Contemporary Mathematics

Hopf Rings Dieudonne Mo dules and E S



Paul G Go erss

This paper is dedicated J Michael Boardman

Abstract The category of graded bicommutative Hopf algebras over the

prime eld with p elements is an ab elian category which is equivalent by work

of Schoeller to a category of graded mo dules known as Dieudonne mo dules

Graded ring ob jects in Hopf algebras are called Hopf rings and they arise

in the study of unstable cohomology op erations for extraordinary cohomology

theories The central p oint of this pap er is that Hopf rings can b e studied by

lo oking at the asso ciated ring ob ject in Dieudonne mo dules They can also b e

computed there and b ecause of the relationship b etween BrownGitler sp ectra

and Dieudonne mo dules calculating the Hopf ring for a homology theory E



comes down to computing E S which Ravenel has done for E BP



From this one recovers the work of Hopkins Hunton and Turner on the Hopf

rings of Landweber exact cohomology theories

The are two ma jor algebraic diculties encountered in this approach The

rst is to decide what a ring ob ject is in the category of Dieudonne mo dules as

there is no obvious symmetric monoidal pairing asso ciated to a tensor pro duct

of mo dules The second is to show that Hopf rings pass to rings in Dieudonne

mo dules This involves studying universal examples and here we pick up an

idea suggested by Bouseld torsionfree Hopf algebras over the padic integers

with some additional structure such as a selfHopfalgebra map that reduces

to the Verschiebung can b e easily classied

An ab elian category A with a set of small pro jective generators is equivalent

to a category M of mo dules over some ring R In addition if A and M are

symmetric monoidal categories and the equivalence of categories A M resp ects

the monoidal structure one can study the ring ob jects in A by studying the ring

ob jects in M The purp ose of this pap er is to develop this observation in the case

where A is the category HA of graded bicommutative Hopf algebras over the

prime eld F The graded ring ob jects in HA are called Hopf rings and they

p

arise naturally when studying unstable cohomology op erations for some cohomology

theory E see

To state some results x a prime p A slight rewording gives the results

at p We will restrict attention to the sort of Hopf algebra that arises in

algebraic top ology namely to Hopf algebras that are skewcommutative and so

Mathematics Subject Classication Primary N S Secondary L

The author was supp orted in part by the National Science Foundation

c

copyright holder

PAUL G GOERSS

that the degree part H of H is the group algebra of an ab elian group We

will call the category of such Hopf algebras HA If X is a p ointed space then

H X F H X HA Schoeller has essentially proved that there is

p

an equivalence of categories

D HA D

where D is the category of graded Dieudonne mo dules An ob ject M D is a

nonnegatively graded ab elian group M so that M is an F vector space and

n p

there are homomorphisms

F M M and V M M

n pn n

np

so that FV V F p and V if n If p do es not divide n we set V

k k

It follows that if n p s p s then p M If H HA then the

n

action of F and V on D H reect the Frobenius and Verschiebung resp ectively of

H

The category HA is a symmetric monoidal category As with the tensor

pro duct of ab elian groups the symmetric monoidal pairing arises by considering

bilinear maps An example of a bilinear map in HA is supplied by considering a

n

ring sp ectrum E The functor X E X is representable in the homotopy category

n

of spaces indeed if E n E then for all CW complexes X one has

n

X E n E X

n m nm

The cuppro duct pairing E X E X E X is induced by a map of spaces

E n E m E n m

and the resulting map of coalgebras

H E n H E m H E n m

is a bilinear map of Hopf algebras One can axiomatize this situation see x or

and following Hunton and Turner we prove in x that given H and K

in HA there is a universal bilinear map

H K H  K

The pairing  HA H A HA is symmetric monoidal the unit is the group

ring F Z

p

Next one would like to calculate D H  K If H H K is a bilinear

pairing of Hopf algebras one obtains a bilinear pairing of Dieudonne mo dules

D D H D H D K

in the sense that D is a bilinear map of graded ab elian groups and

V D x y D V x V y

and

D F x y F D x V y D x F y F D V x y

If M N D there is a universal such bilinear pairing

M N M  N

D

which is easy to write down see Equation Then  is a symmetric monoidal

D

pairing and one of the main results is D H  K D H  D K See Theorem

D

This leads to eective computations

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



We then employ this to study Hopf rings If E is a ring sp ectrum then H E

fH E ng is a Zgraded ring ob ject in HA and hence D H E is a Zgraded ring

ob ject in D This last means that given

x D H E j y D H E k

m n

there is a pro duct x y D H E j k so that

mn

V x y V x V y F x y F x V y x F y F V x y

Such an ob ject will b e called a Dieudonne ring If the ring sp ectrum is homotopy

commutative then this pro duct satises the following skew commutativity formula

nmj k

x y y x

Since

k

D H E k E k E

D H E is actually an E algebra so D H E is an E Dieudonne algebra

To compute this ob ject we use the fact the functor on sp ectra

X D H X

n

which assigns to a space X the degree n part of of D H X is actually part

of a homology theory if n mo d p In fact by if B n is the nth

BrownGitler sp ectrum there is a natural surjection

B n X D H X

n n

which is an isomorphism if n mo d p Thus for a ring sp ectrum E one

obtains a surjection

E B n D H E n k

k n

and if B fB ng is the graded BrownGitler sp ectrum one gets a degree

n

shearing surjection

E B D H E

of bigraded groups In fact this can b e made into a morphism of E Dieudonne

algebras see x In many cases the kernel of this map can b e analyzed Fur

thermore the Snaith splitting of S and the analysis of the summands done by a

variety of authors shows that there is a ltration on E S so that

the asso ciated graded ob ject is E B Since Ravenel has eectively calculated

BP S one can deduce a great deal ab out D H E for Landweber exact theories

In particular one can recover the HopkinsHuntonTurner results which

in this form have a pleasing statement one that in fact succinctly enco des the

RavenelWilson relation of complex oriented theories See Theorem

This pap er is divided into three sections The rst two are devoted to the

algebra of Hopf algebras their asso ciated Dieudonne mo dules and the appropriate

bilinear pairings We sp end most of our energy discussing graded commutative

as opp osed to skew commutative Hopf algebras moving on to skew commutative

Hopf algebras and the top ological applications cited ab ove only in the third section

Because of the splitting principle for skewcommutative Hopf algebras see and

Prop osition the passage from commutative to skewcommutative is easy As

ab ove we denote graded bicommutative Hopf algebras by HA

PAUL G GOERSS

In order to come to grips with some of the algebra involved we sp end a great

deal of time working with universal examples The pro jective generators of HA

include the Hopf algebras

H n F x x x

p k

k i

where n p s p s degx p s all with Witt vector diagonal This is

i

the reduction mo dule p of a Hopf algebra over the padic integers Z

p

CW k Z x x x

s p k

with the unique diagonal so that the Witt p olynomials

i1 i

p p

i

p x px w x

i i

are primitive The CW stands for coWitt The Z Hopf algebra CW k comes

p s

equipp ed with a lift of the Verschiebung that is there is a degree lowering Hopf

algebra map

CW k CW k

s s

which reduces to the Verschiebung H n H n Picking up a thread suggested

by Bouseld it turns out that torsionfree graded connected Hopf algebras over

Z with a lift of the Verschiebung are completely classied by their indecomp os

p

H can b e simply ables Furthermore if H is such a Hopf algebra then D F

p Z

p

computed in terms of QH See Theorem This and other related topics o ccupy

the rst four sections This is the rst part of the pap er

The second part of the pap er is devoted to bilinear pairings developing the

formulas cited ab ove and proving the isomorphism

D H  K D H  D K

D

There is a table of contents at the end of this introduction and a glossary of symbols

b efore the references

This pro ject has it ro ots in a conversation with Bill Dwyer who noted that the

work of Mo ore and Smith shows that the functor

Z H Z

has excellent exactness prop erties when Z is a lo op space Furthermore Dwyer

suggested this fact could b e used to study Hopf rings I knew these exactness

prop erties as the statement that on sp ectra X D H X was part of a ho

mology theory

Much of what is done here can b e greatly generalized The restriction to the

prime eld is dictated by homotopy theory not by algebra and all of the algebraic

results pass to any p erfect eld of characteristic p The internal grading on the

Hopf algebras can probably also b e dropp ed although some care must b e taken

to deal with those Hopf algebras that are neither group rings nor connected in the

sense of

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Contents

Part I Classifying Hopf Algebras

The Dwork Lemma and algebras with a lift of the Frobenius

Hopf algebras with a lift of the Verschiebung

The Implications of a lift of the Frobenius

Dieudonne theory

Part I I Bilinear Maps Pairings and Ring Ob jects

Bilinear maps and the tensor pro duct of Hopf algebras

Bilinear pairings for Hopf Algebras with a lift of the Verschiebung

Universal bilinear maps for Hopf algebras over F

p

Symmetric monoidal structures

Part I I I Hopf rings asso ciated to homology theories

Skew commutatative Hopf algebras

The Hopf ring of complex oriented cohomology theories

The role of E S

Glossary

References

Part I Classifying Hopf Algebras

The Dwork Lemma and algebras with a lift of the Frobenius

This preliminary section introduces the basic to ol we will use for constructing

morphims of Hopf algebras Fix a prime p

Let x x b e a sequence of indeterminants and let Z x x x b e the

p

free graded commutative algebra over the padic integers Z in the indeterminants

p

x Let

i

n1 n

p p

n

p x px w w x w x x x

n n n n n

b e the nth Witt p olynomial Here and b elow x x x

The following is known as the Dwork Lemma

Lemma Let A be a commutative torsionfree Z algebra Suppose A has a

p

ring endomorphism

A A

p

so that x x mo d p Then given a sequence of elements g A n so

n

that

n

g g mo d p

n n

there are unique elements q n so that

n

w q w q q q g

n n n n

p p

n

x Thus q is x p Proof Note that w x x w x

n n n n n

n

determined by the formula

p q

n

p q g w q q

n n n

n

PAUL G GOERSS

n

provided the right hand side of this equation is divisible by p But

p p

n

w q q mo d p q w q

n n n

n

g

n

n

g mo d p

n

Remark There is an obvious graded analog of this result One requires

i

that the indeterminants x have degree p m for some p ositive integer m Then the

i

i

Witt p olynomial w also has degree p m as do the solutions q

i i

Algebras A over Z equipp ed with algebra endomorphisms A A so that

p

p

x x mo d p will b e said to have a lift of Frobenius If A Z T is the free

p

commutative graded algebra on a graded set of generators T then one can dene

p

A A by setting t t for t T

For the next result let y y y b e another set of indeterminants

Corollary There exist unique polynomials

a x y Z x x y y

i p

so that

w a a w x w y

n n n

This is immediate from the Dwork Lemma Note that induction and the equa

tion imply that a is a p olynomial in x x y y

n n n

We use this result to dene a diagonal

Z x x Z x x Z x x

p p p

by

x a x x

i i

Here x x x and similarly for x

Lemma With this diagonal Z x x becomes a bicommutative Hopf

p

algebra over Z The Witt polynomials w x are primitive

p n

Proof That is coasso ciative and co commutative follows from the unique

ness clause of Lemma For example is co commutative b ecause a x y

i

a y x which in turn follows from the uniqueness of the a and the equation

i i

w a x y a x y w x w y w y w x

n n n n n

w a y x a y x

n

The Witt p olynomials are primitive by construction Put another way

w x x w x x

n n

w a x x a x x

n

w x w x

n n

w x w x

n n

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Remark Corollary and Lemma can together b e rephrased as

follows there is a unique Hopf algebra structure on Z x x so that the Witt

p

p olynomials w x are primitive

n

Since a x y is a p olynomial in x x y y the diagonal on the

n n n

Hopf algebra Z x x restricts to a diagonal on Z x x making the

p p n

latter a subHopf algebra of Z x x

p

Witt vectors can b e dened as follows If A is the category of Z algebras

p

and k A then the Witt vectors on k is the set

W k Hom Z x x k

A p

with group structure induced from the Hopf algebra structure on Z x x

p

The group W k acquires a commutative ring structure represented by a map

Z x x Z x x Z x x

p p p

sending w x to w x w x which exists by the Lemma The inclusion

n n n

Z x x x Z x x denes a quotient map to the Witt vectors of

p n p

length n

W k W k Hom Z x x x k

n A p n

Evidently W k lim W k and W k k If k is a p erfect eld W k is the

n

unique complete discrete valuation ring so that W k pW k k in particular if

Z See p k F W F

p p p

The Hopf algebra structure of Lemma passes to a Hopf algebra structure

in the graded case if we use the grading conventions of Remark

It is convenient to have a name for these Hopf algebras

Definition For k let CW k b e the Hopf algebra with under

lying algebra

Z x x

p k

and coalgebra structure dened by Corollary If we are working in a graded

situation write CW k for the graded analog of this Hopf algebra with the degree

m

i

of x equal to p m

i

The Dwork Lemma has a uniqueness clause in it but we are often interested

in a weaker version of uniqueness in particular we will b e interested in knowing

when two algebra maps are equal when reduced mo dulo p This is Lemma

b elow

First some notation Supp ose A is a torsion free algebra equipp ed with an

p

algebra map A A so that x x mo d p Supp ose one has two sequences

n n

of elements in A g and h so that g g mo d p and h h mo d p

n n n n n n

Then there are unique elements q and r in A so that

n n

w q q g

n n n

w r r h

n n n

n

Lemma If for al l n g h mo d p then q r mo d p In other

n n n n

words the two induced maps of algebras

Z x x A

p

agree modulo p

PAUL G GOERSS

k k

n p p nk

Proof Note that if x y mo d p then x y mo d p Then from

n n n

formula we have p q p r mo d p and the result follows

n n

Proposition The Hopf algebra CW has a lift of the Frobenius

which is a Hopf algebra map and so that w w

n n

Let p CW CW be ptimes the identity map in the abelian

group of Hopf algebra maps of CW to itself Then

p

px x mo d p

i

i

Proof Certainly Lemma supplies an algebra map CW

CW so that w w This can b e seen to b e a coalgebra map by applying

n n

the uniqueness clause of Lemma to the two p ossible maps

CW CW CW

To see that is a lift of the Frobenius note that x q where

n n

w q q w x x

n n n n

p

p n

w x x mo d p

n

n

p p

mo d p and x x mo d p So Lemma implies q x

n

n

Virtually the same argument applies using that

p p

pw x x w x x

n n n

n

n

mo d p

The following prop erty of the Hopf algebra CW will b e extremely useful

Let HF b e the category of pairs H where H is a torsion free bicommutative

Hopf algebra over Z and H H is a morphism of Hopf algebras which is a

p

lift of the Frobenius Morphisms in HF must commute with lifts of the Frobenius

Prop osition pro duces a pair CW in HF Hopf algebras withs such lifts

of the Frobenius are fairly rare for example the primitively generated Hopf algebra

Z x cannot supp ort a lift of the Frobenius as there is no primitive which reduces

p

p

to x mo dulo p

For any Hopf algebra H let PH denote the primitives

Proposition Let H HF be a Hopf algebra equipped with a lift of the

Frobenius Then there is a natural isomorphism

Hom CW H PH

HF

given by f f x

Proof Let and denote the lifts of the Frobenius in CW and H

H

resp ectively First note if y f x PH then

n

y f w

n

H

Thus is an injection by the uniqueness clause of Lemma

n

Next if y PH is primitive then so is g y n Since g g

n n H n

H

Lemma supplies f CW H so that

f w g

n n

In particular we have f x y So we need only show f is a morphism in HF that

is f f and that f is a morphism of Hopf algebras For the rst we have

H

f w f w g f w

n n n H n

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



so the uniqueness clause of Lemma applies For the second again apply this

uniqueness clause to the two p ossible comp ositions

CW H H

There is a graded version of this result which yields that there is a natural

isomorphism

CW H PH Hom

m m HF



Here HF are graded torsion free Hopf algebras with a Hopf algebra lift of the

Frobenius and PH is the degree m part of the primitives

m

Hopf algebras with a lift of the Verschiebung

While our primary interest is in ab elian Hopf algebras over elds the natural

generators of that category are the reductions mo dulo p of the Hopf algebras

CW k of the previous section These are not only Hopf algebras over Z but

n p

they supp ort a curious bit of extra structure which we now explore We will work

in the graded case as it is considerably simpler

Let A b e a torsionfree bicommutative Hopf algebra over Z Then A will b e

p

said to have a lift of the Verschiebung if there is a Hopf algebra endomorphism

A A so that for all x A there is a congruence

x x mo d p

where F A F A is the Verschiebung

p p

This is a very restrictive condition on a Hopf algebra For example the divided

p ower algebra on a primitive generator cannot b e equipp ed with a lift of the Ver

schiebung For if x is the primitive generator we would have x py for some

y hence

p p

x p

p

x y mo d p

p

p p

But

x x x mo dp

p p

This kind of simple example can b e expanded into the following observation

A graded Hopf algebra A over a commutative ring k will b e called connected if

k If such a Hopf algebra over Z has a lift of the Verschiebung A A A

p

then necessarily divided degree by p and is the identity on degree

Proposition Let A be a graded connected torsionfree Hopf algebra over

Z equipped with a lift of the Verschiebung If A is nitely generated as an algebra

p

then A is a polynomial algebra

This will b e proved in the next section as it takes us a bit aeld

The next result supplies our main examples Let CW k b e the Hopf algebras

n

of Denition

Proposition The graded Witt Hopf algebras CW k Z x x

n p k

k have a unique lift of the Verschiebung CW k CW k so that

n n

x i

i

x

i

i

PAUL G GOERSS

Proof The Dwork Lemma supplies a map of algebras

Z x x Z x x

p p

so that w pw where w Since the p olynomials pw are primitive

i i i

this yields a morphism of Hopf algebras

CW CW

n n

and a simple induction argument shows x x Since F CW is a

i i p

p olynomial algebra the equation

p p

x px x

i i i

from Prop osition shows that is indeed a lift of the Verschiebung The case of

CW k with k nite follows by restriction

We dene the categories of coalgebras and Hopf algebras we are interested in

Definition The category HV is the category of pairs A where

A is a graded connected torsionfree Hopf algebra over Z and

p

A A is a lift of the Verschiebung

We will call this category the category of Hopf algebras with a lift of the Ver

schiebung

There is an asso ciated notion of a coalgebra with a lift of the Verschiebung

which we will use in Lemma and in section

Definition The category CV of coalgebras with a lift of the Verschiebung

consists of pairs C where

C is a graded connected torsionfree co commutative coalgebra over Z

p

C C is a lift of the Verschiebung

We will call this category the category of coalgebras with a lift of the Verschiebung

The category HV is additive and has all limits and colimits but it is not ab elian

For example the cokernel in HV of the map

p Z x Z x

p p

of primitively generated Hopf algebras with trivial lift of the Verschiebung is simply

Z

p

We now insert a technical lemma that says that the category HV has a set of

generators

Lemma Let H HV Then H is the union of its subobjects H H in

HV which are nitely generated as algebras over Z

p

Proof It is enough to show that if x H is a homogeneous element then

there is an H containing x Let C H b e a nitely generated subcoalgebra with a

lift of the Verschiebung containing x Such a C is easily constructed by a downwards

degree argument Let S C b e the symmetric algebra on the coaugmentation ideal

of C endowed with induced structure as an ob ject in HV Then let H b e the

image of the evident map

S C H

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



If H HV let IH denote the augmentation ideal Then the indecomp os

ables QH I H I H form a torsionfree Z mo dule by Prop osition and Lemma

p

It is in fact a Z V mo dule with

p

V x IH x IH

The action of V is nilp otent on QH for degree reasons This leads to the following

denition

Definition Let M denote the category of p ositively graded Z V mo d

V p

ules M so that

M is torsionfree as a Z mo dule and

p

the action of V divided degree by p that is

V M M

pn n

Implicit in the second statement of this denition is that V on M if p

n

do es not divide n

The main result of this section is the following

Theorem The functor

Q HV M

V

is an equivalence of categories

This will b e proved by a sequence of lemmas To b egin we have the following

result

Lemma For al l H and K in HV the natural map

QH QK Hom H K Hom

HV M

V

is an injection

Proof Let H Q b e the category of graded connected Hopf algebras over Q

p p

Consider the diagram where each of the maps is the obvious one

QH QK Hom Hom H K

M HV

/ V

 

Hom Q H Q K

QQ H QQ K Hom

p p

HQ

p p M p

/ V

Then the left vertical map is an injection b ecause the Hopf algebras are torsion free

and the right vertical map is an injection b ecause of Prop osition and Lemma

The b ottom map is an isomorphism Thus the top map is an injection

Theorem would assert among other things that the natural map of the

previous result is an isomorphism

We now supply an algebraic result Let Z n b e the the graded Z mo dule free

p p

on one generator in degree n

Proposition If M M then there is a natural ismorphism for al l

V

n

M Z n M Hom M QC W Hom

p n Z n M

p V

Suppose n is relatively prime to p and k Then there is a natural

isomorphism

k

Z p n M M k QC W k M Hom Hom

p n Z M

p n

p V

PAUL G GOERSS

Proof If y QC W is the residue class of x then Lemma implies

i n i

that V y y If g M Z n is a Z mo dule homomorphism dene f M

i i p p

CW by

n

X

i

g V xy f x

i

i

i

For degree reasons g V x for at most one i Conversely given a morphism

h M QC W in M one can write

n V

X

g xy hx

i i

i

i

where g M Z and dene a mo dule homomorphism M Z by g The

i p p

p n

functions g f and h g are inverse to each other Part is a simple calculation

with Prop osition

The key to Theorem is the following result

Proposition If H HV then the natural map

QH QH QC W Hom H C W Hom

n HV n M

m V

is an isomorphism

Proof By Lemma we know that this map is an injection To complete

the argument we must prove surjectivity

First note that we may assume that H is nitely generated as a Z mo dule in

p

each degree For by Lemma we may write a general H as colimit

H colim H

where H runs over the nitely generated subob jects in HV of H If the result

holds for H then

lim Hom H C W Hom H C W

HV n HV n

lim Hom QH QC W

HV n

Hom QH QC W

HV n

So assume H is nitely generated in each degree as a Z mo dule Let g

p

QH QC W As Prop osition we may write Hom

M

V

X

g xy g x

i i

i

i

where g QH Z The isomorphism

i p

p n

QH QC W QH H om

M

V

n

sends g to g The Z dual of H which we write H is a Hopf algebra with a lift of

p

P H Lemma supplies the Frobenius given by Since g QH

a morphism of Hopf algebras equipp ed with a lift of the Frobenius

f CW H

n

with f x g Since f commutes with the lifts of the Frobenius Lemma and

P H Dualizing f we the fact g g imply that f w g QH

i i i i

obtain a morphism of Hopf algebras

H CW n

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



If we dene a lift of the Verschiebung on CW by then is a morphism in

n

HV We conclude the argument by identifying CW As a matter of notation

n

if a H and x CW let us write

n

ha xi

for the value of a CW evaluated at x The ab ove considerations imply

n

that

ha w i h a

i i

If we take H to b e CW and g to b e the indentity map we obtain a

n

morphism of Hopf algebras with a lift of Verschiebung

CW CW

n n

with the prop erty that

h x w i h x

i i i i

According to this implies that is an isomorphism To nish we must show

that is an equality

Q g QH QC W

i n

This is equivalent to asserting that for all a QH

g aQ y Qa

i i

in PCW Recall that y is the residue class of x in QC W Since w

i i n i

generates PCW in this degree we may write a x IH and calculate

hQa w i hQx w i

i i

g x g a

i i

g ahQx w i

i i i

hg aQy w i

i i i

The following is the main techinical result b ehind the pro of of Theorem

Theorem The indecomposables functor Q HV M has a right ad

V

joint S Furthermore the natural map M QS M in M is an isomorphism

V

Note that Theorem follows immediately The one natural map M

QS M is an isomorphism by this result the other natural map S QH H is

an isomorphism b ecause of Prop osition and Lemma and the comp osite

QH QS QH QH

forces QS QH QH to b e an isomorphism

Proof We b egin by constructing S M for M nitely generated in each

degree Dene functors

M M

V V

as follows

2

M M M M

n n

np np

where is is understo o d that M if k is a fraction Dene V on M by

k

V x x x x x

PAUL G GOERSS

There is a natural injection M M given by

x x V x V x

Dene M by the formula

2

M M M

n

np np

and V again dened by pro jection There is a natural map

d M M

given by

dx x x x V x x V x

This map is surjective and the sequence

d

M M M

is short exact in M In fact in each degree it is split short exact Furthermore

V

choosing a set of generators over Z for M for each n denes an isomorphism

p n

M QC W

k

The number of times each p ositive integer app ears as a k is nite There is a

similar isomorphism for M

QC W M

k

Dene

S M CW

k

Note that in any given degree this tensor pro duct is the tensor pro duct of nitely

many groups Similary dene S M and note that Prop osition implies

that there is a map in HV

f S M S M

with Qf d Dene S M by the pullback diagram in HV

S M S M

/ f

 

Z

S M p

/

Note that S M is the usual Hopf algebra kernel of f

We must now examine this construction The rst thing to do is to show

QS M QS M is an injection In fact there is a diagram

PS M PS M / /

 

QS M QS M

/

where the vertical maps are the canonical maps from the primitives to indecom

p osables Since these are Hopf algebras in HV Lemma implies that these maps

are injections with torsion cokernel Hence QS M QS M is onetoone

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Next let H b e any ob ject in HV Then there is a diagram with columns exact

 

QH QS M Hom H S M Hom

HV M

/ V

 

Hom H S M Hom QH QS M

M HV /

/ V

 

 

Hom H S M Hom QH QS M

HV M

/ V

Note that for example

QH M QH QS M Hom Hom

M M

V V

and the horizontal maps lab elled as isomorphims are so by Lemma This

diagram implies that

QH QS M Hom H S M Hom

HV M

V

Furthermore if we set H CW k for various n and k Lemma implies that

n

M QS M These two equations together yield the desired adjunction formula

QH M Hom H S M Hom

HV M

V

To complete the pro of we must extend S to M which are not nitely generated in

each degree To do this write such a general M as a ltered colimit of subob jects

M which are nitely generated in each degree then dene

i

S M colim S M

i

Note that this colimit can b e calculated in graded Z mo dules If H HV is nitely

p

generated as an algebra

Hom H S M colim Hom H S M

HV HV i

QH M colim Hom

i M

V

QH M Hom

M

V

as needed In particular setting H CW k for various n and k Lemma

n

implies that M QS M Finally if H is general Lemma implies that H is

the ltered colimit of its nitely generated subob jects which are nitely generated

as algebras Then using an argument similar to that given at the b eginning of

Prop osition it follows that

QH M Hom H S M Hom

HV M

V

Example The Husemoller Splitting Consider the Hopf algebra

Z a a a H BU

p

PAUL G GOERSS

with the degree of a equal to i and

i

X

a a a

k i j

ij k

This represents the functor on graded Z algebras

p

A tAt

where At is the graded p ower series on A with deg t A b ecomes a

i

group under p ower series multiplication If f t a t A then mo dulo p

i

p

p pi

f t a t

i

p

from which it follows that on H B U F pa a or

p i

ip

a a

i

ip

where a if ip is a fraction Now dene H BU H BU to b e the

ip

unique Hopf algebra map inducing

A A

p pi

where f t f t a t Then a a and is a lift of the Frobenius

i i

ip

Thus from Prop osition one has

M

QC W QH BU

n

pn

in M hence one has the Husemoller splitting

V

O

CW H BU

n

pn

To b e fair these arguments are not so dierent that than the original ones

The Implications of a lift of the Frobenius

The main purp ose of this section is to prove Lemma that a Hopf algebra

with a lift of Verschiebung which is also nitely generated as an algebra is a p oly

nomial algebra This requires as examination of the dual Hopf algebra and here I

am very indebted to ideas of Bouseld

To keep the record straight note that some niteness hypothesis is necessary

for example if H is the primitively generated symmetric algebra over Z on Q

p p

with trivial lift of the Verschiebung then H is not a p olynomial algebra

The following is the algebra input It is a consequence of Nakayamas Lemma

Lemma Let f M N be a homomorphism of nitely generated Z

p

modules Then f is an isomorphism if and only if

F f F M F N

p p p

is an isomorphism

This result has the following immediate corollary

Proposition Let A is a graded connected torsionfree Z algebra and

p

suppose A is nitely generated as an algebra Then A is a polynomial algebra if and

only of F A is a polynomial algebra p

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



If H is a nitely generated torsionfree Hopf algebra over Z then we may apply

p

Borels structure theorem to F H and conclude that there is an isomorphism of

p

algebras

O

n

i

p

F a a F H

p i p

i

where n is an integer n Thus F H is a p olynomial algebra if

i i p

and only if the Frobenius is injective Equivalently one need only show that the

Verschiebung on the dual Hopf algebra F H is surjective This will b e proved

p

b elow see Corollary

Because we are considering the dual F Hopf algebra F H we b egin by

p p

H Z Since H has a lift of considering the dual Z Hopf algebra H Hom

p p Z

p

the Verschiebung H H the dual H has a lift of the Frobenius We

now give a short investigation into the category of such ob jects

Let AF b e the category of torsionfree graded connected Z algebra equipp ed

p

with a lift of the Frobenius Let M b e the category of torsionfree p ositively graded

Z mo dules

p

Lemma The augmentation ideal functor I AF M has a left adjoint

S Furthermore for al l nitely M M S M is isomorphic to a polynomial

algebra

Proof First supp ose M is nitely generated Cho ose a homogeneous set of

generators fy g for M If A AF and f M IA is given the Dwork Lemma

i

supplies a unique map

g Z x x x A

i p

n

so that w f y Thus one obtains a unique map in AF

n i

O

Z x x x A g S M

p

i

so that the evident comp osite

M IS M IA

is f The clause stipulating that S M is a p olynomial algebra follows from

Equation

To nish the argument note that the construction of S M is natural in M

at least where so far dened that is for nitely generated M For general M

write M colim M where M M runs over the diagram of nitely generated

submo dules of M Dene S M colim S M since the diagram is ltered the

colimit in AF is isomorphic to the colimit as graded mo dules One easily checks

S M has the requisite universal prop erty

Now let HF b e the category of Hopf algebra with a lift of the Frobenius and CA

the category of graded connected torsion free coalgebras over Z Let J CA M

p

b e the coaugmentation coideal functor that is JC cokerZ C

p

Proposition The forgetful functor HF CA has a left adjoint F Fur

thermore for if C CA if nitely generated as a Z module in each degree then

p

F C is a polynomial algebra indeed as algebras

F C S JC

PAUL G GOERSS

Proof Dene F C to b e the algebra S JC with the diagonal induced by

completing the following diagram using the universal prop erty of S

C C C /

 

JC S JC S JC

S ❴❴❴

/

Then one easily checks F C HF fullls the conclusions of the result

If A is any torsionfree Z algebra equipp ed with a lift of the Frobenius dene

p

a function A A by the formula

p

x x p x

Compare for the following result

Lemma For al l x and y in A

p p

xy xy x y p x y

For al l x and y in A

p

X

p

pi i

x y x y x y

i

p

i

Proof These are b oth consequences of the fact that is an algebra map

Note that these formulas imply that if A is equipp ed with an augmentation

A Z then the augmentation ideal is closed under

p

The op eration may clarify the structure of free ob jects in AF Compare

Lemma Suppose M is a free Z module in each degree Let fy g M be

p i

a set of generators for M Then there is an isomorphism of Z algebras

p

n

S M Z y j n

p i

Proof By the construction of Lemma it is sucient to consider the case

where M has one generator in degree m Then

S M Z x x x CW

p m

in AF where w w Thus one need only show

n n

x x

n n

mo dulo p and indecomp osables The formula

p

w w w p w

n n n

n

and the formulas of Lemma imply

n n

p x p x

n n

n n n pn n

mo dulo p and decomp osables Since p p p p p Lemma

implies

n n

p x p x

n n

n

mo dulo p and decomp osables as required

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Next let H HF b e a Hopf algebra with a lift of the Frobenius and let

F H F H b e the Verschiebung If x H let fxg denote the class of x

p p

in F H H pH

p

Lemma For al l x H there is a congruence

f xg fxg

modulo the decomposables in F H

p

Proof First supp ose H is a p olynomial algebra Then F H is a p olynomial

p

p

algebra We will use the formula y py where p is p times the identity

map in the ab elian group of Hopf algebra endomorphisms of F H Thus it is

p

sucient to show

p

pf xg fxg

mo dulo decomp osables Since p is a morphism of Hopf algebras and commutes

with the lift of the Frobenius one has p x px Now px px z where

z IH Hence

p

X

p

pi i

px z px px z

i

p

i

The last term is zero mo dulo p Lemma implies that IH IH Hence

px px mo dulo p and indecomp osables But

p p

px px p x p p x

p

p p

p p p

x p x p p x

p

p

x mo d p

For the general case of H x x H and let C H b e a coalgebra so that

x C and so that C is nitely generated over Z Consider the induced map

p

F C H given by Corollary Then F C is a p olynomial algebra and this

result follows from the naturality of and

This has the following immediate consequence

Proposition Let H be a graded connected torsionfree Hopf algebra over

Z equipped with a lift of the Frobenius The the Verschiebung

p

F H F H

p p

is surjective

Proof The previous result shows that is surjective on indecomp osables

This implies is surjective

For completeness we now add

Corollary Let H HV be a graded connected torsionfree Hopf algebra

over Z equipped with a lift of the Frobenius If H is nitely generated as an algebra

p

then H is a polynomial algebra

Proof Combining Lemma with the remarks following that result we need

only show that the Verschiebung is surjective on F H This follows from the

p

previous result and the fact that H has a lift of the Frobenius

PAUL G GOERSS

Dieudonne theory

Positively graded bicommutative Hopf algebras over a p erfect eld form an

ab elian category with a set of pro jective generators As such this category is equiv

alent to a category of mo dules over some ring Dieudonne theory says which mo d

ules over which ring In this section we use the results of the previous sections to

elucidate the case k F Here the main classication result is due to Schoeller

p

the main advance is that we gave a formula of Theorem for computations

Let HA b e the category of graded connected bicommutative Hopf algebras

over F We describ e a go o d set of generators for this category In this context a

p

set of ob jects fA g is a set of generators if every ob ject is a quotient of a copro duct

of the A

k

Let n b e a p ositive integer n p m where p m Consider the

torsion free Hopf algebra CW k Z x x x of Denition It has the

m p k

Witt vector diagonal Dene

F x x x H n F CW k

p k p m

There are many pro ofs of the following result A very conceptual argument due to

Fabien Morel can b e found in

Lemma The Hopf algebras H n are projective in HA and form a set of

generators Furthermore is H n H n is the Verschiebung then

x x

i i

Here and elsewhere in this do cument one takes x to b e zero

Of all the morphisms in HA b etween the various H n we emphasize two The

rst is in the inclusion

v H n F x x F x x H pn

p k p k

For the second we note that Prop osition implies that there is a unique map

of Hopf algebras f H pn H n so that the following diagram commutes

f

H pn n / H ■■ ■■

■ v

■■

p ■

■$ 

H pn

Note that by construction v f p and that Prop osition also implies that

f v p

Definition If H HA the Dieudonne mo dule D H is the graded ab elian

group fD H g with

n n

D H Hom H n H

n HA

and homomorphisms

F f D H D H

n pn

and

V v D H D H

pn n

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



It follows from the remarks on v and f ab ove that FV V F p Also

Prop osition implies that the order of the identity in Hom H n H n is

HA

s s s

p if n p k with p k hence we also have p D H This suggests

n

the following denition

Definition Let D b e the category of graded mo dules M equipp ed with

s

s

op erators F and V so that FV V F p and p M if k p This is

p k

the category of Dieudonne mo dules

Thus we have a functor D HA D

Some familiar functors on Hopf algebras can b e recovered from the Dieudonne

mo dule Let b e the doubling functor on graded mo dules that is M M

pn n

and M of p do es not divide k Then for example if M is a Dieudonne

k

mo dule V denes a homomorphism of graded mo dules V M M

Lemma Let H HA then there is an exact sequence of graded abelian

groups

V

PH D H D H

There is also an exact sequence

F

D H D H QH

Proof For notice there is a short exact sequence of Hopf algebras

v

F H n H pn F x F

p p k p

where v denes V

Part is proved in a similar manner after the introduction of some auxiliary

technology See for example

Proposition Let f H K be a morphism in HA If the induced

homomorphism D f D H D K is an isomorphism then f is an isomorphism

Proof From Lemma we have that b oth Qf QH QK and P f PH

PK are isomorphisms Hence f is an isomorphism

A crucial calculational result is the following

Proposition The homomorphism

D Hom H n H m Hom D H n D H m

HA D

is an isomorphism

This is a consequence of a more general calculation which we give b elow in

Theorem See Corollary An immediate consequence of Lemma and

Prop osition is the following result which is Schoellers Theorem The pro of

follows from standard techniques in ab elian category theory See for example the

pro of of Mitchells Theorem in

Theorem The functor D HA D has a right adjoint U and the pair

D U form an equivalence of categories

As mentioned Prop osition follows from a much more general calculation

To set the stage let R b e the graded ring

R Z V F V F p p

PAUL G GOERSS

where the degree of V is and the degree of F is Then an ob ject M D

may b e regarded as a graded R mo dule provided one adopts the convention that

for a R and x M

dega

degax p x

With this convention D is exactly the category of p ositively graded R mo dules

provided we agree that ax if deg ax is a fraction Similarly we have a category

of p ositively graded Z V mo dules M with deg V and the same convention

p

on degrees Let us call this category D The category M of Denition is

V V

the full sub category of D with torsion free ob jects There is a forgetful functor

V

D D and it has a left adjoint given by

V

M R M

Z V

p

Now let H HV b e a Hopf algebra with a lift of the Verschiebung We wish to

give a formula to calculate D F H Since H is determined by QH M D

p V V

wed like this formula to b e functorial in QH

We can dene a natural homorphism QH D F H as follows Write

p

k

n p m with p m Then Theorem and Prop osition supply an

isomorphism

QH Hom CW k H

n HV m

and the morphism V QH QH is induced by the inclusion CW k

pn n m

CW k In degree n dene to b e the comp osition

m

Hom CW k H Hom H n F H D F H QH

HV m HA p n p n

Then is a morphism in D and hence it extends to a natural map of Dieudonne

V

mo dules

R QH D F H

H p

Z V

p

The following is the main result of this section

Theorem The map is a natural isomorphism of Dieudonne modules

H

for al l H HV

This will b e proved b elow after some preliminary calculations

Lemma Let M D be torsionfree and R Z V F V F p the

V p

Dieudonne ring Then for al l s

s

R M Tor

Z V

p

Proof Because Tor commutes with ltered colimits we may assume M is

nitely generated Then if M n M is the submo dule of elements in degree n

or less one has exact sequences

M n M n M nM n

in M and M nM n is isomorphic to a direct sum of mo dules which Z n

V p

s

meaning a single copy of Z in degree n If Tor R Z n for all n then

p p

Z V

p

a simple induction argument nishes the pro of

Dene a complex of right Z V mo dules

p

M M

y Z V x Z V R

s p s p

s s

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



s

with x F and y x p x V This is a pro jective

s s s s

resolution of R as a right Z V mo dule Tensoring with Z n yields a complex

p p

M M

y Z n x Z n R Z n

s p s p p

Z V

p

s s

One calculates that y x p Hence is an injection and the result

s s

follows

We also need a calculation

Lemma Let F x be the primitively generated Hopf algebra on an element

p

of degree n and f H n F x the unique map of Hopf algebras so that f x x

p k

and f x for i s Then D F x is the free module over F F on f

i p p

Proof Since the Verschiebung is zero on F x Lemma and the denition

p

of V on D F x show that V on D F x The result now follows from

p p

Lemma

We now give the pro of of Theorem

Proof Since b oth the source and target of commute with ltered colimits

H

we may assume that H is nitely generated and hence a p olynomial algebra See

Prop osition Since QH has a nite ltration

M M M QH

s

so that M M is a direct sum mo dules of the form Z n Theorem implies

k k p

that H has a nite ltration

Z H H H H

p s

so that QH M and Z H is a primitively generated p olynomial algebra

k k p H k

k +1

with indecomp osables isomorphic to M M Both the source and target of

k k

are exact on this ltration here we use Lemma and we are reduced to

H

the case where H is a primitively generated p olynomial algebra Since b oth the

source and target of commute with copro ducts we may assume H Z x with

H p

degx n Then it is a matter of direct calculation The mo dule QH Z n

p

Hom CW k H and we can choose as generator of the latter group the map

HV m

CW k Z x x x Z x H

m p k p

sending x to x and x to if i k Let f D F H b e the reduction Now

k i n p

R QH is the free mo dule over F F R Z on f Hence the result

p p

Z V Z V

p p

follows from Lemma

k

Now write n p m with m p and let D H n b e the identity

n n

Corollary Let H n F x x x be one of the projective gen

p k

erators of HA Then for al l Dieudonne modules M then the natural map

D H n M M Hom

n D



given by sending f to f is an isomorphism Furthermore

n

R K n D H n

Z V

p

i

where K n is the Z V module with K n unless m p k i s

p m

i

i

Zp Z and V is onto K n

p s

PAUL G GOERSS

Proof This follows from Theorem and Prop osition

Example Lets calculate the Dieudonne mo dule of H BU H B U F

p

We could use Example and Theorem but heres another way more suited

to a later application Consider the coalgebra H C P Z This is dual to

p

Z x which has a lift of the Frobenius given by the algebra H C P Z

p p

p

x x Thus H C P Z has a lift of the Verschiebung indeed if we dene

p

i

H C P Z to b e dual to x then where if pi is a

i i p i

pi pi

fraction Now

H B U Z S H C P Z Z b b

p p p

where we write b for the image of Theorem now implies that

i i

D H BU Z R H C P

p

Z V

p

with V In particular if we also write b for the image of in D H BU

i i i i

pi

then

i

D H BU Zp Z

i

k

generated by b where i k if i p j with j p Furthermore V b b

i i

ip

and this forces F b pb

i pi

Part I I Bilinear Maps Pairings and Ring Ob jects

Bilinear maps and the tensor pro duct of Hopf algebras

First we establish a categorical framework for tensor pro ducts then sp ecialize

to our main interestthe category of Hopf algebras over over a eld Much of the

ideas ab out tensor pro ducts in the category of coalgebras can b e found in

Let C b e a category and A C a subcategory of ab elian ob jects in C Thus

for all A A

op

F Hom A C Ab

A C

is a functor to ab elian groups Morphisms f A B in A induce natural trans

formations F F of groupvalued functors

A B

We will assume for simplicity that

b oth C and A have all limits and colimits

the forgetful functor A C has a left adjoint S

Definition Let A B and C b e ob jects in A A morphism A B C

in C is a bilinear map if for all X C the induced map

F X F X F X

A B C

is a natural bilinear map of ab elian groups

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



This is equivalent to demanding that the following diagrams commute

m

B

A B B B

A /

T A

 

A B A B PP tt9 C PP tt PPP tt

P m

C

PP tt

P' tt

C C

m

A

A B A B

A /

T B

 

A B A B PP tt9 C PP tt PPP tt

P m

C

PP tt

P' tt

C C

Here m A A A is the multiplication is the diagonal and T is the twist

A A

map interchanging factors

We now dene tenor pro ducts The symbol will b e reserved for the tensor

pro duct of mo dules over rings and hence for the pro duct of coalgebras

Definition A tensor pro duct of A B A is an initial bilinear map

A B A  B in A Sp ecically if A B C is any bilinear map there

is a unique morphism A  B C in A making the diagram

A B

 B / A ❋❋ ✇ ❋❋ ✇✇

❋❋ ✇✇

❋ ✇ ❋❋ ✇

# {✇✇

C

commute in C

Remark If tensor pro ducts exist they are unique up to isomorphism in

A Also there is then a natural transformation of groupvalued functors

F F X F X F X F X

A B A B

A  B

This need not b e an isomorphism consider the case where C is the category of sets

A the category of ab elian groups and A B Z

The assumptions made ab ove make it easy to show tensor pro ducts exist See

Proposition Under the assumptions of any two objects A B A

have a tensor product A  B in A

Proof This is an adaptation of the case where A is the category of R mo dules

for some ring R Dene A  B to b e the colimit of the diagram in A

f f

A B

S A A B S A B S A B B  

g g

A B

PAUL G GOERSS

where f is adjoint to

A

m

A

A A B A B S A B

and g is adjoint to

A

m

A A B A B A B S A B S A B S A B

where the rst map is T The morphism f and g are dened

B B B

similarly Let A B A  B b e dened by the comp osite

A B S A B A  B

it is bilinear by construction Also if A B C is any bilinear map the

induced morphism

S A B C

factors uniquely through A  B

We now b egin to sp ecialize to the case where A is a category of bicommutative

Hopf algebras over a commutative ring k and C is a category of coalgebras In this

case a bilinear map is a morphism of coalgebras

H H K

where H H and K are Hopf algebras It is convenient to write x y for x y

Lemma For Hopf algebras the fol lowing formulas hold

For al l x y H and z H

X

x z y z xy z

i

i

i

where z z z A similar formula holds for x H y z H

i

i

If H is the unit of H regarded as an algebra and x H then

x x where H k is the augmentation

Proof Part merely rewrites in formulas what it means to b e bilinear For

k More generally given A and C as in let it is sucient to show k  H

A b e the terminal ob ject Then for all A A  A To see this notice

that if A C is a bilinear map the induced bilinear map

F X F X F X

A C

is the trivial map since it factors through F X F X and F X Now

A

consider the pair

F A F A

A A

where A is the unique map This maps to in F A so there is a diagram

C

A /

 

A

/ C

where is the unit map in C from to C This implies that the unique map

A is the initial bilinear map as required

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



For the next few results we will stipulate only that C is a category of co com

mutative coalgebras over a ring k and that A is a category of bicommutative Hopf

algebras over k We will abbreviate this by saying A is a category of Hopf algebras

and C is a category of coalgebras

Lemma Let A be a category of Hopf algebras and C a category of coalge

bras Then for any diagram in A the natural map

colimA  B colim A  B

is an isomorphism

Proof This result is true in much more general contexts however it do es

require that the ab elian category A satisfy certain axioms which can b e uncovered

by examining the argument

f f

2 1

A b e a diagram of Hopf algebras A First consider pushouts Let A

A that is the algebra A A mo dulo the Then the pushout in A is A

A

12

ideal generated by elements of the form

f ax y x f ay

Supp ose one has a diagram

f

2

A B A B

/

f

2 1

 

A B C

/

1

where and are bilinear Dene a bilinear map

A B C A

A

12

by

X

x b y c x y b

i i

i

where b b c The asso ciativity of the diagonal the formula and

B i i

Lemma imply is welldened Now take

C A  B A  B

A  B

12

and and the evident bilinear maps Then there is a resulting bilinear map

A B A  B A  B A

A

A  B

12

12

We leave it as an exercise to show has the requisite universal prop erty to prove

the result in this case

Since the result is true for pushouts it is true for nite copro ducts and co

equalizers I claim it is true for all copro ducts In fact I claim it is true for ltered

colimits so the claim ab out copro ducts follows from the fact that any copro duct

is the ltered colimit of its nite subcopro ducts To see that it is true for ltered

colimits one uses the construction given in the pro of of Prop osition and the fact

that in this case the left adjoint S commutes with ltered colimits in the sense

that for any diagram of coalgebras the natural map in C

A C C

S C colim C S C S colim colim

is an isomorphism that is the forgetful functor from A to C makes ltered colimits

PAUL G GOERSS

Finally the result is true for all colimits b ecause there is for any diagram A

in A a co equalizer diagram

a a

A  A colim A

Hence the general result follows from the result on copro ducts and co equalizers

We next calculate the tensor pro duct of free ob jects

Corollary Let A be a category of Hopf algebras and K A Then the

functor A A given by H H  K has a right adjoint

Proof This is a consequence of the sp ecial adjoint functor theorem and

the fact that  K commutes with all colimits

It is appropriate to call this functor homK so that one has a formula

Hom H  K H Hom H homK H

A A

It would interesting to have a concrete description of this homomorphism ob ject

and its Dieudonne mo dule

Bilinear pairings for Hopf Algebras with a lift of the Verschiebung

We now examine the case where A HV the category of connected torsion

free Hopf algebras over Z equipp ed with a lift of the Verschiebung and C CV

p

the category of connected torsion free coalgebras over Z also equipp ed with a lift

p

of the Verschiebung These ob jects were discussed in some detail in section The

forgetful functor HV CV has left adjoint S if C CV S C is the symmetric

algebra on

JC kerf C Z g

p

with diagonal induced from C and lift of the Verschiebung given by extending the

left on C Thus HV has the pairing 

The indecomp osables functor Q denes a functor from HV to the category M

V

of graded Z V mo dules M which are torsionfree as Z mo dules If x M then

p p k

V x M wwhere we mean V x if p do es not divide k In Theorem it was

k p

noted that this indecomosables functor has a right adjoint S and in Theorem

we showed that these two functors give an equivalence of categories

Now let H H and K b e in HV and

H H K

b e a bilinear map in CV Then b ecause is a morphism in CV commutes with

the lifts of the Verschiebung by denition Lemma now implies that induces

a pairing

Q QH QH QK

where writing x y for Qx y one must have

V x y V x V y

Thus we should give QH QH the structure of an ob ject in M by dening

V

V x y V x V y

Now let H S M and H S N b e two ob jects in HV

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Proposition There is a natural isomorphism in HV

S M  S N S M N

Proof We dene a bilinear pairing S M S N S M N and then

we will show it has the correct universal prop erty

First note that if C is a coalgebra with a lift of the Verschiebung one can regard

the coaugmentation ideal JC as an ob ject in M and the functor J CV M

V V

has as right adjoint S where we forget the algebra structure To see this let

S CV HV b e left adjoint to the forgetful functor Then

Hom C S M Hom S C S M

CV HV

Hom QS C M

MV

J C M Hom

M

V

With this in hand let

S M S N K S QK

b e any bilinear map in CV The adjoint

J S M S N QK

factors by Lemma as

Q

J S M S N J S M J S N M N QK

where Q is an in equation This supplies a factoring of



S Q

S M S N S M N S QK K

If we can show the rst map is bilinear well b e done To do this we give another

construction of

Note there is an obvious bilinear pairing

J C M N J C N Hom J C M Hom Hom

M M M

V V V

sending a pair f g to the comp osition

f g

J

JC J C C JC JC M N

Thus we get a bilinear pairing

Hom C S M Hom C S N Hom C S M N

CV CV CV

and hence a bilinear pairing

S M S N S M N

Note that is obtained by applying to the two pro jections S M S N

S M and S M S N S N Hence the morphism of coalgebras is

adjoint to the morphism in M

V

M S N JS M JS N M N J S

where the last map is pro jection onto the indecomp osables Since and are

adjoint to the same map and since is bilinear must b e bilinear

PAUL G GOERSS

The universal bilinear map

S M S N S M N

induces a map via equation

Q QS M QS N QS M N

which ts into the following diagram

QS M QS N M N / QS

 

N M N

M /

The vertical maps are induced by the isomorphism of functors QS

Thus we have proved

Corollary If H and H are two Hopf algebras in HV the universal

bilinear map

H H H  H

induces an isomorphism in M

V

QH QH QH  H

In the next result we are concerned with free Hopf algebras If C CV is a

torsionfree coalgebra with a lift of the Verschiebung let S C b e the free Hopf

algebra in HV on C It is the symmetric algebra on the coaugmentation ideal of

C equipp ed with the copro duct and the lift of the Verschiebung induced from C

Given C and C in CV let C C b e the algebraic smash pro duct of C and

C This is dened as follows First C C is dened by the pushout diagram

Z Z

C C

p p /

 

Z

C C

p

/

where Z Z Z is addition Then C C is dened by the pushout diagram

p p p

C C C C

/

 

Z

C C

p

/

If S C S C K is a bilinear map in HV then the comp osite

C C S C S C K

factors in CV through C C by Lemma Thus one gets a map in HV

S C C K

Corollary The induced map in HV

S C C S C  S C

is an isomporphism

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Proof There is a natural isomorphism QS C JC where JC is the coaug

mentation ideal of C The result now follows from Corollary and Theorem

Universal bilinear maps for Hopf algebras over F

p

Let HA b e the category of graded connected Hopf algebras over F The

p

purp ose of this section is to calculate the Dieudonne mo dule of the target of the

universal bilinear map H K H  K as a functor of D H and D K in HA In

summary there will b e an induced pairing D H D K D H  K which while

not an isomorphism can b e mo died in a simple way to pro duce an isomorphism

The exact result is b elow in Theorem

The rst observation is this Let H and K b e ob jects in HV the category of

graded connected torsionfree Hopf algebras over the Z equipp ed with a lift of the

p

Verschiebung Let H K H  K b e the universal bilinear map in that category

Lemma Suppose H and K in HV are nitely generated as Z modules in

p

each degree Then the induced map

F H  F K F H  K

p p p

is an isomorphism

Proof By construction F H  F K is a quotient of of the symmetric

p p

algebra S F H F K See Prop osition This fact and Lemma

p p

imply that if fx g is a homogeneous basis of QF H F QH and fy g is a

i p p j

homogeneous basis of QF K then fx y g spans QF H  F K

p i j p p

But Corollary implies that the elements x y are linearly indep endent in

i j

QF H  K Hence

p

QF H  F K QF H  K

p p p

is an isomorphism Since H  K is a p olynomial algebra by Prop osition the

result follows

Remark The nite type hypothesis can b e removed by a ltered colimit

argument or by an application of Theorem b elow

In the following result we are primarily concerned with generators H n of

HA Recall from Equation that

H n F CW k F x x

p m p k

k

where n p m with p m For simplicity write

Gn QC W k

m

Corollary Suppose H and K in HV are nitely generated as Z modules

p

in each degree Then there is an isomorphism

R QH QK D F H  F K

p p

Z V

p

In particular if H n HA are the projective generators

R Gn Gm D H n  H m

Z V

p

is an isomorphism

PAUL G GOERSS

Proof The rst isomorphism follows from Theorem and Corollary

The second isomorphism follows from the rst

We now can dene the pairing D H D K D H  K using the metho d of

Gn M universal examples By Prop osition if M M then Hom

V M

V

M Let Gn corresp ond to the identity By abuse of notation also write

n n n

for the reduction of this class in

n

D H n R Gn

Z V

p

Then Corollary says the function

Hom D H n M M

D n

given f f is an isomorphism In particular if H HA the natural isomor

n

phism

Hom H n H D H

HA n

is dened by f D f

n

Now let D H n  H m R Gn Gm b e the evident

n m

F V

p

class If H H K is a bilinear pairing of ob jects in HA we get a pairing

D H D H D K

n m nm

as follows If x D H and y D H we get a diagram

n m

H n  H m n H m

H /

g

f f

x y

 

H H

K

/

where D f x and D f y and g is the unique Hopf algebra map

x n y m

lling the diagram Then

x y D g

n m

Notice that element D H n  H m is represented by the map g in

n m nm

the ab ove commutative diagram

Lemma This pairing is bilinear and induces a pairing of graded modules

D H D H D K

Proof It is sucient to examine the universal example

H n H n H m H n H n  H m

If i D H n H m D H n D H n we need

n n n

n n

m m m

n n n n

But the bilinear map of is the reduction mo dulo p of a bilinear map of ob jects

j k

in HV Indeed write n p s and m p t where p s p t Then the

pairing of is the reduction of pairing

CW j CW j CW k CW j CW j  CW k

s s t s s t

which induces a pairing on indecomp osables which is an isomorphism

Gn Gn Gm QCW j CW j  CW k

s s t

Here the formula is obvious

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Because the isomorphism

QC W j QC W k QCW j  CW k

s t s t

is an isomorphism in M similar metho ds imply

V

Lemma If H H K is a bilinear pairing of Hopf algebras over F

p

then the pairing

D H D H D K

has the property that

V x y V x V y

The op erator F b ehaves dierently In fact if F is supp osed to reect the

p

Frobenius and V the Verschiebung then one calculates

p p

x y x y

using Lemma Then one might exp ect

Lemma The pairing D H D H D K has the property that

F x y F x V y and x F y F V x y

Proof Again one need only calculate

F F V

n m n m

k j

in D H n  H m To do this write m p t and n p s where s p

t p Let CW Z x x and CW Z y y b e the Hopf

s p t p

algebras of Denition Then

CW j Z x x x CW

s p j s

and similarly CW k CW Now

t t

r

Z i p s

p

QC W

s i

otherwise

and x we confuse x with its image in the indecomp osables generates the mo dule

r r

r

of indecomp osables QC W Also V x x by Lemma Finally

s p s r r

QC W j CW is a split injection of Z mo dules and x

s s p n j

Notice that H n  H m F CW  F CW is an injection

p s p t

since

QC W j QC W k QC W QC W

s t s t

is a split injection and b oth source and target are p olynomial algebras Thus

D H n  H m D F CW  F CW

p s p t

is an injection Now one calculates

F F x y FV x y

n m k j k j

px y

k j

FV x y

k j

F x V y

k j

F V

n m

PAUL G GOERSS

These results prompt the following denitions A function f M M N

of graded ab elian groups will b e called a graded pairing if

deg f x y deg x degy

A graded pairing f M M N of Dieudonne mo dules will b e called bilinear

in D if it is bilinear as a pairing of graded ab elian groups and

V f x y f V x V y

F f x V y f F x y f x F y F f V x y

Then the content of Lemmas is that a bilinear map of Hopf algebras over

F induces a bilinear pairing in D

p

D H D H D K

A universal bilinear pairing in D written M M M  M is an initial

D

bilinear pairing in D out of M M in the obvious sense If it exists it is unique

We will show it exists and give a concrete description

Indeed for the purp oses of the next few paragraphs dene a Dieudonne mo dule

M  N as follows For the graded tensor pro duct M N then this is a Z V

D p

mo dule with

V x y V x V y

Our grading conventions are the exp onential conventions given in Equation Let

R Z F V and

p

M  N R M N K

D

Z V

p

where K is the subDieudonne mo dule generated by the relations

F x V y F x y

F V x y x F y

If M M N is any bilinear pairing in D there is a morphism of Dieudonne

mo dules M  M N Also the function

D

M M M  M

D

given by x y x y is a bilinear pairing in D One now easily checks that has

the requisite universal prop erty so D has universal bilinear maps

The following is the main result of this section

Theorem Let H and K be Hopf algebras over F Then the induced map

p

of Dieudonne modules

D H  D K D H  K

D

is an isomorphism

Proof We reduce to a sp ecial case Let S CA HA b e left adjoint to

the forgetful functor Then the Hopf algebras S F C with C CV nitely

p

generated in each degree generate HA This follows from the fact that the Hopf

algebras H n F CW k generate HA so the Hopf algebras F H H HV

p m p

with H nitely generated in each degree generate HA and HV is in turn generated

by Hopf algebras of the form S C with C CV nitely generated in each degree

See Lemma

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Thus we may write equations

colim S F C H and colim S F C K

p p

for suitable diagrams where C and C are ob jects in CV Since b oth the source

and target of the natural map

D H  D K D H  K

D

commute with colimits see Lemma we may assume that H S F C

p

and K S F C with C and C in CV and nitely generated in each degree

p

To prove the result in this case Lemma implies there is a natural isomor

phism

F S C  S C S F C  S F C

p p p

Then Corollary and Theorem complete the calculation

D S F C  S F C D F S C  S C

p p p

D F S C C

p

R JC JC

Z V

p

R JC  R JC

D

Z V Z V

p p

The pairings  on HA and  on D are symmetric and the isomorphism of

D

Theorem reects the symmetry Note that if t H H H H is the

switch map the comp osite

t

H H H  H H H

is bilinear and induces an isomorphism

t H  H H  H

The following can b e proved by reducing to the case of universal examples as

in Lemma

Lemma Let H and H be Hopf algebras in HA Then the fol lowing dia

gram commutes

D H  H

D H  D H

D

/

D t

t  

 

D H  H

D H  D H

D

/

A similar statement holds for HV

Example Supp ose M N are Dieudonne mo dules and V is surjective on

M and N Then one can dene the structure of a Dieudonne mo dule on M N

as follows First V x y V x V y Second if x M write x V z Then one

denes

F x y z F y

If V z V z then z z w where pw Write y V y Then

z F y z F y w FV y z F y

PAUL G GOERSS

so F is welldened One easily checks V F FV p Now consider the inclusion

j M N M  N

D

To start this is only a morphism of Z V mo dules However

p

j F x y z F y F x y

F j x y

so this is a morphism of Dieudonne mo dules I claim it is an isomorphism To see

this we use the universal prop erty of M  N Supp ose f M N K is a

D

bilinear pairing in D Then we need only show there is a unique map of Dieudonne

mo dules

g M N K

making the obvious diagram commute It is required then that g x y f x y

It follows that g commutes with V That g commutes with F follows exactly as in

Equation

Symmetric monoidal structures

The categories of bicommutative Hopf algebras considered in the previous two

sections are nearly symmetric monoidal categories what is missing is a unit ob ject

e so that e  H H  e H In order to supply this ob ject we must extend the

category somewhat

Let H b e a nonnegatively graded Hopf algebra over a commutative ring k

Then H H the elements of degree form a Hopf algebra over k With an eye

to top ological applications we insist that H b e a group ring Put another way let

X H Hom k H

CA

k

where CA is the category of nonnegatively graded coalgebras over k Then X H

k

is an ab elian group and evaluation at k denes an injection X H H and

hence an injection of Hopf algebras

k X H H

where k X H is the group algebra We will insist that k X H H b e an

isomorphism and say H is grouplike in degree

More generally if C CA we can dene

k

X C Hom k C

CA

k

and get an injection k X C C where k X C the free k mo dule on X C is

now only a coalgebra If this is an isomorphism we will say C is setlike in degree

H Finally if H is a nonnegatively graded Hopf algebra over k let H k

c H

0

b e the connected comp onent of the identity Then there is a natural isomorphism

H H H

c k

Such a splitting fails for coalgebras The map H H is an isomorphism if and

c

only if H is connected

We b egin with a category of torsionfree Hopf algebras over Z p

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Definition Let HV b e the category of torsionfree bicommutative Hopf

algebras

i grouplike in degree

ii are equipp ed with a Hopf algebra map lifting the Verschiebung and

iii in degree

The last hypothesis is inno cuous as for all H HV the Verschiebung

F H F H is the identity in degree zero The underlying coalgebra category

p p

will b e called CV it consists of torsionfree coalgebras C over Z equipp ed with

p

a coalgebra map C C lifting the Verschiebung Again we ask that in

degree

Lemma The forgetful functor HV CV has a left adjoint F

Proof Let C CV and S C b e the symmetric algebra on C endowed with

the obvious diagonal This is not yet a Hopf algebra In fact let X X C Then

there is an isomorphism

Z N X S C

p

where N X is the free ab elian monoid on X Set

FC Z ZX S C

p Z NX

p

Note that FC is an appropriate group completion of S C

To classify HV as a category of mo dules we extend the category M of

V

to b e the category of graded ab elian section as follows dene a new category M

V

groups M equipp ed with a shift map V M M so that M is a torsion free

pn n n

then the elements M M Z mo dule if n and V M M If M M

c p

V

M M of p ositive degree form a subob ject and the natural isomorphism M

c

denes an equivalence of categories

M M Ab

V

V

Here Ab is the category of ab elian groups This will reect the isomorphism of

Equation

We next dene a functor Q HV M by the equation

V

Q H QH X H

c

k

In degree n this functor is representable If n let n p s with n s and

CW k the Witt vector Hopf algebra of Denition Then we have by Theorem

s

and Prop osition that

Hom CW k H H om + CW k H QH Q H

HV s c s c n

HV

If n then

+

Q H X H Hom Z Z H

p

HV

The splitting of Equation and Theorem immediately imply

Proposition The functor Q denes an equivalence of categories Q

HV M

V

Now we turn to bilinear maps Prop ositions and Lemma imply that

the category HV has universal bilinear maps H K H  K Here is the rst

result on these

PAUL G GOERSS

Lemma Let A and B be abelian groups Then in HV

Z A  Z B Z A B

p p p Z

Let H HV be connected and A an abelian group Then Z A  H is

p

connected and

A QH T QZ A  H

Z p

where T A QH is the torsion subgroup

Z

For al l H HV there is a natural isomorphism

Z Z  H H

p

Proof Part is immediate from the universal prop erty and the requirement

that ob jects on HV are grouplike in degree

We next prove Let Z Z Z b e the generator Write Z multiplicatively

p

If f Z Z H K is any bilinear map let h H K b e the map hx

p

f x x This is a Hopf algebra map since is grouplike See Lemma

Also f is determined by h and bilinearity The claim is that there is a bilinear map

g making the following diagram commute

g

Z Z H H p / ✉✉ ✉✉ f ✉✉ ✉✉ h

 z✉✉

K

If so the result will follow If x H write the nfold diagonal of x

x x x

n i in

Then g is dened by bilinearity and

x x x n

i i in

n

x n

g x

x x n

i in

Here H H is the canonical antiautomorphism arising from the fact that as

a group ob ject H must supp ort inverses One easily checks g is bilinear

For part we rst prove Z A  H is connected Given any bilinear map

p

Z A H K we get a diagram of bilinear maps

p

Z A H K p /

 

Z A Z

K

p p

/

hence a diagram of Hopf algebras

Z A  H K

p /

 

Z

K p

/

since Z A  Z Z is an isomorphism Hence factors through K Now let

p p p c

b e universal bilinear map Z A H Z A  H then and Z A  H

p p p

is connected

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



To prove the assertion on indecomp osables note that part implies the result

for A Z For general A take a free resolution

F F A

Then since QZ  H commutes with all colimits one gets a short exact sequence

p

in M

V

F QH F QH QZ A  H

Z Z p

Since all ob jects in M are torsion free the result follows

V

This last result suggests how to dene and analyze universal bilinear maps in

M A bilinear pairing in M

V

V

f M M N

is a bilinear map of graded ab elian groups that is deg f x y degx deg y

so that

V f x y f V x V y

There is a universal bilinear map M N M  N Write M M M where

M c

M and M are the elements of degree and p ositive degree resp ectively Then

c

M  N is nearly M N mo dulo torsion in p ositive degrees

M

N M N M N T M N T M M  N

c c c c Z M

p

where T and T are the torsion submo dules Lemma and Corollary now

imply

Proposition Let H K H  K be the universal bilinear pairing in

the equivalence of categories Then there is a bilinear HV and Q HV M

V

pairing in M

V

Q H Q K Q H  K

and the induced map in M

V

Q H  MQ K Q H  K

is an isomorphism

For the following result we will say that a functor F C e C e is

an equivalance of categories with symmetric monoidal structure if F is an equiva

lence of categories F e e and there are natural isomorphisms

F X F Y F X Y

Corollary The category of Hopf algebras HV with pairing  is a

symmetric monoidal category with unit e Z Z

p

The category M with pairing  is a symmetric monoidal category with unit

M

V

e Z concentrated in degree

Q HV M is an equivalence of categories with symmetric monoidal

V

structure

Similar results hold over F Let HA b e the category of bicommutative Hopf

p

algebras over F grouplike in degree and let CA b e the category of co commu

p

p

tative coalgebras over F which are setlike in degree Then

p

Lemma The forgetful functor HA CA has a left adjoint

The corresp onding category of Dieudonne mo dules is easily dened

PAUL G GOERSS

Definition An ob ject M D is a nonnegatively graded ab elian group

equipp ed with op erators V M M F M M so that

pn n n pn

V F FV p

M D and

c

V M M

Notice as in Equation M D may b e regarded as a Z V F V F p

p

mo dule

Dene H n HA by H F Z and H n F x x x

p p s

k

CW k for n p m with m p Then one has a functor D HA D

m

given by

+

D H Hom H n H

n

HA

with V and F given as in Denition for n or Dention for n

Proposition The functor D HA D is an equivalence of cate

gories

Lemma and Prop osition imply HA has universal bilinear pairings

H K H  K Just as in Lemma one has

Lemma Let A and B be abelian group Then in HA

F A  F B F A B

p p p Z

Let H HA be connected and A an abelian group Then F A  H is

p

connected and

D Z A  H A D H

p

For al l H HA there is a natural isomorphism F Z  H H

p

To characterize bilinear pairings via Dieudonne mo dules we introduce the no

tion of a D bilinear map It is a mild generalization of the notion introduced in

section A pairing f M N K of ob jects in D is a bilinear pairing in D

if it is a bilinear pairing of graded ab elian groups and

f V x V y V f x y

f F x y F f x V y and f x F y F f V x y

There is a universal bilinear pairing

M N M  N

D

In fact we can write

M  + N R M N K

ZV

D

where R ZV F VF p and K is the submo dule generated by

F x V y F x y and F V x y x F y

This mimics the construction M  N of the previous section In fact if we write

D

M M M and N N N where the and c indicate elements of degree

c c

and p ositive degree resp ectively we have

Lemma There is a natural isomorphism in D

+

M  N M N M N M N M  N

Z Z c c Z c D c D

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Proof One has

+ + + + +

M  N M  N M  N M  N M  N

c c c c

D D D D D

+

M  N so one must identify the other three summands I Now M  N

c D c c c

D

+

claim M N M  N is an isomorphism To see this note that M N

Z c c Z c

D

has a structure of a Dieudonne mo dule with V x y V x V y x V y and

F x y x F y The bilinear pairing M N M N is a bilinear pairing

c Z c

in D and any bilinear pairing in D factors through this one Hence M N

Z c

has the required universal prop erty The other summands are handled the same

way

Now let H H K b e a bilinear pairing in HA The one gets an induced

pairing

D H D H D K

Indeed there are universal maps

H n m H n  H m

given by Equation for n and m and by Lemma for n or

m If f H n H and g H m H represent x D H and y D H

n m

resp ectively then x y is represented by

f  g

H n m H n  H m H  H K

Lemma This pairing is a bilinear pairing in D

Proof If n and m this follows from Lemmas If n

or m one only has to check the universal examples Thus for example the

pairing is bilinear b ecause one has a diagram



F Z F Z  H n F Z  H n

p p

p /

 

 

n H n H n

H /

using Lemma or rather the pro of of that statementsee Lemma and

the fact that  commutes with copro ducts

That V x V y V x y follows from the diagram

 v

F Z  H n F Z  H pn

p

/ p

 v 

n H pn

H /

where v induces V see Denition This implies F x y F x V y b ecause

F x px and

px y px y FV x y F x V y

Finally x F y F V x y F x y by using a diagram similar to of Equation

using f H pn H n which denes F in place of the morphism v

That completed note that Lemmas and Theorem will now imply

the following result The pro of is the same as for Prop osition

PAUL G GOERSS

Proposition Let H K H  L be the universal bilinear pairing in

HA Then the induced bilinear pairing in D

D H D K D H  K

induces an isomorphism

+

D H  D K D H  K

D

We now can write down the analog of Corollary

Corollary The categories HA and D are symmetric monoidal cat

egories and the equivalence of categories

D HA D

is an equivalence of categories with symmetric monoidal sturcture

Part I I I Hopf rings asso ciated to homology theories

Skew commutatative Hopf algebras

If X is a double lo op space the homology Hopf algebra H X H X F is

p

not commutative but skew commutative meaning that if x H X and y H X

m n

then

mn

xy y x

Similary the homology coalgebra of a space Y is skew co commuative This turns

out to b e only a mild variation on the commutative case considered up to now and

this section adapts Dieudonne theory and the theory of bilinear pairings to this

new situation

To make the denitions precise we work with nonnegatively graded vector

spaces over the eld F p The case p is the commutative case Then the

p

signed twist map of the tensor pro duct of two such ob jects

t V W W V

mn

is given on homogeneous elements by tx y y x with x V and

m

y W A skew commutative coalgebra over F is a coasso ciative coalgebra C

n p

setlike in degree zero so that the following diagram commutes

C C C ❋ / ❋❋ ❋❋ ❋❋ t ❋

❋# 

C C

The category of such will b e written CA Similary a skew commutative Hopf

algebra H is a Hopf algebra in the category of skew commutative coalgebras so that

the multiplication satises the formula of Equation The category of such Hopf

algebras will b e written HA

The reason this adaptation is simple is the following result known as the split

ting principle See

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Proposition Let H be a skew commutative Hopf algebra Then there is

a natural isomorphism in HA

H H H

ev odd

where H is concentrated in even degrees and H is an exterior algebra on prim

ev odd

itive generators in odd degrees

Notice that a skew commutative Hopf algebra concentrated in even degrees

is in fact commutative In eect Prop osition implies that there is a natural

equivence of categories

HA HA V

where V is the category of nonnegatively graded F vector spaces and HA is the

p

category of commutative Hopf algebras which are grouplike in degree zero

The splitting principle immediately translates into a Dieudonne theory for this

situation

Definition The category D of skew commutative Dieudonne mo dules

has as ob jects nonnegatively graded ab elian groups M equipp ed with homomor

phisms

F M M and V M M

n pn pn n

so that

FV V F p

V M M

k k

if n p m and m p then p M and

n

pM for all n

n

Note that the category of skew commutative Dieudonne mo dules can b e re

garded as a category of mo dules over the ring R ZV F VF p sub ject

to the exp onential grading conventions of Equation and the requirement that

V x F x if x is of o dd degree

Prop osition the assumption that our Hopf algebras are grouplike in degree

zero and Schoellers Theorem now immediately imply

Proposition There is an equivalence of categories

D HA D

The functor H D H is representable just as in the commutative case

n

k

Indeed if n p m with p m then

H n H D H Hom H n H Hom

n HA ev HA



where H n F CW k is the Witt vector Hopf algebra of Denition A

p m

similar remark holds in degree zero with H F Z If n m then let

p

F n b e the exterior algebra on a single primitive generator of degree n Then

p

F n P H F n H PH D H Hom Hom

p p n n F HA

p 

Here PH are the primitives If we write H n H n and H m

F m then these formulas combine to read

p

H n H Hom D H

HA n 

PAUL G GOERSS

We now come to bilinear pairings The forgetful functor from HA CA

has a left adjoint it is the free skewcommutative algebra functor suitably group

completed in degree zero Compare Lemma As a result Prop osition implies

that HA has bilinear pairings Wed like to compute these via Dieudonne mo dules

The following is the crucial result Let S denote the free skew commutative

algebra functor If W is a vector spaces the S W can b e made into a primitively

generated skew commutative Hopf algebra

Proposition Let V be the primitively generated exterior algebra on a

vector space V concentrated in odd degrees If H HA is connected then there

is a natural ismorphism of Hopf algebras

V  H S V QH

Proof Because  H commutes with colimits by Lemma we may

assume that V F n for some o dd integer n Let x F n a generator If

p p

F n H K

p

is any bilinear map then the formulas of Lemma imply that for all y and z in

the augmentation ideal of H

x y z x y z

Furthermore for any y the element x y K is primitive In fact there is a

factoring of

q

F n QH F n H p p / PPP PPP PPP PPP

P( 

PK

where q is the comp osite

F n H I F n IH QF n QH F n QH

p p p p

The vertical map extends to a Hopf algebra map S F n QH K and one

p

checks that

q

F n H F n QH S F n QH

p p p

is bilinear The result follows from the uniqueness of the universal bilinear maps

Exterior algebras and group rings b ehave in the exp ected way The pro ofs are

the same as those of Lemma

Lemma Let V a primitively generated exterior algebra on a vector space

concentrated in odd degrees

There is a natural isomorphism F Z  V V

p

A V For al l abelian groups A F A  V

p

Prop osition and Lemma also allow one to come to term with H  V

In fact if t H H H H is the signed switch map then t is an isomorphism

of Hopf algebras and the comp osition

t

H H H  H H H

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



is bilinear and induces an ismorphism

t H  H H  H

We now notice that the bilinear pairings dened on the various categories of

Dieudonne mo dules of the previous sections extends to D as well Regarding an

ob ject in D as a R ZV F VF p mo dule we dene for M N D

M R M N K M 

D

ZV



where K is the submo dule generated by

F x V y F x y and F V x y x F y

This mimics the construction M  N of the two previous sections The tech

D

niques of section now imply that for all H H HA there is a natural map of

Dieudonne mo dules

D H D H  H D H 

D



The exp ected result is the following

Theorem This natural map

D H D H  H D H 

D



is an isomorphism in D

Proof As in the pro of of Prop osition one splits up the bilinear pairing

in D and reduces the result to previous calculations

If M D then there is a natural splitting in D

M M M

ev odd

where M and M are the elements of even and o degrees resp ectively This

ev odd

reects the splitting principle of Prop osition From this it follows that

N M N F N M F M N M N N M  M 

ev odd ev ev ev ev odd odd odd ev D D

+ 

The result now follows from the splitting principle Prop osition the analagous

result for HA Prop osition Prop osition Lemma and Lemma which

says that for H HA there is a natural isomorphism

QH D H F D H

It is worth p ointing out that the natural isomorphism of the previous result

reects the skew symmetric nature of the category HA In the following let t

stand for any of the signed switch maps that is if the degree of x is m and the

degree of y in n then

nm

tx y y x

Also let t also b e the induced isomorphism of Hopf algebras

t H  H H  H

For the following compare Lemma

PAUL G GOERSS

Lemma Let H and H be Hopf algebras in HA Then the fol lowing

diagram commutes

D H D H 

D H  H

D

 /

D t

t  

 

D H D H 

D H  H

D

 /

Proof It is only necessary to check this for the universal examples H H n

and H H m of Equation Compare the pro of of Lemma If n and m

are even this follows from Lemma If either n or m is o dd the result follows

from Prop osition and Lemma

The Hopf ring of complex oriented cohomology theories

Let E b e a multiplicative cohomology theory represented by a ring sp ectrum

E Dene spaces E n by the formula

n

E n E

The spaces E n are generalized EilenbergMac Lane spaces in the sense that if X

is a CW complex then there is a natural isomorphism

n

E X X E n

The cup pro duct pairings

E m E n E m n

induce bilinear maps

H E m H E n H E m n

and hence D H E fD H E ng is a graded ring ob ject in the category D

nZ

of skewcommutative Dieudonne mo dules This means that the ring multiplication

satises the formulas of Lemmas Such an ob ject will b e called a Dieudonne

ring Since

n n

D H E n E n E E E

n

D H E is an E algebra and the op erators V and F act on E as the identity and

multiplication by p We will call such an ob ject an E Dieudonne algebra

We will b e particularly interested in homotopy commutative ring sp ectra E

This implies that that for all integers n and m there is a homotopy commutative

diagram

E m E n m n

/ E

mn T

 

n E m E m n

E /

where the horizontal maps are the cup pro duct maps T is top ological switch map

and is

mn

nm

id E n m E n m

mn

This observation and Lemma immediately imply the following result

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Lemma Let E be a homotopy commutative ring spectrum Suppose x

D H E m and y D H E n Then

i j

ij mn

x y y x D H E m n

ij

Now supp ose E is complex oriented thus there is a chosen element

x E C P C P E

so that the comp osite

x

E S C P C P

represents the multiplicative identity in E E The map x induces a

morphism of coalgebras

H C P H E

hence by adjointness a morphism of Hopf algebras

H BU S H C P H E

Since H BU is the reduction mo dulo p of a torsionfree Hopf algebra over Z with

p

a lift of the Verschiebung we have as in Example

D H BU R QH BU Z

p

Z V

p

In fact if H C P Z is the standard generator we obtain an induced

i i p

element b D H BU under the comp osition

i i

e e

H C P Z H B U Z R QH B U Z D H BU

p p p

Z V

p

Furthermore V b b Note that in H C P C P Z

pi i p

X

i j k

j k i

We will also use later that if m C P C P C P is the multiplication then

j k

m

j k j k

j

The complex orientation induces a map

D x D H BU D H E

E

Dene b D H E by

i

i

E

b D x b

i

i

E

If there is no ambiguity we may abuse notation and write b for b

i

i

Now dene an E Dieudonne algebra with underlying E algebra

R E E b b

with E in Dieudonnedegree and b in bidegree i where i is the Dieudonne

i

degree We require V b b and V on E and that V b e multiplicative This

pi i

and the formulas of Lemma determine the action of F The existence of the

E

elements b determine a morphism of E Dieudonne algebras

i

R E D H E

This brings us to the RavenelWilson relation Let

x y E x y F

PAUL G GOERSS

b e the formal group law for E and let

X

E i

b t bt

i

i

b e the evident p ower series over the ring D H E Since D H E is an E algebra

we can form the p ower series in two variables

bs bt D H E s t

F

and in this context the RavenelWilson relation b ecomes

Proposition Over D H E there is a formula

bs bt bs t

F

Proof The argument here is not so dierent than the one in the idea of

using the total unstable op eration as an organizing principle is due to Neil Strick

land

For any CW complex X there is a total unstable op eration

E X Hom H X H E

X CA

n

sending f E X X E n to

f H X H E n

this is continuous E algebra homomorphism and natural in X If

m C P C P C P

is the H space multiplication we then get a commutative diagram

Hom H C P H E E x E C P

/ CA



Homm

E m 

 

E x y Hom H C P C P H E E C P C P

/ CA

so

Homm x E mx

Since is an E algebra homomorphism

E mx x y x y

F F

where x is the coalgebra map

x

 1

H E H C P H C P C P

and y uses instead of On the other hand

Homm x

is the comp osite

x m

 

H E H C P H C P C P

Given any map of coalgebras

C P H E n f H C P

it extends to a map of Hopf algebras

f S H C P C P H E n

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



and the isomorphisms

Hom H C P C P H E

CA

Hom S H C P C P H E

HA

+

Hom D SH C P C P D H E

D

are isomorphisms of E algebras

Since S H C P C P is the mo d p reduction of S H C P C P Z

p

which has a lift of the Frobenius we can calculate by Theorem

D S H C P C P R QS H C P C P Z

p

Z V

p

e

R H C P C P Z

p

Z V

p

Let b D S H C P C P b e the image of

ij i j

ij

Then naturality and implies that

D S m D S H C P C P D S H C P

ij

sends b to b Similarly if is pro jection on the rst factor

ij ij

j

D S D S H C P C P D S H C P

is given by sending b to b if j and otherwise There is an analogous formula

ij i

involving Now let f H C P C P H E b e either of the maps of

and

D f D S H C P C P D H E

the induced map Then D f extends to a map of E Dieudonne algebras

E b D H E

ij

E

which we will also call D f Let b D f b and consider the p ower series

ij

ij

over D H E

E i j

bs t b s t

ij

Note i or j Using the expression for f given in we have

X X

E i j i j

bs t b s t D f b s t

ij

ij

X

i j

D x D S m b s t

ij

X

i j

i j

b s t D x

ij

j

X

i j

i j E

s t bs t b

ij

j

This rewrites bs t in one way Next using the expression for f given in

and that the isomorphisms of are E algebra maps we have

bs t D x bs t D y bs t

F

However since x involves pro jection on the rst factor

X

i j

D x bs t D x b s t

ij

X

i j

D x D S b s t

ij

X

i

D x b s bs

i

PAUL G GOERSS

Similarly D y bs t bt so bs t bs bt Combining the two expres

F

sions for bs t yields the result

Now let I E b b R E b e the ideal of relations forced by requiring

that bs t bs bt Then we get an induced map

F

R R E I D H E

E

of E Dieudonne algebras Under favorable circumstances this is almost an iso

morphism but not quite To see whats missing note that R E is concentrated

in bidegrees s t with b oth s and t even To account for o dd degree groups we

pro ceed as follows

PH S b e the If S S is the stable sphere let e D H S

image of the generator of S under the Hurewicz map For any sp ectrum X

there is a bilinear pairing

D H S D H X H S X H X

and we can dene a degree raising map

D H X D H X

n

by

x e x

Note that since V e we have by Lemma and Theorem

V e x e F x

Hence there is a factorization by Lemma

e

D H X

D X

/ O





PH X

QH X

/

The map lab eled is the homology susp ension induced by the evaluation

X X

To see this note that there is a commutative diagram

X

X

S /

 

X

S X

/

So at p there is a diagram of bilinear pairings of Hopf algebras

H X

e H X /

 

H X

H S H X

/

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



which induces a diagram using Prop osition

H X

e  H X S QH X

/

 

H X

H S  H X

/

and the top map in this diagram is induced by the homology susp ension The

argument at p is similar

n n

If E is a ring sp ectrum and X E then E E n and the

natural map S E n E n ts into a diagram

E n

S E n

/

1

 

E n E n

E /

where S E is the susp ension of the unit and the b ottom map is the cup

pro duct pairing Let e D H E b e the image of e under D H If E is

E

complex oriented then there is a diagram

E E

S S S S / /

 x 

E

C P

S / /

where x is the complex orientation By denition of x the b ottom comp osite must

b e the unit in E E It follows that

E

e b D H E

E

and hence we can extend to a map of E Dieudonne algebras

R ee b D H E

E

The equation e b also app ears in and

The main theorems on Hopf rings can now b e rephrased as follows

Theorem Suppose E is a Landweber exact homology theory with coef

cient ring E concentrated in even degrees Then

R ee b D H E

E

is an isomorphism

There is another pro of of this fact in the next section

It might b e worth p ointing out that the ring R has an interpretation in the

E

language of formal groups If is an E Dieudonne algebra the formal group law

over E passes to a formal group law over via the ring homomorphism E

This might b e called the E formal group law over Then the set of E Dieudonne

algebra homomorphisms

R

E

is in onetoone corresp ondence with homomorphisms of the additive formal group

law over to the E formal group law Compare Paul Turners interpretation of

QB P BP

PAUL G GOERSS

Example Supp ose E K is complex oriented K theory hence K

Z where K The formal group law for K is

x y x y xy

F

so the equation bs t bs bt implies

F

i j

b b b

i j ij

j

in D H K

Thus multiplication by induces isomorphisms

n

D H K D H K n

n

D H K D H K n

and with the padic valuation

D H K D H Z BU

i i

j

Zp i j

i j

The generator of D H K is b

j i

i j

D H K D H U

i i

Zp i j

The generator of D H K is e if j and eb if j

j j

Example Because of the connection b etween the element e D H E

and the homology susp ension given in Equation one can use Theorem

to give a description of H E for certain E Indeed there is an isomorphism

D H E H E

e

where D H Ee in degree k is the colimit of

e e

D H E n D H E n

nk nk

Note that E D H E n maps to H E this is the image of the Hurewicz

n n

homomorphism E H E If we let a H E b e the image of b note the

i i i

shift in indices then we get a surjective map

E a a p a H E

and the kernel is determined by the relation

as t as at

F

P

i

where a a t In short a is a strict isomorphism b etween F mo dulo p

i

i

and the additive formal group law that is a is an exp onential for F after reducing

mo d p Compare Corollary of

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



The role of E S

If X is a sp ectrum then H X is a graded bicommutative Hopf algebra and

one can study the functor

X D H X n

n

The following was proved in

Proposition There is a spectrum B n and a natural surjection

B n X D H X

n n

which is an isomorphism if n mo d p

As the notation suggests the sp ectra B n are the BrownGitler sp ectra This

is to say if p

i

H B n AAfS q i ng

or if p

i

H B n AAf P pi ng

and furthermore if B n H ZpZ classies the generator of H B n then the

induced map

B n Z H Z

n n

is surjective for all CW complexes Z

The group homomorphisms

V D H X D H X and F D H X D H X

pn n n pn

are induced resp ectively by maps

np

B pn B n

and

np

B n B pn

The map is very familiar as it is the one that ts into the Mahowald exact

sequencethe cobration sequence

np

B pn B pn B n

explored at p in and and at o dd primes in The map is less

familiarit is for example zero in cohomology

At p the maps

V D H X D H X

n n

and

F D H X D H X

n n

are induced by

n

B n B n

and

n

B n B n

resp ectively but the maps are not uniquely determined

PAUL G GOERSS

If X and Y are sp ectra there is a natural bilinear pairing in HA

H X H Y H X Y

induced by the map

X Y X Y

adjoint to the evaluation X Y X Y Thus we get a pairing

D H X D H Y D H X Y

n m nm

This yields pairings

B n B m B n m

which are uniquely determined if n m mo d p and in particular if n and

m are b oth even These pairings are also familiar at least in cohomology as

H B n m H B n H B m

sends the generator to the tensor pro duct of the two generators

The ambiguity in the denition of when n m mo d p can b e removed

by noting that there there are canonical maps B n B n which is a weak

equivalence if n is even Thus if n is o dd we could require B n B m

B n m to b e the comp osite

B n B m B n B m B n m B n m

This said the ob ject B fB ng b ecomes a graded commutative ring sp ec

n

trum

Now let E b e a ring sp ectrum representing a cohomology theory E with pro d

ucts Then

nk

E B n B n E B n E

k k n

so there is a surjective homomorphism

E B n D H E n k

k n

This induces a surjection

h E B D H E

which skews degrees Note that when k we get an isomorphism

k

E B D H E k E E

k k

E and makes the homomorphism h is This is the standard isomorphism E

E mo dules It is for this reason that we will sp eak of E a morphism of E

Dieudonne algebras in the sequel rather than E Dieudonne algebras

The prop erties of the map h of Equation are recorded in the following

sequence of results

Proposition At primes p the bigraded E module E B is an E

Dieudonne algebra At the prime E B has homomorphisms

V E B n E B n and F E B n E B n

satisfying the formulas of Lemmas

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Proposition At odd primes the map

h E B D H E

is a surjective homomorphism of E Dieudonne algebras At the prime the map

h respects the homomorphisms V and F

Remark At p D H E has op erators

V D H E D H E

n n

and

F D H E D H E

n n

which are not unambiguously dened in E B

The failure of the B n X D H X to b e an isomorphism

n n

can b e measured by the op erator e introduced in Equation Since B n

B n we have a diagram

B n X

D H X

n

n

/

e

 

D H X

B n X

n

n

/

and we see that these kernel of B n X D H X is isomorphic

n n

to the kernel of

e D H X D H X

n n

We now b egin our analysis of sp ecic ring sp ectra The following result will

allow a careful examination of the kernel of E B D H E in some cases for

example any Landweber exact theory with co ecients in even degrees

For the following compare and

Lemma Suppose E is a ring spectrum with E torsion free and concen

trated in even degrees Suppose further that E B n is concentrated in even degrees

Then for al l k

H E k is concentrated in even degrees

On H E k the Frobenius is injective and the Verschiebung surjective

There is an isomorphism of primitively generated Hopf algebras

H E k QH E k

Proof These will b e proved together The surjection

E B n D H E k

nk n

shows D H E k is concentrated in o dd degrees and hence H E k is

an exterior algebra Let E n E n b e the comp onent of the basep oint Then

the RothenbergSteenro d sp ectral sequence

H E k



QH E k Tor F F H E n

p p

is a sp ectral sequence of Hopf algebras and hence must collapse Since the Ver

schiebung is surjective at E on this sp ectral sequence it is on H E n and hence

on H E n

PAUL G GOERSS

Similarly consider

H E k



Tor F F H E k

p p t st

s

We work at p The argument for p is similar Let QH E k denote

the indecomp osables and Q and Q the even and o dd degree subvector spaces of

QH E k Let L QH E k b e the rst derived functor of Q applied to H E k

Then L QH E k L Q is concentrated in degrees congruent to zero mo dulo p

Then

H E k



Tor F F Q Q L Q

p p

where

H E k



QH E k Tor F F

p p

and

H E k



L Q Tor F F

p p

are the primitives If Q or L Q the lowest degree nonzero class in Q

L Q would pro duce a nonzero even degree class in H E k a contradiction

so Q L Q and

QH E k Q

The sp ectral sequence of Equation now collapses proving part Since the

mo dule of indecomp osables QH E k is in even degrees part follows Half of part

has already b een proved and since L QH E k the Frobenius is injective

on H E k See In fact the kernel of F in Lemma is exactly L Q

Now let D D H E b e the Dieudonne ring

E

D fD H E ng

E m

If E is concentrated in even degrees Let e D H E b e the susp ension element

of Equation

Proposition Suppose E is a ring spectrum with E torsion free and

concentrated in even degrees Suppose E B n is concentrated is even degrees for

al l n Then the natural map of E Dieudonne algebras

D ee b D H E

E

is an isomorphism

Proof Since D H E n by Lemma the induced map

m

D ee b D H E n

E n

is an isomorphism The result will follow once we show

eD H E n f e x x D H E n g D H E n

is an isomorphism Since eF x F V e x and D H F D H QH

eD H E n QH E n

and the result follows from Lemma

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



If E B n is concentrated in even degrees for all n then we may dene

E B ev fE B ng fE B ng

m

and the homomorphism of E Dieudonne algebras

E B D H E

restricts to an isomorphism of E Dieudonne algebras

E B ev D

E

Calculating the source of this map is where E S comes in

Write S for the susp ension sp ectrum of the space S with a disjoint

basep oint The MayMilgram ltration fF S g of S susp ends to a ltration

k

fF F S g of S The Snaith splitting implies this stable ltration is

k k

trivial S F F At a prime p the ltration quotients are Brown

k k k

Gitler sp ectra Sp ecically at p

k

F F B k

k k

and if p

np

B n if k pn

np

F F

B n if k pn

k k

if k mo d p

This previous paragraph summarizes work of In short the

is a regraded version of the graded sp ectrum asso ciated graded sp ectrum of S

B fB ng

It is convenient for our purp oses to eliminate the o dd BrownGitler sp ectra

If we write S hi for the universal cover of S and F S hi for the universal

k

cover of F S then fF S hig is a ltration of S hi and we get a ltration

k k

F S hi g of the susp ension sp ectrum Let B b e the asso ciated graded fF

k

k

sp ectrum Then Equation and a homology calculation shows that at any prime

k p

B f B k g

In short B is a regraded version of B ev In particular there is a degree shearing

isomorphism of E mo dules

E B ev E B

We next observe that the lo op space multiplication on S hi gives S hi the

structure of a ring sp ectrum and since the MayMilgram ltration is multiplicative

gives B the structure of a graded ring sp ectrum I dont know whether B ev

B as graded ring sp ectra this would settle Ravenels Conjecture for

example However the standard calculations show that

H B ev H B

as F algebras We can use this fact and Ravenels Adams sp ectral sequence calcu

p

lations to calculate

D D H BP

BP

Let R b e the ring of Equation E

PAUL G GOERSS

Theorem The graded BP modules BP B n are concentrated in even

degrees and the natural maps

BP B ev D R

BP BP

are isomorphisms of BP Dieudonne algebras

Proof Let I p v v BP b e maximal ideal If M is an BP

mo dule we can lter M by p owers of I and form the asso ciated graded ob ject

E M Then

Ext F H BP Ext F F E BP

p p p

A E

where A is the dual Steenro d algebra and E F A If X is a sp ectrum

p BP

then the Adams sp ectral sequence

Ext F H X BP X

p

E

is a sp ectral sequence of E BP mo dules The calculations of Theorem of

combined with Theorem of which is algebraic and precedes their

computation of H BP now imply

F H B E BP B E Ext E R

p BP

E

all as E BP algebras As a result we have a commutative square of E BP algebras

E BP B ev

E D H BP

/

O O

E R E BP B BP

/

This nishes the pro of

Landweber exactness now implies the following result which nishes the pro of

of Theorem

Corollary Let E be any Landweber exact theory with coecients con

centrated in even degrees Then E B n is in even degrees and the natural maps

E B ev D R

E E

are isomorphisms of E Dieudonne algebras

Proof Note that Landweber exactness implies that E is torsion free Since

B is plo cal

M

n

BP B ev MU B ev

pn

and the result follows from Theorem For general E we have a commutative

diagram

R R E

MU B ev E

E MU MU

MU 

 /



 

E B ev

D / E

2 3

HOPF RINGS DIEUDONNE MODULES AND E S



Glossary

Categories of Hopf Algebras

HA graded connected bicommutative Hopf algebras over F section

p

HA graded bicommutative Hopf algebras over F that are grouplike in

p

degree section

HA graded skewcommutative Hopf algebra over F that are grouplike in

p

degree for example the homology of a double lo op space section

HV graded connected torsionfree Hopf algebra over Z equipp ed with a lift

p

of the Verschiebung section

HV graded torsionfree Hopf algebra over Z equipp ed with a lift of the

p

Verschiebungand grouplike in degree section

HF graded connected Hopf algebras equipp ed with a lift of the Frobenius

sections and

Categories of Mo dules

D Dieudonne mo dules for HA section

D Dieudonne mo dules for HA section

D m Dieudonne mo dules for HA section

p

M Torsionfree graded Z mo dules with an endomorphism V for example

V p

QH H HV section

M the analog of M for HV

V

V

D similar to M dropping the torsionfree hypothesis section

V V

Certain Hopf Algebras

CW k and CW k the torsion free Hopf algebras with Wittvector diagonal

n

Denition

H n the pro jective generators of HA Lemma

H BU Examples and

References

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PAUL G GOERSS

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Department of Mathematics Northwestern University Evanston Illinois

Email address pgoerssmathnwuedu