GS U for each Glinear isometry

f U U The external smash pro duct is suitably asso ciative commutative

and unital on the p oint set level hence we may iterate and form an external smash

j j

pro duct GS U GS U for each j the rst external smash p ower

j

b eing the identity functor For each Glinear isometry f U U we have an

j

asso ciated internal smash pro duct f GS U GS U We allow the case

j here GS fg GT the only linear isometry fg U is the inclusion i

and i is the susp ension Gsp ectrum functor At least if we restrict attention to

XXI I I BRAVE NEW EQUIVARIANT FOUNDATIONS

tame Gsp ectra the functors induced by varying f are all equivalent see Theorem

j

b elow Thus varying Glinear isometries f U U parametrize equivalent

internal smash pro ducts

There is a language for the discussion of such parametrized pro ducts in various

mathematical contexts namely the language of op erads that was intro duced

for the study of iterated lo op space theory in Let L j denote the space

j j

U U of linear isometries U U Here we allow all linear isometries not I

just the Glinear ones and G acts on L j by conjugation Thus the xed p oint

G j

space L j is the space of Glinear isometries U U The symmetric group

acts freely from the right on L j and the actions of G and commute The

j j

equivariant homotopy typ e of L j dep ends on U If U is complete then for

G L j is empty unless e and contractible otherwise That

j j

is L j is a universal G bundle We have maps

j

L j j L k L j L j

k k

dened by

g f f g f f

k k

These data are interrelated in a manner co died in the denition of an op erad

and L is called the linear isometries Gop erad of the universe U When U is

complete L is an E Gop erad

There is a twisted halfsmash pro duct

E E L j n

j

into which we can map each of the j fold internal smash pro ducts f E E

j

Moreover if we restrict attention to tame Gsp ectra then each of these maps into

the twisted halfsmash pro duct is an equivalence The twisted halfsmash

pro ducts L n E and L n E E are the starting p oints for the construction

of the category of L sp ectra and the denition of its smash pro duct We shall

return to this p oint in the next section after saying a little more ab out twisted

smash pro ducts of Gsp ectra

Supp ose given Guniverses U and U and let I U U b e the Gspace of linear

isometries U U with G acting by conjugation Let A b e an unbased G

space together with a given Gmap A I U U We then have a twisted

halfsmash pro duct functor

n GS U GS U

When A has the homotopy typ e of a GCW complex and E GS U is tame

dierent choices of give homotopy equivalent Gsp ectra n E For this reason

TWISTED HALFSMASH PRODUCTS

and b ecause we often have a canonical choice of in mind we usually abuse

notation by writing A n E instead of n E Thus we think of A as a space over

I U U

When A is a p oint is a choice of a Glinear isometry f U U In

this case the twisted halfsmash functor is just the change of universe functor

f GS U GS U see XI I Intuitively one may think of n E

as obtained by suitably top ologizing and giving a Gaction to the union of the

E as a runs through A Another intuition is that nonequivariant sp ectra a

the twisted halfsmash pro duct is a generalization to sp ectra of the untwisted

functor A X on based Gspaces X This intuition is made precise by the

following untwisting formula that relates twisted halfsmash pro ducts and shift

desusp ensions

V Proposition For a Gspace A over I U U and an isomorphism V

of indexing Gspaces where V U and V U there is an isomorphism of

Gsp ectra

X A X A n

0

V V

that is natural in Gspaces A over I U U and based Gspaces X

The twistedhalf smash pro duct functor enjoys essentially the same formal prop

erties as the space level functor A X For example we have the following

prop erties whose space level analogues are trivial to verify

Proposition The following statements hold

i There is a canonical isomorphism fid g n E E

U

ii Let A I U U and B I U U b e given and give B A the

comp osite structure map

I U U I U U I U U

B A

Then there is a canonical isomorphism

B n A n E B A n E

iii Let A I U U and B I U U b e given and give A B the

comp osite structure map

I U U I U U I U U U U

A B

XXI I I BRAVE NEW EQUIVARIANT FOUNDATIONS

Let E and E b e Gsp ectra indexed on U and U resp ectively Then there

is a canonical isomorphism

A n E B n E A B n E E

iv For A I U U E GS U and a based Gspace X there is a canonical

isomorphism

A n E X A n E X

The functor A n has a right adjoint twisted function sp ectrum functor

F A GS U GS U

which is the sp ectrum level analog of the function Gspace F A X Thus

GS U E F A E GS U A n E E

The functor A n E is homotopypreserving in E and it therefore preserves

homotopy equivalences in the variable E However it only preserves homotopies

over I U U in A Nevertheless it very often preserves homotopy equivalences

in the variable A The following central technical result is an easy consequence

of Prop osition and XI I It explains why all j fold internal smash pro ducts

are equivalent to the twisted halfsmash pro duct

Theorem Let E GS U b e tame and let A b e a Gspace over I U U If

A A is a homotopy equivalence of Gspaces then n id A n E A n E

is a homotopy equivalence of Gsp ectra

Since A n E is a GCW sp ectrum if A is a GCW complex and E is a GCW

sp ectrum this has the following consequence

Corollary Let E GS U have the homotopy typ e of a GCW sp ectrum

and let A b e a Gspace over I U U that has the homotopy typ e of a GCW

complex Then A n E has the homotopy typ e of a GCW sp ectrum

LMS Chapter VI

J P May The Geometry of Iterated Lo op Spaces Springer Lecture Notes in Mathematics

Volume

THE CATEGORY OF LSPECTRA

The category of Lsp ectra

Return to the twisted halfsmash pro duct of We think of it as a canonical

j fold internal smash pro duct However if we are to take this p oint of view

seriously we must take note of the dierence b etween E and its fold smash

pro duct L n E The space L is a monoid under comp osition and the

formal prop erties of twisted halfsmash pro ducts imply a natural isomorphism

L L n E L n L n E

where on the right L L is regarded as a Gspace over L via the

comp osition pro duct This pro duct induces a map

L L n E L n E

and the inclusion fg L induces a map E L n E The functor

L given by LE L n E is a monad under the pro duct and unit We

therefore have the notion of a Gsp ectrum E with an action LE E of L

the evident asso ciativity and unit diagrams are required to commute

Definition An Lsp ectrum is a Gsp ectrum M together with an action of

the monad L Let GS L denote the category of Lsp ectra

The formal prop erties of GS L are virtually the same as those of GS since

L is a contractible Gspace so are the homotopical prop erties For tame G

sp ectra E we have a natural equivalence E id E LE For Lsp ectra M that

are tame as Gsp ectra the action LM M is a weak equivalence Taking

n

the L S as sphere Lmo dules we obtain a theory of GCW L sp ectra exactly like

the theory of GCW sp ectra The functor L preserves GCW sp ectra We let

hGS L b e the category that is obtained from the homotopy category hGS L

by formally inverting the weak equivalences and nd that it is equivalent to the

homotopy category of GCW Lsp ectra The functor L GS GS L and

the forgetful functor GS L GS induce an adjoint equivalence b etween the

stable homotopy category h GS and the category hGS L

L X and the obvious Via the untwisting isomorphism L n X

pro jection L S we obtain a natural action of L on susp ension sp ectra

However even when X is a GCW complex X is not of the homotopy typ e of

a GCW L sp ectrum and it is the functor L and not the functor that is

left adjoint to the zeroth space functor GS L T

The reason for intro ducing the category of Lsp ectra is that it has a wellb ehaved

op eradic smash pro duct which we dene next Via instances of the structural

maps of the op erad L we have b oth a left action of the monoid L and a

XXI I I BRAVE NEW EQUIVARIANT FOUNDATIONS

right action of the monoid L L on L These actions commute with

each other If M and N are L sp ectra then L L acts from the left on

the external smash pro duct M N via the map

L n M L n N L L n M N

M N

To form the twisted half smash pro duct on the left we think of L L as

mapping to I U U via direct sum of linear isometries The smash pro duct over

L of M and N is simply the balanced pro duct of the two L L actions

Definition Let M and N b e Lsp ectra Dene the op eradic smash pro d

uct M N to b e the co equalizer displayed in the diagram

L

nid

L L L n M N L n M N

M N

L

id n

Here we have implicitly used the isomorphism

L n L L n M N L L L n M N

given by Prop osition ii The left action of L on L induces a left action

of L on M N that gives it a structure of Lsp ectrum

L

We may mimic tensor pro duct notation and write

M N M N L n

L

L L

This smash pro duct is commutative and a sp ecial prop erty of the linear isome

tries op erad rst noticed by Hopkins implies that it is also asso ciative There

is a function Lsp ectrum functor F to go with it is constructed from the

L L

external and twisted function sp ectra functors and we have the adjunction

GS LM F M M GS LM M M

L L

The smash pro duct is not unital However there is a natural map

L

S M M

L

of Lsp ectra that is always a weak equivalence of sp ectra It is not usually an

isomorphism but another sp ecial prop erty of the linear isometries op erad implies

that it is an isomorphism if M S or if M S N for any L sp ectrum N Thus

L

any Lsp ectrum is weakly equivalent to one whose unit map is an isomorphism

This makes sense of the following denition in which we understand S to mean

the sphere Gsp ectrum indexed on our xed chosen Guniverse U

AND E RING SPECTRA AND S ALGEBRAS A

1 1

Definition An S mo dule is an Lsp ectrum M such that S M

L

M is an isomorphism The category GM of S mo dules is the full sub category of

S

GS L whose ob jects are the S mo dules For S mo dules M and M dene

M M M M and F M M S F M M

S L S L L

Although easy to prove one surprising formal feature of the theory is that the

functor S GS L GM is right and not left adjoint to the forgetful

L S

functor it is left adjoint to the functor F S This categorical situation dictates

L

our denition of function S mo dules It also dictates that we construct limits of

S mo dules by constructing limits of their underlying Lsp ectra and then applying

the functor S as indicated in XXI Ix The free S mo dule functor F

L S

GS GM is dened by

S

F E S LE

S L

It is left adjoint to the functor F S GM GS and this is the functor

L S

that we denoted by U in XXI Ix From this p oint the prop erties of the category

S

of S mo dules that we describ ed in XXI Ix are inherited directly from the go o d

prop erties of the category of Lsp ectra

A and E ring sp ectra and S algebras

We dened S algebras and their mo dules in terms of structure maps that make

the evident diagrams commute in the symmetric monoidal category of S mo dules

There are older notions of A and E ring sp ectra and their mo dules that May

Quinn and Ray intro duced nonequivariantly in the equivariant generaliza

tion was given in LMS Working equivariantly an A ring sp ectrum is a sp ectrum

R together with an action by the linear isometries Gop erad L Such an action

is given by Gmaps

j

L j n R R j

j

such that appropriate asso ciativity and unity diagrams commute If the are

j

equivariant then R is said to b e an E ring sp ectrum Similarly a left mo dule

j

M over an A ring sp ectrum R is dened in terms of maps

j

L j n R M M j

j

in the E case we require these maps to b e equivariant It turns out that

j

the higher and are determined by the and for j That is we have

j j j j

the following result which might instead b e taken as a denition

XXI I I BRAVE NEW EQUIVARIANT FOUNDATIONS

Theorem An A ring sp ectrum is an L sp ectrum R with a unit map

S R and a pro duct R R R such that the following diagrams

L

commute

id id

o

R S S R R R

L L L

M

M

q

M

q

M

q

M q

M

q

M

q

M q

M

q

M

q

M q

M

q

xq

R

and

id

R R R R R

L L L

id

R

R R

L

R is an E ring sp ectrum if the following diagram also commutes

R R R R

L L

I

I

u

u I

I

u

u I

I

u

u I

u I

u I

zz u

R

A mo dule over an A or E ring sp ectrum R is an Lsp ectrum M with a map

R M M such that the following diagrams commute

L

id id

R M S M R R M R M

and

L L L L L

N

N

N

N

N

N

N

id

N

N

N

N

N

R M

M M

L

This leads to the following description of S algebras

Corollary An S algebra or commutative S algebra is an A or E ring

sp ectrum that is also an S mo dule A mo dule over an S algebra or commutative

S algebra R is a mo dule over the underlying A or E ring sp ectrum that is also

an S mo dule

In particular we have a functorial way to replace A and E ring sp ectra and

their mo dules by weakly equivalent S algebras and commutative S algebras and

their mo dules

Corollary For an A ring sp ectrum R S R is an S algebra and

L

S R R is a weak equivalence of A ring sp ectra and similarly in

L

the E case If M is an Rmo dule then S M is an S Rmo dule and

L L

ALTERNATIVE PERSPECTIVES ON EQUIVARIANCE

S M M is a weak equivalence of Rmo dules and of mo dules over

L

S R regarded as an A ring sp ectrum

L

Thus the earlier denitions are essentially equivalent to the new ones and earlier

work gives a plenitude of examples Thom Gsp ectra o ccur in nature as E ring

Gsp ectra For nite groups G multiplicative innite lo op space theory works

as it do es nonequivariantly however the details have yet to b e fully worked out

and written up that is planned for a later work This theory gives that the

Eilenb ergMac Lane Gsp ectra of Green functors the Gsp ectra of connective real

and complex K theory and the Gsp ectra of equivariant algebraic K theory are

E ring sp ectra As observed in XXI I it follows that the Gsp ectra of p e

rio dic real and complex K theory are also E ring Gsp ectra Nonequivariantly

many more examples are known due to recent work mostly unpublished of such

p eople as Hopkins Miller and Kriz

J P May with contributions by F Quinn N Ray and J Tornehave E ring spaces and

1

E ring sp ectra Springer Lecture Notes in Mathematics Volume

1

J P May Multiplicative innite lo op space theory J Pure and Applied Algebra

Alternative p ersp ectives on equivariance

We have develop ed the theory of Lsp ectra and S mo dules starting from a xed

given Guniverse U However there are alternative p ersp ectives on the role of the

universe and of equivariance that shed considerable light on the theory Much of

this material do es not app ear in the literature and we give pro ofs in Section

after explaining the ideas here Let S denote the sphere Gsp ectrum indexed on a

U

Guniverse U The essential p oint is that while the categories GS U of Gsp ectra

of S mo dules do not all indexed on U vary as U varies the categories GM

U S

U

such categories are actually isomorphic These isomorphisms preserve homotopies

and thus pass to ordinary homotopy categories However they do not preserve

weak equivalences and therefore do not pass to derived categories which do vary

with U This observation rst app eared in a pap er of Elmendorf and May but we

shall b egin with a dierent explanation than the one we gave there

We shall explain matters by describing the categories of Gsp ectra and of LG

sp ectra indexed on varying universes U in terms of algebras over monads dened

on the ground category S S R of nonequivariant sp ectra indexed on R

Abbreviate notation by writing L for the monoid L I R R Any G

universe U is isomorphic to R with an action by G through linear isometries The

action may b e written in the form g x f g x for x R where f G L is

XXI I I BRAVE NEW EQUIVARIANT FOUNDATIONS

a homomorphism of monoids To x ideas we shall write R for the Guniverse

f

determined by such a homomorphism f For a sp ectrum E we then dene

E G n E G

f

where the twisted halfsmash pro duct is determined by the map f The multi

plication and unit of G determine maps G G E G E and E G E

f f f f

that give G a structure of monad in S As was observed in LMS the category

f

GS R of Gsp ectra indexed on R is canonically isomorphic to the category

f f

S G of algebras over the monad G Of course we also have the monad L in S

f f

with LE L n E by denition a nonequivariant L sp ectrum is an algebra over

this monad

Proposition The following statements ab out the monads L and G hold

f

for any homomorphism of monoids f G L I R R

i L restricts to a monad in the category S G of Gsp ectra indexed on R

f

f

ii G restricts to a monad in the category S L of Lsp ectra indexed on R

f

iii The comp osite monads LG and G L in S are isomorphic

f f

Moreover up to isomorphism the comp osite monad LG is indep endent of f

f

L S G L of LGsp ectra indexed Corollary The category GS R

f

f

Up is isomorphic to the category S LG of GL sp ectra indexed on R on R

f

f f

to isomorphism this category is indep endent of f

The isomorphisms that we shall obtain preserve spheres and op eradic smash

pro ducts and so restrict to give isomorphisms b etween categories of S mo dules

Corollary Up to isomorphism the category GM of S mo dules is

S U

U

indep endent of the Guniverse U

1

mo dule on a naive Gsp ectrum is so rich that it en Thus a structure of S

R

compasses an S action on a Gsp ectrum indexed on U for any universe U This

U

richness is p ossible b ecause the action of G on U can itself b e expressed in terms

of the monoid L

There is another way to think ab out these isomorphisms which is given in

Elmendorf and May and which we now summarize It is motivated by the denition

of the op eradic smash pro duct

ALTERNATIVE PERSPECTIVES ON EQUIVARIANCE

Definition Fix universes U and U write L and L for the resp ective

monads in GS U and GS U and write L and L for the resp ective Gop erads

0

U

For an L sp ectrum M dene an L sp ectrum I M by

U

0

U

M I M I U U n

UU I

U

0

U

That is I M is the co equalizer displayed in the diagram

U

nid

0

U

I U U n M I U U n I U U n M

I M

U

id n

Here I U U n M M is the given action of L on M We regard the

pro duct I U U I U U as a space over I U U via the comp osition map

I U U I U U I U U

Prop osition ii gives a natural isomorphism

I U U I U U n M I U U n I U U n M

This makes sense of the map n id in the diagram The required left action of

0

U

I U U on I M is induced by the comp osition pro duct

U

I U U I U U I U U

which induces a natural map of co equalizer diagrams on passage to twisted half

smash pro ducts

Proposition Let U U and U b e Guniverses Consider the functors

0

U

I GS U L GS U L and GT GS U L

U U

0

U

i I is naturally isomorphic to

0

U U U

00 0 00

U U U

ii I I is naturally isomorphic to I

0

U U U

U

iii I is naturally isomorphic to the identity functor

U

0

U U

Therefore the functor I Moreover is an equivalence of categories with inverse I

0

U U

0 0 0

U U U

the functor I is continuous and satises I I M X M X for Lsp ectra

U U U

0

U

and M and based Gspaces X In particular it is homotopy preserving and I

U

U

induce inverse equivalences of homotopy categories I

0

U

Since the co equalizer dening Now supp ose that U R and U R

0

f f

0

U

I is the underlying nonequivariant co equalizer with a suitable action of G we

U

0

U

is naturally isomorphic see that with all group actions ignored the functor I

U

to the identity functor on S L In this case the equivalences of categories of

XXI I I BRAVE NEW EQUIVARIANT FOUNDATIONS

the previous result are natural isomorphisms and tracing through the denitions

one can check that they agree with the equivalences given by the last statement of

Corollary Therefore the following result which applies to any pair of universes

U and U is an elab oration of Corollary

Proposition The following statements hold

0

U

0

i I S is canonically isomorphic to S

U U

U

ii For Lsp ectra M and N there is a natural isomorphism

0 0 0

U U U

0

I N I M I M N

L L

U U U

iii The following diagram commutes for all Lsp ectra M

0 0

U U

0 0

I S M I S M

L U U L

U U

N

N o

N o

N

o

N

o

N

o

N

o

N

o

N

o

0

N

o

U

N

o

o I w w

U

0

U

I M

U

0

U

0

iv M is an S mo dule if and only if I mo dule M is an S

U U

U

0

U U

restrict to inverse equivalences of categories and I Therefore the functors I

0

U U

b etween GM and GM that induce inverse equivalences of categories b etween

S S

0

U

U

hGM and hGM

S S

0

U

U

This has the following consequence which shows that on the p ointset level

our brave new equivariant algebraic structures are indep endent of the universe in

which they are dened

0

U

Theorem The functor I GM is monoidal If R is an GM

S S

0

U U

U

0 0

U U

0

S algebra and M is an Rmo dule then I algebra and I R is an S M is an

U U

U U

0

U

I Rmo dule

U

The ideas of this section are illuminated by thinking mo del theoretically We

1

where G acts trivially on R We can fo cus attention on the category GM

R

reinterpret our results as saying that the mo del categories of S mo dules for vary

U

1

but given a mo del ing universes U are all isomorphic to the category GM

R

we have the isomorphism structure that dep ends on U Indeed for any U R

f

1

R

1

of categories I and we can transp ort the mo del category GM GM

U R

U

1

which we call structure of GM to a new mo del category structure on GM

U R

1

the U mo del structure on GM

R

1

R

The essential p oint is that I do es not carry the cobrant sphere S mo dules

U

U

n n

1

mo dules The weak S S LS to the corresp onding cobrant sphere S

U L R

S U

THE CONSTRUCTION OF EQUIVARIANT ALGEBRAS AND MODULES

equivalences in the U mo del structure are the maps that induce isomorphisms on

1

R n

1

mo dule maps out of the U spheres GH I homotopy classes of S S

R

U S

U

1

mo dules by using these U spheres as the We dene U cell and relative U cell S

R

domains of their attaching maps The U cobrations are the retracts of the relative

1

mo dules and the U brations are then determined as the maps that U cell S

R

satisfy the right lifting prop erty with resp ect to the acyclic U cobrations

A D Elmendorf and J P May Algebras over equivariant sphere sp ectra Preprint

The construction of equivariant algebras and mo dules

The results of the previous section are not mere esoterica They lead to homo

topically wellb ehaved constructions of brave new equivariant algebraic structures

from brave new nonequivariant algebraic structures The essential p oint is to un

derstand the homotopical b ehavior of p ointset level constructions that have de

sirable formal prop erties We shall explain the solutions to two natural problems

in this direction

First supp ose given a nonequivariant S algebra R and an Rmo dule M for

deniteness we supp ose that these sp ectra are indexed on the xed p oint universe

G

U of a complete Guniverse U Is there an S algebra R and an R mo dule M

G G G G

whose underlying nonequivariant sp ectra are equivalent to R and M in a way that

preserves the brave new algebraic structures In this generality the only obvious

G

candidates for R and M are i R and i M where i U U is the inclusion

G G

In any case we want R and M to b e equivalent to i R and i M However the

G G

change of universe functor i do es not preserve brave new algebraic structures

Thus the problem is to nd a functor that do es preserve such structures and yet is

equivalent to i A very sp ecial case of the solution of this problem has b een used

by Benson and Greenlees to obtain calculational information ab out the ordinary

cohomology of classifying spaces of compact Lie groups

Second supp ose given an S algebra R with underlying nonequivariant S

G G

algebra R and supp ose given an Rmo dule M Can we construct an R mo dule

G

M whose underlying nonequivariant Rmo dule is M Note in particular that

G

the problem presupp oses that up to equivalence the underlying nonequivariant

sp ectrum of R is an S algebra and similarly for mo dules We are thinking of

G

MU and MU and the solution of this problem gives equivariant versions as

G

MU mo dules of all of the sp ectra such as the BrownPeterson and Morava K

G

theory sp ectra that can b e constructed from MU by killing some generators and

inverting others

The following homotopical result of Elmendorf and May combines with Theo

XXI I I BRAVE NEW EQUIVARIANT FOUNDATIONS

rem to solve the rst problem In fact it shows more generally that up to

isomorphism in derived categories any change of universe functor preserves brave

new algebraic structures Observe that for a linear isometry f U U and

S mo dules M GM we have a comp osite natural map

U S

U

U

M f M I U U n M I

0

U

of Gsp ectra indexed on U where the rst arrow is induced by the inclusion

ff g I U U and the second is the evident quotient map

Theorem Let f U U b e a Glinear isometry Then for suciently

wellb ehaved S mo dules M GM those in the collection E of XXI I the

U S S

U U

0

U

natural map f M I M is a homotopy equivalence of Gsp ectra indexed

U

on U

Rememb er that E includes the q cobrant ob jects in all of our categories

S

U

of brave new algebras and mo dules We are entitled to conclude that up to

equivalence the change of universe functor f preserves brave new algebras and

G

mo dules The most imp ortant case is the inclusion i U U If we start

from any nonequivariant q cobrant brave new algebraic structure then up to

equivalence the change of universe functor i constructs from it a corresp onding

equivariant brave new algebraic structure

Turning to the second problem that we p osed we give a result due to May

that interrelates brave new algebraic structures in GM and M G Its starting

U

U

U

p oint is the idea of combining the op eradic smash pro duct with the functors I

0

U

We think of U as the basic universe of interest in what follows

Definition Let U U and U b e Guniverses For an L sp ectrum M and

an L sp ectrum N dene an L sp ectrum M N by

L

U U

N M I M N I

00 0

L L

U U

The formal prop erties of this pro duct can b e deduced from those of the functors

U

together with those of the op eradic smash pro duct for the xed universe U In I

0

U

U

0

mo dules to S mo dules and the smash takes S particular since the functor I

0

U U

U

pro duct over S is the restriction to S mo dules of the smash pro duct over L

U U

we have the following observation

Lemma The functor GS U L GS U L GS U L restricts

L

to a functor

GM GM GM

S S S S

0 0 0

U U

U U

THE CONSTRUCTION OF EQUIVARIANT ALGEBRAS AND MODULES

This allows us to dene mo dules indexed on one universe over algebras indexed

on another

00

algebra and let M GM Say Definition Let R GM b e an S

S S U

0 0 0

U U

0

U

that M is a right Rmo dule if it is a right I Rmo dule and similarly for left

00

U

mo dules

U

to To dene smash pro ducts over R in this context we use the functors I

0

U

index everything on our preferred universe U and then take smash pro ducts there

00

algebra let M GM b e a Definition Let R GM b e an S

S S U

0 00

U U

000

b e a left Rmo dule Dene right Rmo dule and let N GM

U

U U

N M U I M N I

000 0

R

I R

U U

00

U

These smash pro ducts inherit go o d formal prop erties from those of the smash

pro ducts of Rmo dules and their homotopical prop erties can b e deduced from the

homotopical prop erties of the smash pro duct of Rmo dules and the homotopical

U

as given by Theorem prop erties of the functors I

0

U

G

U Write S for the sphere Gsp ectrum Now sp ecialize to consideration of U

G

G

indexed on U and S for the nonequivariant sphere sp ectrum indexed on U We

take S mo dules to b e in GM and S mo dules to b e in M G in what follows

G U

U

Theorem Let R b e a commutative S algebra and assume that R is

G G G

split as an algebra with underlying nonequivariant S algebra R Then there is

a monoidal functor R M GM If M is a cell Rmo dule then

G R R R

G

R M is split as a mo dule with underlying nonequivariant Rmo dule M The

G R

functor R induces a derived monoidal functor D GD Therefore if

G R R R

G

M is an Rring sp ectrum in the homotopical sense then R M is an R ring

G R G

Gsp ectrum

The terms split as an algebra and split as a mo dule are a bit technical

and we will explain them in a moment However we have the following imp ortant

example see XVx for the denition of MU

G

Proposition The Gsp ectrum MU that represents stable complex cob or

G

dism is a commutative S algebra and it is split as an algebra with underlying

G

nonequivariant S algebra MU

We shall return to this p oint and say something ab out the pro of of the prop o

sition in XXVx We conclude that for any compact Lie group G and any MU

M This mo dule M we have a corresp onding split MU mo dule M MU

U G G G M

XXI I I BRAVE NEW EQUIVARIANT FOUNDATIONS

allows us to transp ort the nonequivariant constructions of XXI Ix into the equiv

ariant world For example taking M BP or M K n we obtain equivariant

BrownPeterson and Morava K theory MU mo dules BP and K n Moreover

G G G

if M is an MU ring sp ectrum then M is an MU ring Gsp ectrum and M is

G G G

asso ciative or commutative if M is so

We must still explain our terms and sketch the pro of of Theorem The notion

of a split Gsp ectrum was a homotopical one involving the change of universe

functor i and neither that functor nor its right adjoint i preserves brave new

algebraic structures We are led to the following denitions

Definition A commutative S algebra R is split as an algebra if there is

G G

U

a commutative S algebra R and a map I R R of S algebras such that

G

G G

U

U

R is a nonequivariant equivalence of sp ectra and the natural map i R I

G

U

is an equivariant equivalence of Gsp ectra We call R the or more accurately

an underlying nonequivariant S algebra of R

G

Since the comp osite is a nonequivariant equivalence and the natural map

R i i R is a weak equivalence provided that R is tame R is weakly equiva

lent to i R with Gaction ignored Thus R is a highly structured version of the

G

underlying nonequivariant sp ectrum of R Clearly R is split as a Gsp ectrum

G G

with splitting map

We have a parallel denition for mo dules

Definition Let R b e a commutative S algebra that is split as an algebra

G G

with underlying S algebra R and let M b e an R mo dule Regard M as an

G G G

U

I Rmo dule by pullback along Then M is split as a mo dule if there is an

G

G

U

U U

Rmo dules such that is a M M of I Rmo dule M and a map I

G G

G

U U

U

M nonequivariant equivalence of sp ectra and the natural map i M I

G

U

is an equivariant equivalence of Gsp ectra We call M the or more accurately

an underlying nonequivariant Rmo dule of M

G

Again M is a highly structured version of the underlying nonequivariant sp ec

trum of M and M is split as a Gsp ectrum with splitting map The

G G

ambiguity that we allow in the notion of an underlying ob ject is quite useful it

allows us to use Theorem and q cobrant approximation of S algebras and of

Rmo dules to arrange the condition on in the denitions if we have succeeded

in arranging the other conditions

For the pro of of Theorem Denition sp ecializes to give the required

functor R and it is clearly monoidal We may as well assume that our

G R

given underlying nonequivariant S algebra R is q cobrant as an S algebra Let

COMPARISONS OF CATEGORIES OF LGSPECTRA

M b e a cell Rmo dule By Theorem the condition on in the denition of an

underlying Rmo dule is satised Dene

U U U U

M M M R U I R U I I M id I

G G G G

G G

I R I R

U U U U

G G

U U

U

Clearly is a map of I Rmo dules and it is not hard to prove that it is an equiv

G

U

alence of sp ectra Thus M is split as a mo dule with underlying nonequivariant

G

Rmo dule M That is the main p oint and the rest follows without diculty

D J Benson and J P C Greenlees Commutative algebra for cohomology rings of classifying

spaces of compact Lie groups Preprint

A D Elmendorf and J P May Algebras over equivariant sphere sp ectra Preprint

J P May Equivariant and nonequivariant mo dule sp ectra Preprint

Comparisons of categories of LGsp ectra

We prove Prop osition and Corollary here The pro of of Prop osition

is based on the comparison of certain monoids constructed from the monoids G

G b e the and L and the homomorphism f G L Thus let G n L and L o

f f

left and right semidirect pro ducts of G and L determined by f As spaces

G n L G L and L o G L G

f f

and their multiplications are sp ecied by

g mg m g g f g mf g m

and

m g m g mf g m f g g g

There is an isomorphism of monoids

G n L L o G

f f

sp ecied by

g m f g mf g g

there is also an isomorphism of monoids

G n L G L

f

sp ecied by

g m g f g m

its inverse takes g m to g f g m Let

G L L

XXI I I BRAVE NEW EQUIVARIANT FOUNDATIONS

b e the pro jection We regard G L as a monoid over L via and we regard

G n L and L G as monoids over L via the comp osites and

f f

so that and are isomorphisms over L Using Prop osition we see that for

sp ectra E S the map induces a natural isomorphism

LG E G n L n E L o G n E G L E

f f f f

and the map induces a natural isomorphism

G LE G n L n E G L n E G LE

f f

In the domains and targets here the units and pro ducts of the given monoids

determine natural transformations and that give the sp ecied comp osite monad

structures to the displayed functors S S Elementary diagram chases on the

level of monoids imply that the displayed natural transformations are welldened

isomorphisms of monads If f is the trivial homomorphism that sends all of G to

L then G n L G L Thus in we are comparing the monad for the

f

to the monad determined by R regarded as a trivial Guniverse Guniverse R

f

The conclusions of Prop osition follow and Corollary follows as a matter of

category theory

The following two lemmas in category theory may or may not illuminate what is

going on The rst is proven in EKMM and shows why Corollary follows from

Prop osition The second dictates exactly what elementary diagram chases

are needed to complete the pro of of Prop osition

b e a monad in a category C and let T b e a monad in the Lemma Let S

category C S of Salgebras Then the category C ST of Talgebras in C S is

isomorphic to the category C TS of algebras over the comp osite monad TS in C

Here the unit of TS is the comp osite id S TS given by the units of S

and T and the pro duct on TS is the comp osite TSTS TTS TS where the

second map is given by the pro duct of T and the rst is obtained by applying T

to the action STS TS given by the fact that T is a monad in C S In our

applications we are taking T to b e the restriction to C S of a monad in C This

requires us to start with monads S and T that commute with one another

Lemma Let S and T b e monads in C Supp ose there is a natural isomor

COMPARISONS OF CATEGORIES OF LGSPECTRA

phism ST TS such that the following diagrams commute

ST and T SST

B

B

B

T

B

B

B

S

B

B

TSS TS ST TS STS

T

Then T restricts to a monad in C S to which the previous lemma applies Supp ose

further that these diagrams with the roles of S and T reversed also commute as

do the following diagrams

1

SS

S

STST SSTT SST ST and id T ST

ST

TSTS TTSS TTS TS id S TS

T TT

Then ST TS is an isomorphism of monads Therefore the categories

C TS C ST and C TS are b oth isomorphic to the category C ST

Here for the rst statement if X is an Salgebra with action then the required

T

action of S on TX is the comp osite STX TSX TX

XXI I I BRAVE NEW EQUIVARIANT FOUNDATIONS

CHAPTER XXIV

Equivari ant Algebra Brave New

by J P C Greenlees and J P May

Intro duction

We shall explain how useful it is to b e able to mimic commutative algebra

in equivariant top ology Actually the nonequivariant sp ecializations of the con

structions that we shall describ e are also of considerable interest esp ecially in

connection with the chromatic ltration of stable homotopy theory We have dis

cussed this in an exp ository pap er GM and that pap er also says more ab out the

relevant algebraic constructions than we shall say here We shall give a connected

sequence of examples of brave new analogues of constructions in commutative al

gebra The general pattern of how the theory works is this We rst give an

algebraic denition We next give its brave new analogue The homotopy groups

of the brave new analogue will b e computable in terms of a sp ectral sequence that

starts with the relevant algebraic construction computed on co ecient rings and

mo dules The usefulness of the constructions is that they are often related by a

natural map to or from an analogous geometric construction that one wishes to

compute Lo calization and completion theorems say when such maps are equiva

lences

The AtiyahSegal completion theorem and the Segal conjecture are examples of

this paradigm that we have already discussed However very sp ecial features of

those cases allowed them to b e handled without explicit use of brave new alge

bra the force of Bott p erio dicity in the case of K theory and the fact that the

sphere Gsp ectrum acts naturally on the stable homotopy category in the case of

cohomotopy We shall explain how brave new algebra gives a coherent general

XXIV BRAVE NEW EQUIVARIANT ALGEBRA

framework for the study of such completion phenomena in cohomology and anal

ogous lo calization phenomena in homology We have given another exp osition of

these matters in GM which says more ab out the basic philosophy We shall

describ e the results in a little greater generality here and so clarify the application

to K theory We shall also explain the relationship b etween lo calization theorems

and Tate theory which we nd quite illuminating

GM J P C Greenlees and J P May Completions in algebra and top ology In Handb o ok

of Algebraic Top ology edited by IM James North Holland pp

GM J P C Greenlees and J P May Equivariant stable homotopy theory In Handb o ok of

Algebraic Top ology edited by IM James North Holland pp

cal and Cech cohomology in algebra Lo

Supp ose given a ring R which may b e graded and which need not b e No etherian

and supp ose given a nitely generated ideal I If R is graded

n

the are required to b e homogeneous

i

For any element we may consider the stable Koszul co chain complex

K R R

concentrated in co degrees and Notice that we have a b er sequence

K R R

of co chain complexes

We may now form the tensor pro duct

K K K

n n

It is clear that this complex is unchanged if we replace some by a p ower and

i

it is not hard to check the following result

Lemma If I then K is exact Up to quasiisomor

n

phism the complex K dep ends only on the ideal I

n

Therefore up to quasiisomorphism K dep ends only on the radical

n

of the ideal I and we henceforth write K I for it

Following Grothendieck we dene the lo cal cohomology groups of an Rmo dule

M by

H R M H K I M I

BRAVE NEW VERSIONS OF LOCAL AND CECH COHOMOLOGY

It is easy to see that H R M is the submo dule

I

k

M fm M jI m for some p ositive integer k g

I

of I p ower torsion elements of M If R is No etherian it is not hard to prove

that H R is eaceable and hence that lo cal cohomology calculates the right

I

derived functors of It is clear that the lo cal cohomology groups vanish ab ove

I

co degree n in the No etherian case Grothendiecks vanishing theorem shows that

they are actually zero ab ove the Krull dimension of R Observe that if I then

H R M this is a restatement of the exactness of K I

I

The Koszul complex K comes with a natural map K R the

tensor pro duct of such maps gives an augmentation K I R Dene

the Cech complex C I to b e Ker The name is justied in GM By

insp ection or as an alternative denition we then have the b er sequence of

co chain complexes

K I R C I

We dene the Cech cohomology groups of an Rmo dule M by

CH R M H C I M

I

We often delete R from the notation for these functors The b er sequence

gives rise to long exact sequences relating lo cal and Cech cohomology and these

reduce to exact sequences

H M M CH M H M

I I I

together with isomorphisms

i i

CH M H M

I I

A Grothendieck notes by RHartshorne Lo cal cohomology Springer Lecture notes in mathe

matics Vol

Brave new versions of lo cal and Cech cohomology

Turning to top ology we x a compact Lie group G and consider Gsp ectra

indexed on a complete Guniverse U We let S b e the sphere Gsp ectrum and

G

we work in the category of S mo dules Fix a commutative S algebra R and

G G

consider Rmo dules M We write

G G n

M M M

n n G

XXIV BRAVE NEW EQUIVARIANT ALGEBRA

G G G

Thus R is a ring and M is an R mo dule

G

Mimicking the algebra for R we dene the Koszul sp ectrum K by the

b er sequence

K R R

Here suppressing notation for susp ensions R ho colimR R

it is an Rmo dule and the inclusion of R is a mo dule map therefore K is an

Rmo dule Analogous to the ltration at the chain level we obtain a ltration of

K by viewing it as R CR

Next we dene the Koszul sp ectrum of a sequence by

n

K K K

n R R n

Using the same pro of as in the algebraic case we conclude that up to equivalence

K dep ends only on the radical of I we therefore denote

n n

it K I We then dene the homotopy I p ower torsion or lo cal cohomology

mo dule of an Rmo dule M by

M K I M

I R

In particular R K I

I

To calculate the homotopy groups of M we use the pro duct of the ltrations

I

of the K given ab ove Since the ltration mo dels the algebra precisely there

i

results a sp ectral sequence of the form

G G G s

M M R H E

I t

st I st

r r r

E with dierentials d E

srtr st

Remark In practice it is often useful to use the fact that the algebraic lo cal

cohomology H R M is essentially indep endent of R Indeed if the generators of I

I

come from a ring R in which they generate an ideal I via a ring homomorphism

R R then H R M H R M In practice we often use this if the

I I

0

G

ideal I of R may b e radically generated by elements of degree This holds for

G G

any ideal of S since the elements of p ositive degree in S are nilp otent

Similarly we dene the Cech sp ectrum of I by the cob er sequence of Rmo dules

K I R C I

LOCALIZATION THEOREMS IN EQUIVARIANT HOMOLOGY

We think of C I as analogous to EG We then dene the homotopical lo calization

or Cech cohomology mo dule asso ciated to an Rmo dule M by

M I C I M

R

In particular RI C I Again we have a sp ectral sequence of the form

s G G G

E CH R M M I

t

st I st

r r r

with dierentials d E E

st srtr

The lo calization M I is generally not a lo calization of M at a multiplica

tively closed subset of R However the term is justied by the following theorem

from GM x Recall the discussion of Bouseld lo calization from XXI Ix

G

the Theorem For any nitely generated ideal I of R

n

map M M I is Bouseld lo calization with resp ect to the Rmo dule RI

or equivalently with resp ect to the wedge of the Rmo dules R

i

Observe that we have a natural cob er sequence

M M M I

I

relating our I p ower torsion and lo calization functors

Lo calization theorems in equivariant homology

For an Rmo dule M we have the fundamental cob er sequence of Rmo dules

EG M M EG M

Such sequences played a central role in our study of the Segal conjecture and

Tate cohomology for example and we would like to understand their homotopical

b ehavior In favorable cases the cob er sequence mo dels this sequence and

so allows computations via the sp ectral sequences of the previous section The

relevant ideal is the augmentation ideal

G G

I Kerres R R

In order to apply the constructions of the previous section we need an assumption

G

It will b e satised automatically when R is No etherian

Assumption Up to taking radicals the ideal I is nitely generated That

is there are elements I such that

n

q

p

I

n

XXIV BRAVE NEW EQUIVARIANT ALGEBRA

Under Assumption it is reasonable to let K I denote K The

n

canonical map K I R is then a nonequivariant equivalence Indeed this is

a sp ecial case of the following observation which is evident from our constructions

G G H

Lemma Let H G let R and let res R Then

i i i

H

regarded as a mo dule over the S algebra Rj

H H

K j K

n H n

G

Therefore if Ker res then the natural map K R is an

i n

H

H equivalence

Here the last statement holds since K R If we take the smash pro duct

of with the identity map of EG we obtain a Gequivalence of Rmo dules

EG K I EG R Working in the derived category GD we may invert

R

this map and comp ose with the map

EG K I S K I K I

induced by the pro jection EG S to obtain a map of Rmo dules over R

EG R K I

Passing to cob ers we obtain a compatible map

EG R C I

Finally taking the smash pro duct over R with an Rmo dule M there results a

natural map of cob er sequences

EG M

M

EG M

M

M I

M

I

Clearly is an equivalence if and only if is an equivalence When the latter

holds it should b e interpreted as stating that the top ological lo calization of M

away from its free part is equivalent to the algebraic lo calization of M away

from I We adopt this idea in a denition Recall the homotopical notions of

an Rring sp ectrum A and of an Amo dule sp ectrum from XXI I we tacitly

assume throughout the chapter that all given Rring sp ectra are asso ciative and

commutative

LOCALIZATION THEOREMS IN EQUIVARIANT HOMOLOGY

Definition The lo calization theorem holds for an Rring sp ectrum A if

id EG A EG R A C I A

A R R

is a weak equivalence of Rmo dules that is if it is an isomorphism in GD It is

R

equivalent that

id EG A EG R A K I A

A R R

b e an isomorphism in GD

R

In our equivariant context we dene the Ahomology of an Rmo dule M by

GR G

A M M A

R

n n

G G

compare XXI I This must not b e confused with A X X A which

n n

GR G

is dened on all Gsp ectra X When A R A is the restriction of A to

G GS

G

thought of as a theory dened on S is A Rmo dules When R S A

G G

mo dules In general for Gsp ectra X we have the relation

GR G

F X A X A

R

where the free Rmo dule F X is weakly equivalent to the sp ectrum X R The

R

H R

isomorphism for all subgroups H lo calization theorem asserts that is an A

H R

acyclic for all H Observe that the of G and thus that the cob er C is A

denition of implies that C is equivalent to EGK I We are mainly interested

in the case A R but we shall see in the next section that the lo calization theorem

holds for K regarded as an S ring sp ectrum although it fails for S itself The

G G G

conclusion of the lo calization theorem is inherited by arbitrary Amo dules

Lemma If the lo calization theorem holds for the Rring sp ectrum A then

the maps

EG M M and EG M M I

I

of are isomorphisms in GD for all Amo dules M

R

Proof C M is trivial since it is a retract in GD of C A M

R R R R

When this holds we obtain the isomorphism

G G G

M M EG EG M

I

on passage to homotopy groups Here in favorable cases the homotopy groups on

the right can b e calculated by the sp ectral sequence When M is split and

G is nite the homology groups on the left are the reduced homology groups

XXIV BRAVE NEW EQUIVARIANT ALGEBRA

M BG dened with resp ect to the underlying nonequivariant sp ectrum of M

see XVIx We also obtain the isomorphism

G G G

M I M EG EG M

the homotopy groups on the right can b e calculated by the sp ectral sequence

More generally it is valuable to obtain a lo calization theorem ab out EG X

G

for a general based Gspace X obtaining the result ab out BG by taking X to

b e S To obtain this we simply replace M by M X in the rst equivalence of

the previous lemma If M is split we conclude from XVIx that

G AdG

M EG X EG X M

G

where AdG is the adjoint representation of G Thus we have the following

implication

Corollary If the lo calization theorem holds for A and M is an Amo dule

sp ectrum that is split as a Gsp ectrum then

AdG G

M X M EG X

I G

for any based Gspace X Therefore there is a sp ectral sequence of the form

AdG G G s

X M EG X M R H E

t st G

I st

Completions completion theorems and lo cal homology

The lo calization theorem also implies a completion theorem In fact applying

the functor F M to the map we obtain a cohomological analogue of Lemma

R

To give the appropriate context we dene the completion of an Rmo dule

M at a nitely generated ideal I by

F K I M M

R

I

We shall shortly return to algebra and dene certain lo cal homology groups

H R M that are closely related to the I adic completion functor In the top o

I

logical context it will follow from the denitions that the ltration of K I gives

rise to a sp ectral sequence of the form

st

I t G

E H R M M

s G G st I

st srtr

with dierentials d E E Here if R is No etherian and M is

r

r r G G

nitely generated then

G

M M

I G I

COMPLETIONS COMPLETION THEOREMS AND LOCAL HOMOLOGY

Again a theorem from GM x gives an interpretation of the completion

functor as a Bouseld lo calization

G

Theorem For any nitely generated ideal I of R the map

n

M M is Bouseld lo calization in the category of Rmo dules with resp ect to

I

the Rmo dule K I or equivalently with resp ect to the smash pro duct of the

Rmo dules R

i

Returning to the augmentation ideal I we have the promised cohomological

implication of the lo calization theorem the case M A is called the completion

theorem for A

Lemma If the lo calization theorem holds for the Rring sp ectrum A then

the map

F EG M F K I M F EG R M M

R R

I

is an isomorphism in GD for all Amo dule sp ectra M

R

Proof F C M is trivial since any map C M factors as a comp osite

R

C C A M A M

R R

and similarly for susp ensions of C

When this holds we obtain the isomorphism

G

M EG M

G I

on passage to homotopy groups If M is split the cohomology groups on the

right are the reduced cohomology groups M BG dened with resp ect to the

underlying nonequivariant sp ectrum of M see XVIx

To obtain a completion theorem ab out EG X for a based Gspace X we

G

replace M by F X M in the previous lemma If M is split then

G

F EG X M M EG X

G

Corollary If the lo calization theorem holds for A and M is an Amo dule

sp ectrum that is split as a Gsp ectrum then

F X M M EG X

G

I G

for any based Gspace X Therefore there is a sp ectral sequence of the form

st

I t st

E H R M X M EG X

G

s G G