The Homology of Homotopy Inverse Limits

by Paul G. Goerss

1

Abstract: The homology of a homotopy inverse limit can be studied by a spectral

sequence with has E

2

term the derived functors of limit in the category of coalgebras.

These derived functors can be computed using the theory of Dieudonn�e modules if one has

a diagram of connected abelian Hopf algebras.

One of the standard problems in homotopy theory is to calculate the homology of a

given type of inverse limit. For example, one might want to know the homology of the

inverse limit of a tower of �brations, or of the pull-back of a �bration, or of the homotopy

�xed point set of a group action, or even of an in�nite product of spaces. This paper

presents a systematic method for dealing with this problem and works out a series of

examples.

It simpli�es the foundational questions present when dealing with inverse limits to

work with simplicial sets rather than topological spaces. So this paper is written simpli-

cially; that is, a space is a simplicial set. As usual this doesn't a�ect the homotopy theory.

Homology is with F

p

coe�cients.

Here are some simple examples of the type of result we obtain. An abelian Hopf

algebra is one for which both diagonal and multiplication are commutative.

Theorem A. Let fX

�

g be an arbitrary set of connected nilpotent spaces and suppose for

all �, H

�

X

�

is an abelian Hopf algebra. Then there is a natural isomorphism

H

�

(

Y

�

X

�

)

�

=

Y

�

H

�

X

�

:

This is a companion to a result of Bous�eld's [3, 4.4] where slightly di�erent hypotheses

yield a slightly di�erent result. The product

Q

�

H

�

X

�

is in the category of graded connected

coalgebras. If the set is �nite it is simply the graded tensor product. If the set is in�nite

something more sophisticated is required. See section 1 for limits of coalgebras.

1

The author was partially supported by the National Science Foundation.

1

Theorem B. Consider a tower of �brations over the natural numbers

X

1

X

2

X

3

� � �

so that for all i, X

i

is connected and H

�

X

i+1

! H

�

X

i

is a morphism of abelian Hopf

algebras. Suppose each X

i

is simple and lim

1

�

1

X

i

= 0. Then under either of the following

conditions, there is an isomorphism

H

�

(limX

i

)

�

=

lim

i

H

�

X

i

:

1) for all i, H

�

X

i

is of �nite type;

2) the tower H

�

X

1

H

�

X

2

� � � is Mittag-Le�er.

These are two of the usual conditions for vanishing of lim

1

for abelian groups. Again

lim

i

H

�

X

i

must be calculated in the category of coalgebras. Under various conditions,

however, this will be isomorphic to the inverse limit as vector spaces. One common example

of this is if the tower fH

�

X

i

g

i

is pro-isomorphic, as a tower of Hopf algebras, to a constant

tower. This is a stronger condition than Mittag-Le�er.

Behind these results is a technique which I will now explain. In general, given a small

category I and an I-diagram X of spaces there is a map of graded coalgebras

H

�

(holim

I

X) ! lim

I

H

�

X

where holim is the Bous�eld-Kan homotopy inverse limit and the limit on the right is in the

category of coalgebras. There is no reason to think that it is either injective or surjective;

however, it is the edge homomorphism of a spectral sequence. This spectral sequence is a

variant of one due to Anderson [1] and studied by Bous�eld in [3]. The major advantage of

our variant is that the E

2

term can be identi�ed and computed by non-abelian homological

algebra.

Let CA

+

be the category of graded connected coalgebras over F

p

and CA

I

+

the category

of I-diagrams in CA

+

. The point of this paper is that the functor

lim

I

: CA

I

+

! CA

+

has right derived functors R

s

CA

lim

I

, s � 0, so that R

0

CA

lim

I

�

=

lim

I

and the bigraded vector

space R

�

CA

lim

I

C is a bigraded, cocommutative coalgebra. Then we have

2

Theorem C. Let X : I ! S be an I-diagram of connected spaces. Then there is a natural

second quadrant homology spectral sequence.

R

�

CA

lim

I

H

�

X ) H

�

holim

I

X

p

:

Here X

p

: I ! S is the I-diagram obtained by applying the Bous�eld-Kan p-comple-

tion functor to X. Because this is a second quadrant spectral sequence, convergence is

a problem. For this we appeal to [3] or [13]. This accounts for some of the hypotheses

of Theorems A and B. Beyond this there is the problem of computing R

�

CA

lim

I

H

�

X. For

example Theorems A and B are based on the assertion that R

s

CA

lim

I

H

�

X = 0 for s > 0

under the listed hypotheses. It is here that we use the assumption that H

�

X : I ! CA

+

is actually a diagram of abelian Hopf algebras. The category of abelian Hopf algebras is

equivalent to a category of modules called Dieudonn�e modules and one can pass from the

ordinary derived functors of lim

I

for modules to R

�

CA

lim

I

, at least in some cases. We give

examples, at least when lim

I

s

= 0 for s > 1 as modules. Even in this case R

s

CA

lim

I

may not

be zero for any s � 0. Using these computations we make some calculations of the spectral

sequence in cases where R

�

CA

lim

I

H

�

X is not concentrated in degree zero. Included is an

example of homotopy �xed points. Finally, there is the problem of relating H

�

holimX to

H

�

holimX

p

. Again there are techniques in [3] that apply.

Despite the title of this paper and the general thrust of this introduction, this is pri-

marily a work in non-abelian homological algebra { �ve of the six sections are devoted to

the de�nitions and calculations of R

�

CA

lim

I

. Only the sixth and last section contains homo-

topy theoretic applications. Sections 1 and 2 are devoted to the de�nition and homotopical

algebra foundations of R

�

CA

lim

I

{ these derived functors are, in fact, the cohomotopy groups

of a homotopy inverse limit of cosimplicial coalgebras, although this language is avoided

until section 2. Section 3 is on Dieudonn�e modules, Section 4 gives some calculations, and

Section 5 contains the proofs of some technical lemmas.

The paper began as a meditation on Example 4.4 of [3], spurred on by a conversation

with John Hunton. In fact, the proof of Theorem A given here may be regarded as a wildly

expanded version of Bous�eld's argument, and, in general, this paper owes a great debt

to [3]. Finally, a conversation with Brooke Shipley on bicosimplicial spaces was useful for

widening my scope. Several results for [13] make this paper go much more smoothly.

x1. Derived functors of limits in the category of coalgebras

Let CA be the category of graded cocommutative coalgebras over a �eld k. Later,

we will specialize to the case where k = F

p

. As always in homotopy theory applications,

3

commutativity is with a sign. Thus if C 2 CA and x 2 C has diagonal

�x = �y

i

z

i

then

�y

i

z

i

= �(�1)

jy

i

j jz

i

j

z

i

y

i

where jwj is the degree of w. Let CA

+

� CA denote the full sub-category of connected

coalgebras. Thus C 2 CA

+

if and only if C

0

�

=

F

p

.

A useful fact is:

Lemma 1.1. Let C 2 CA. Then C is isomorphic to the �ltered colimit of its �nite

dimensional sub-coalgebras.

Proof: This is clear in C 2 CA

+

. The general case follows from [14], p. 46. �

Now let I be a small category and CA

I

the category of I-diagrams in CA. By de�nition,

lim

I

: CA

I

! CA

is right adjoint to the constant diagram functor. Such limits exist for all I. To see this,

�rst suppose I is �ltered. Then if C : I ! CA is an I-diagram we can form the vector

space limit lim

I

k

C. This is not, in general, a coalgebra; however, it is a complete coalgebra

in the following sense. For i 2 I, let

E(i) = kerflim

I

k

C ! C(i)g:

Then lim

I

k

C is a complete topological vector space with respect to the neighborhood base

for zero fE(i)g. Then there is a coproduct

lim

I

k

C ! lim

I

k

C

b

lim

I

k

C

where

b

denotes the completion of the tensor product with respect to fE(i)E(j)g. This

is because

lim

I

k

C

b

lim

I

k

C

�

=

lim

I

k

(C C):

A sub-coalgebra D � lim

I

k

C is a sub-vector space equipped with a coproduct D! D D

making D a coalgebra and such that the evident diagram commutes:

D w

u

D D

u

lim

k

C w lim

k

C lim

k

C:

4

Then, for the limit in the category of coalgebras,

(1.2.1) lim

I

C = colim

�

D

�

whereD

�

runs over all sub-coalgebras over lim

I

k

C. Because of Lemma 1.1, we could equally

require the D

�

to be �nite. Next, suppose I is discrete, so that lim

I

=

�

I

. In the object

set of I is �nite, then

�

I

C =

I

C:

Then, in general,

(1.2.2)

�

I

C = lim

J

�

J

C

where lim is in the category CA and J runs over the �ltered diagram of �nite sub-categories

of I.

Proposition 1.3. The category CA has all limits.

Proof: We now have products, by 1.2.2, so to get all limits we need only supply pull-

backs. However, the pull-back of a diagram

C

1

! C

12

C

2

is the cotensor product C

1

�

C

12

C

2

. �

Because of the colimit in formula 1.2.1, lim

I

will not, in general, preserve surjections,

even in the case of (in�nite) products 1.2.2. Thus one will have derived functors. There

are several possible de�nitions of these. After some preliminaries, we will give a de�nition

in (1.8) which, while complicated, arises in homotopy theory applications.

Let J : CA

+

! n

+

k be the \coaugmentation coideal" functor from connected coal-

gebras to positively graded vector spaces. Thus JC is nothing more than the elements of

positive degree. The functor J has a right adjoint S with SW =

L

n�0

S

n

W where S

0

W = k

and

S

n

W = (W � � � W

| {z }

n

)

�

n

:

where �

n

acts by permuting factors, up to signs required by the grading. If W is of �nite

type, (SW )

�

is the free graded commutative algebra on W

�

and, hence, if char(k) 6= 2, is

a tensor product of polynomial and exterior algebras.

5

Let

�

S = S � J : CA

+

! CA

+

be the composite functor. It is the underlying functor

of a triple on CA

+

. Thus if C 2 CA

+

, one has a canonical cosimplicial resolution

(1.4) � : C !

�

S

�

C:

In particular, �

�

�

S

�

C

�

=

C with isomorphism induced by �. This can be seen by applying J

to (1.4) and noting that the resulting cosimplicial vector space has a natural contraction.

This resolution can be used to de�ne derived functors. For example, if P is the

primitive element functor

(1.5) R

s

PC = �

s

P

�

S

�

C:

These derived functors were one of the main topics of [4]. We also use this resolution to

de�ne derived functors of limit. But we need another preliminary.

Fix a small category I. The classifying space of the category I is the simplicial set

BI with 0-simplices the objects of I and n-simplices, n � 1, strings of arrows in I

*

i

= i

n

! i

n�1

! � � � ! i

1

! i

0

with face and degeneracy operators given by

d

0

*

i

= i

n

! i

n�1

! � � � ! i

1

d

j

*

i

= i

n

! � � � ! i

j+1

! i

j�1

! � � � ! i

0

d

n

*

i

= i

n�1

! � � � ! i

1

! i

0

;

s

j

*

i

= i

n

! � � � ! i

j

1

! i

j

! � � � ! i

0

:and

Note that for d

j

, i

j+1

! i

j�1

is the composition. Now let C be a category with products

and X : I ! C an I-diagram. Then we get a cosimplicial objects

�

�

X in C with

(1.6)

�

n

X =

�

*

i

2BI

n

X(i

0

):

with coface and codegeneracy maps induced by those in BI, with the evident twist in d

0

:

the composite

�

*

i

2BI

n�1

X(i

0

)

d

0

�!

�

*

j2BI

n

X(j

0

)

�

*

j

�!X(j

0

)

6

�ts into a commutative diagram

�

*

i

X(i

0

)

u

�

d

0

*

j

w

d

0

�

*

j

X(j

0

)

u

�

*

j

X(j

1

) w X(j

0

)

:

Note that if A : I ! ab is a diagram of abelian groups, then

(1.7) lim

I

s

A

�

=

�

s

�

�

A:

See [5, p. 305].

Now let C : I ! CA

+

be an I-diagram of connected coalgebras. Then for each i 2 I,

one can form the augmented cosimplicial coalgebra

� : C(i)!

�

S

�

C(i)

and the naturality of this construction yields an I-diagram of augmented cosimplicial

coalgebras. Hence one obtains a bicosimplicial coalgebra �

�

�

S

�

C which has a diagonal

cosimplicial coalgebra diag�

�

�

S

�

C.

Definition 1.8. Let C : I ! CA

+

be an I-diagram of coalgebras. Then the derived

functors of lim

I

are de�ned by

R

s

CA

lim

I

C = �

s

diag(�

�

�

S

�

C):

Remarks 1.9:

1) The augmentation � : C !

�

S

�

C induces a map

�

�

C ! diag�

�

�

S

�

C

which may or may not be an equivalence. See Example 1.10 below and Corollary 4.9.

2) Since R

�

CA

lim

I

C is the cohomotopy of a cosimplicial coalgebra, it is a bigraded

cocommutative coalgebra. The switch sign depends on the total degree.

3) R

0

CA

lim

I

C

�

=

lim

I

C since �

0

diag�

�

�

S

�

C is an equalizer and limits commute.

The remainder of this section is devoted to two examples.

Example 1.10: If I has �nite object set and C : I ! CA

+

is an I-diagram { so C is a

�nite diagram { then

� : �

�

C ! diag�

�

�

S

�

C

7

is a �

�

isomorphism. To see this, �lter the bicosimplicial coalgebra

�

�

�

S

�

C = f�

n

�

S

m

Cg

by degree in n to get a spectral sequence

E

n;m

1

= �

m

(�

n

�

S

�

C)) �

n+m

diag�

�

�

S

�

C:

Since I is �nite, BI

n

is �nite, so �

n

�

S

�

C =

N

*

i

BI

n

�

S

�

C(i

0

) and the Eilenberg-Zilber

Theorem says

E

n;m

1

=

�

0 m > 0

�

n

C m = 0

:

The claims follows. Thus one has

�

�

(�

�

C)

�

=

R

�

CA

lim

I

C:

If I is �nite and discrete, one has R

n

CA

lim

I

C = 0 for n > 0 and R

0

CA

lim

I

C

�

=

�

I

C

�

=

N

I

C.

If I is the category with diagram 1! 12 2, so and I-diagram is of the form

C

1

! C

12

C

2

one has

R

�

CA

lim

I

C

�

=

�

�

(�

�

C)

�

=

Cotor

�

C

12

(C

1

; C

2

)

as we would hope.

Example 1.11: Let I be discrete, but not necessarily �nite. Then I claim the other

augmentation

�

I

�

S

�

C ! �

�

�

S

�

C

induces a �

�

isomorphism

�

I

�

S

�

C ! diag �

�

�

S

�

C

so that

�

�

(

�

I

�

S

�

C)

�

=

R

�

CA

(

�

I

)C:

To see this, �lter �

�

�

S

�

C = f�

n

�

S

m

Cg by degree in m, to get a spectral sequence

E

n;m

1

= �

n

�

�

�

S

m

C ) �

n+m

diag �

�

�

S

�

C:

8

For �xed m,

�

S

m

C = (

�

S � � � � �

�

S)C, with the composition taken m+ 1 times. Then

�

�

�

S

m

C

�

=

S(�

�

J �

�

S

m�1

C)

since S is a right adjoint and �

�

on the inside is in the category of vector spaces. Then

�

n

�

�

J �

�

S

m�1

C

�

=

lim

I

n

(J �

�

S

m�1

C)

�

=

(

0 n > 0

�

I

J �

�

S

m�1

C n = 0

since these derived functors are in vector spaces. Thus �

�

J �

�

S

m�1

C has a cosimplicial

contraction and

E

n;m

1

=

(

0 n > 0

�

I

�

S

m

C n = 0

where this product is in CA

+

. The result follows.

x2. The homotopical foundations of R

�

CA

lim

I

.

The point of this section is to demonstrate that the derived functors of limits in

the category CA can be given a homotopical foundation exible enough to be useful in

computations. Afterward we supply some examples.

Let cCA

+

be the category of cosimplicial coalgebras. We will show that R

�

CA

lim

I

are

the cohomotopy groups of a homotopy inverse limit in cCA

+

.

To begin with cCA

+

is a simplicial model category in the sense of Quillen [11, II.2].

This is a result of [7]; see also [2], although the latter reference is not explicit about the

simplicial structure. To spell out the details, �rst note that cCA

+

is a simplicial category

by the results of Quillen. Indeed if K 2 S is a simplicial set and C

�

2 cCA

+

then

hom(K;C

�

) 2 cCA

+

is the cosimplicial coalgebra with

(2.1.1) hom(K;C

�

)

n

=

�

k2K

n

C

n

with the induced face and codegeneracy operators. There is also an object C

�

K uniquely

determined by the natural isomorphism

(2.1.2) Hom

eCA

+

(C

�

K;D

�

)

�

=

Hom

cCA

+

(C

�

;hom(K;D

�

)):

9

Finally, there is a mapping space functor on cCA

+

with n-simplices given by

(2.1.3) map

cCA

+

(C

�

;D

�

) = Hom

cCA

+

(C

�

;hom(��

�

n

;D

�

)):

Now de�ne a morphism f : C

�

! D

�

to be

2.1.4) a weak equivalence if �

�

f is an isomorphism;

2.1.5) a co�bration if the induced map Nf : NC

�

! ND

�

of normalized cochain com-

plexes is an injection in positive degrees;

2.1.6) a �bration if it has the right lifting property with respect to all maps which are at

once co�brations and weak equivalences.

Then one has:

Theorem 2.2 [5]. With these de�nitions, cCA

+

becomes a simplicial model category.

Part of the proof is to describe �brations. A morphism f : C

�

! D

�

is almost-free if

for all n � 0 there are vector spaces W

n

and maps �

o

: W

n

! W

n�1

, 0 � i � n � 1, so

that

2.3.1) there are isomorphisms C

n

�

=

D

n

S(W

n

) that �t into commutative diagrams

C

n

u

f

w

�

=

D

n

S(W

n

)

u

�

1

D

n

w

=

D

n

;

2.3.2) the following diagrams commute

C

n

u

�

=

w

s

i

C

n�1

u

�

=

D

n

S(W

n

) w

s

i

s�

i

D

n�1

S(W

n�1

):

Proposition 2.4. Almost-free maps are �brations in cCA

+

and any �bration is a retract

of an almost-free map.

If the unique map " : C

�

! k is a �bration, C

�

will be called �brant; if " is almost-free,

C

�

will be called almost-free. Since any retract of a coalgebra of the form S(W ) is of the

form S(W

0

) (by [4], Proposition 4.2), one has

Corollary 2.5. An object 2.5 is �brant if and only if it is almost-free.

10

Example 2.6: Let C 2 CA

+

be regarded as a constant cosimplicial coalgebra. Then the

canonical map � : C !

�

S

�

C is a co�bration and a weak equivalence, and

�

S

�

C is �brant.

Because cCA

+

is a simplicial model category with all products and coproducts, it has

homotopy inverse limits in the style of Bous�eld and Kan [5, XI, x4�]. To recapitulate,

let I be a small category. For i 2 I, let I # i be the category of objects over I and let

B(I # i) be its classifying space. Each of these spaces is contractible since I # i has a

terminal object. The assignment

i 7! B(I # i)

is an I-diagram of spaces. Now let C

�

: I ! cCA

+

be an I-diagram of cosimplicial

coalgebras. Then holimC

�

= holim

I

C

�

is de�ned by an equalizer diagram in cCA

+

(2.7) holimC

�

!

�

i

hom(B(I # i); C

�

(i))

d

1

=)

d

0

�

i!i

0

hom(B(I # i); C

�

(i

0

))

where d

0

and d

1

are induced respectively by

hom(B(I # i

;

C

�

(i)) ! hom(B(I # i); C

�

(i

0

))

and

hom(B(I # i

0

); C

�

(i

0

))! hom(B(I # i); C

�

(i

0

)):

However, combining this de�nition with the description of hom(K;C

�

) given in 2.1.1 one

has

(2.8) holimC

�

�

=

diag(�

�

C

�

):

where �

�

is the cosimplicial construction of the previous section.

One would like holimC

�

to have the following invariance property: if C

�

(i) ! D

�

(i)

is a weak equivalence for all i, then

holimC

�

! holimD

�

should be a weak equivalence. This fails in general; however, one has:

Lemma 2.9. Let C

�

! D

�

be a morphism of I-diagrams in cCA

+

and suppose C

�

(i) !

D

�

(i) is a weak equivalence for all i 2 I. Then if C

�

(i) and D

�

(i) are �brant for all i,

holimC

�

! holimD

�

is a weak equivalence.

11

Proof: Write the bicosimplicial vector space

�

�

C

�

= f�

p

C

q

g:

Filtering by degree in p gives a spectral sequence

E

p;q

1

= �

q

(�

p

C

�

) ) �

p+q

diag(�

�

C

�

)

�

=

�

p+q

holimC

�

:

Thus we need only show �

p

C

�

! �

p

D

�

is a weak equivalence for all p. Since every

object of cCA

+

is co�brant, any weak equivalence between �brant objects is a homotopy

equivalence, which, in cCA

+

is the same as a cosimplicial homotopy equivalence in the

sense of [10]. Thus �

p

C

�

! �

p

D

�

is a cosimplicial homotopy equivalence, hence a weak

equivalence. �

For this reason one demands, before taking a homotopy inverse limit in cCA

+

, that

one has a diagram of �brant objects. One of the results of [7] is that for all C

�

2 cCA

+

there is a natural co�bration and weak equivalence C

�

! D

�

with D

�

�brant. Thus if

C

�

: I ! cCA

+

is an I-diagram D

�

: I ! cCA

+

is an I-diagram of �brant objects and one

sets

(2.10) R holimC

�

= holimD

�

:

The boldface R means Quillen's total right derived functor.

Corollary 2.11. If C

�

1

! C

�

2

is a morphism of I-diagrams in cCA

+

and C

�

1

(i) ! C

�

2

(i)

is a weak equivalence for all i, then

RholimC

�

1

! RholimC

�

2

is a weak equivalence.

Proof: Combine Lemma 2.9 with the de�nition 2.10. �

More is true however.

Lemma 2.12. Let C

�

: I ! cCA

+

be an I-diagram of cosimplicial coalgebras and let

C

�

! C

�

1

be a morphism of I diagrams so that

C

�

(i) ! C

�

1

(i)

12

is a weak equivalence for all i and C

�

1

(i) is �brant for all i. Then there is a weak equivalence

RholimC

�

' holimC

�

1

:

Proof: Consider the diagram

C

�

w

u

D

�

u

C

�

1

w D

�

1

where D

�

and D

�

1

are the natural �brant replacements indicated above. Then Lemma 2.9

guarantees weak equivalences

holimC

�

1

! holimD

�

1

holimD

�

:

�

Example 2.13: Let C : I ! cCA

+

be an I-diagram of coalgebras. We may regard

each C(i) as a constant cosimplicial coalgebra. Then, by Example 2.6, � : C !

�

S

�

C is

a morphism of I-diagrams in cCA

+

so that C(i) !

�

S

�

C(i) is a weak equivalence with

�

S

�

C(i) �brant. Hence

RholimC = holim

�

S

�

C

�

=

diag(�

�

�

S

�

C)

�

�

RholimC

�

=

R

�

CA

lim

I

C:and

As a �rst example of the kind of exibility we've acquired, I supply:

Example 2.14: Let C

1

; C

2

: I ! CA

+

be two I-diagrams. Then there is a natural

isomorphism of bigraded coalgebras

R

�

CA

lim

I

(C

1

C

2

)

�

=

R

�

CA

lim

I

C

1

R

�

CA

lim

I

C

2

:

As a result of H : I ! CA

+

is actually a diagram of Hopf algebras, R

�

CA

lim

I

H is a

bigraded Hopf algebra. If H is commutative, so is R

�

CA

lim

I

H.

To see the isomorphism, I claim there is a weak equivalence

Rholim(C

1

C

2

) ' RholimC

1

RholimC

2

of cosimplicial coalgebras. Here the tensor product is taken level-wise; the result above

follows from the Eilenberg-Zilber Theorem.

To see the claim, let C

1

: I ! cCA

+

and C

2

: I ! cCA

+

be any two I-diagrams.

Choose morphisms of I-diagrams

C

1

! D

1

C

2

! D

2

13

where D

1

(i) and D

2

(i) are �brant for all i, and C

1

(i) ! D

1

(i) and C

2

(i) ! D

2

(i) are

weak equivalences. Then C

1

C

2

! D

1

D

2

has the same properties: D

1

(i) D

2

(i) is

�brant, since products of �brant objects are �brant and C

1

(i) C

2

(i) ! D

1

(i) D

2

(i) is

a weak equivalence. Hence, by Lemma 2.12 and the fact that holim commutes with �nite

products, we have weak equivalences

Rholim(C

1

C

2

) ' holim(D

1

D

2

)

' holimD

1

holimD

2

' RholimC

1

RholimC

2

:

�

x3. Recollections on Dieudonnn�e modules.

Let HA

+

be the category of graded, bicommutative Hopf algebras { hereinafter known

as abelian Hopf algebras because an object inHA

+

is an abelian group object in CA

+

. The

point of the next two sections is to address the problem of computing R

�

CA

lim

I

H where

H : I ! HA

+

is an I-diagram of abelian Hopf algebras. There are several advantages

to this situation. One is the existence of the classi�cation scheme known as Dieudonnn�e

modules. Another is the natural presentation proof of Proposition 4.1 below. See 4.3 and

Remark 4.8.

We begin by recalling Schoeller's work on Dieudonnn�e modules [12]. The point is that

HA

+

is an abelian category with a set of projective generators. As such it is equivalent

to a category of modules over a ring. Dieudonnn�e theory says which modules over which

ring. Let us assume the ground �eld k = F

p

.

Definition 3.1. Suppose p > 2. Let D be the category of positively graded Z

p

modules

M equipped with homomorphisms

V :M

2pk

!M

2k

F :M

2k

!M

2pk

so that

1) pM

2k+1

= 0 for all k and if (k; p) = 1, p

s+1

M

2p

s

k

= 0;

2) V F (x) = px and FV (y) = py.

If p = 2, D is the category of graded Z

2

modules with homomorphisms V : M

2k

! M

k

and F :M

k

!M

2k

so that V F = FV = 2 and

2

s+1

M

2

s

(2t+1)

= 0:

14

Theorem 3.2 [12]. There is an equivalence of categories

D

�

: HA

+

!D:

The category D is the category of Dieudonnn�e modules; if H 2 HA

+

, then D

�

H is

the associated Dieudonnn�e module. The homomorphisms V and F are to be taken as

reecting the Verschiebung and Frobenius (p

th

power map) respectively.

In the sequel, I'll often state results for p > 2 and leave the case p = 2 implicit.

Example 3.3: The category D has an obvious set of projectives; indeed, for all n > 0,

there is a module F (n) 2 D so that there is a natural isomorphism

Hom

D

(F (n);M)

�

=

M

n

where M

n

denotes the elements of degree n. If n is odd F (n) = Z=pZconcentrated in

degree n and if n = 2p

s

k with (p; k) = 1

F (n)

t

�

=

8

<

:

Z=p

s+1

Z t = 2p

i

k; i � s

Z=p

i+1

Z t = 2p

i

k; i � s

0 otherwise

and

F

i

: F (n)

n

! F (n)

2p

s+i

k

V

i

: F (n)

n

! F (n)

2p

s�i

k

are surjective for all i.

Let H(n) be the unique (up to isomorphism) Hopf algebra so that D

�

H(n)

�

=

F (n).

Then if n is odd H(n)

�

=

�(x) where � denotes the exterior algebra and the degree of x is

n. If n = 2p

s

k with (p; k) = 1,

H(n)

�

=

F

p

[x

0

; x

1

; : : : ; x

s

]

where the degree of x

i

is 2p

i

k andH(n) has the \Witt vector diagonal." The Verschiebung

is described by �x

i

= x

i�1

. See [12]. Of course, the H(n) are the projective generators of

HA

+

and Schoeller identi�es them as such before proving her theorem. ThenD

�

: HA

+

!

D is given, in degree n, by

D

n

H = Hom

HA

+

(H(n);H):

15

The operators V and F are described by V = '

�

and F =

�

where ' : H(2n) ! H(2pn)

is the inclusion and : H(2pn) ! H(2n) is the unique map making the diagram commute

H(2n)

u

'

H(2pn)

[

[

[

[]

w

[p]

H(2pn)

where [p] is p times the identity in HA

+

. In particular x

i

= x

p

i�1

.

Example 3.4: Again assume p > 2. Let n = 2k (with no assumption on (p; k)) and

G(n) 2 D the module with

G(n)

t

=

�

Z=p

i+1

Z t = 2p

i

k = p

i

n

0 otherwise

and V is surjective. Then if n = 2p

s

k with (p; k) = 1, there is a short exact sequence in D

0! F (n) ! G(2k)

V

s+1

���!G(2p

s+1

k)! 0

If A(n) is the unique (up to isomorphism) Hopf algebra with D

�

A(n)

�

=

G(n) then

A(n)

�

=

F

p

[x

0

; x

1

; x

2

; : : : ]

with deg(x

i

) = p

i

n and the \in�nite Witt vector" diagonal. Thus there is a short exact

sequence of Hopf algebras

F

p

! H(n)! A(2k)! A(2p

s+1

k)! F

p

if n = 2p

s

k with (k; p) = 1.

The deeper ideas about HA

+

rely on the following result. Let n

+

Ab be the category

of positively graded abelian groups.

Proposition 3.5. The forgetful functor D ! n

+

Ab has a right adjoint J .

Proof: The argument, a variant on the proof of the special adjoint functor theorem

(which, by the way, would su�ce in proof) is adequately covered in [9] in the context of

unstable modules. So I'll give only an outline. Assume p > 2.

Let N 2 n

+

Ab. To construct J(N) it is su�cient to assume N is concentrated in a

single degree, say k. Now if J(N) exists, then

J(N)

n

�

=

Hom

D

(F (n); J(N))

�

=

Hom

Ab

(F (n)

k

;N

k

)

16

so one simply de�nes

J(N)

n

= Hom

Ab

(F (n)

k

;N

k

):

The operators V and F in D are induced by maps ' : F (2pn) ! F (2n) and : F (2n) !

F (2pn) respectively as representing objects. These induce V and F on J(N). Next notice

that if M 2 D one has maps

Hom

D

(M;J(N))! Hom

Ab

(M

k

;Hom

Ab

(F (k)

k

;N

k

))

�

=

Hom

Ab

(M

k

F (k)

k

;N

k

)

�

=

Hom

n

+

Ab

(M;N)

since M

k

!M

k

F (k)

k

is an isomorphism. By construction this map is an isomorphism

if M = F (n) for some n. Since both source and target send sums to products and are left

exact as functors from D

op

to Ab, the result follows by analyzing projective resolutions.�

Remark 3.6: Again take p > 2 and let N 2 n

+

Ab be concentrated in a single degree k.

Then

J(N)

n

= Hom

Ab

(F (n)

k

;N

k

)

is often zero. If k is odd, F (n)

k

= 0 unless n = k, thus J(N)

n

= 0 unless n = k and

J(N)

k

= Hom

Ab

(Z=pZ;N

k

). If k = 2p

i

k

0

, with (k

0

; p) = 1, then F (n)

k

= 0 unless

k = 2p

s

k

0

. Hence J(N)

n

= 0 unless n = 2p

s

k and

J(N)

2p

s

k

0

= Hom

Ab

(Z=p

a

Z;N

k

)

where

a =

�

s+ 1 s � i

i+ 1 s � i

:

Corollary 3.7. The categories D and HA

+

have enough injectives.

Proof: For the category D this follows from Proposition 3.5 and the fact that n

+

Ab has

enough injectives. The result follows for HA

+

because this latter category is equivalent to

D. �

Example 3.8: The functor D ! Ab given by

M 7! Hom

Ab

(M

k

;Q=Z)

is representable. Let N(k) 2 n

+

Ab be the module isomorphic to Q=Zin degree k. Then

if J(k) = J(N(k)),

Hom

D

(M;J(k))

�

=

Hom

Ab

(M

k

;Q=Z):

17

The object J(k) is evidently injective since Q=Z is an injective abelian group. Also J(k)

can be described explicitly using Remark 3.6. Assume p > 2. If k is odd J(k)

�

=

Z=pZin

degree k. If k = 2p

i

k

0

with (k

0

; p) = 1, then J(k)

n

= 0 unless n = 2p

s

k

0

and

J(k)

2p

s

k

0

�

=

�

Z=p

s+1

Z; s � i

Z=p

i+1

Z; s � i

and V is always surjective, either by calculation or Corollary 3.15 below.

We now examine the implications for Hopf algebras.

Definition 3.9. A coalgebra D 2 CA

+

is injective if any diagram of coalgebras

C

0

w

i

u

C

1

i

i

i

ik

D

with i an inclusion can be completed.

Proposition 3.10. For a coalgebra D 2 CA

+

the following are equivalent.

1) D

�

=

S(W ) for some graded vector space W ;

2) D is injective;

3) R

s

PD = 0 for s � 1; and

4) R

1

PD = 0.

Proof: That 1) implies 2) follows from adjointness. For 2) implies 3), notice that if D is

injective, the inclusion D !

�

S(D) has a coalgebra retraction. That 3) implies 4) is clear.

Finally 4) implies 1) is Proposition 4.2 of [4]. �

The connection to Hopf algebras is supplied by:

Lemma 3.11. Let H 2 HA

+

be an injective Hopf algebra. Then H is an injective coalge-

bra.

Proof: The forgetful functor HA

+

! CA

+

has a left adjoint that preserves injections

{ namely, the symmetric algebra functor endowed with the evident diagonal. Hence the

forgetful functor preserves injectives. �

We now show how to recover derived functors of primitives on abelian Hopf algebras

from the Dieudonnn�e module. If M 2 D let �M 2 D be its \double". If p = 2,

(�M)

t

=

�

M

k

t = 2k

0 t = 2k + 1

18

where V and F induced from M . If p > 2

(�M)

t

=

�

M

2k

t = 2pk

0 otherwise

with again V and F induced from M . The homomorphism V induces a homomorphism

in D,

V :M ! �M:

Proposition 3.12. Let H 2 HA

+

. Then there is a natural exact sequence

0! PH ! D

�

H

V

!�D

�

H ! R

1

PH ! 0:

Proof: Assume p > 2. If n is odd or n = 2k with (p; k) = 1, then (�D

�

H)

n

=

(R

1

PH)

n

= 0, so here we are asserting (PH)

n

= D

n

H. If n is odd, one has

(PH)

n

�

=

Hom

HA

+

(�(x);H)

�

=

D

n

H

where deg(x) = n; if n = 2k, (k; p) = 1,

(PH)

n

�

=

Hom

HA

+

(F

p

[x

0

];H)

�

=

D

n

H

where the degree of x

0

is n. So we may assume n = 2p

s

k, s > 1, with (k; p) = 1. Then

there is a short exact sequence of Hopf algebras

(3.12.1) F

p

! H(

n

p

)

'

!H(n) ! F

p

[x

s

]! F

p

where H(n)

�

=

F

p

[x

0

; : : : ; x

s

] is the projective Example 3.3 and ' is the inclusion. Since

Hom

HA

+

(H(n);H) = D

n

H

and ' de�nes V we get an exact sequence

0! Hom

HA

+

(F

p

[x

s

];H) ! D

n

H

V

!D

n

p

H ! Ext

1

HA

+

(F

p

[x

s

];H) ! 0:

Since Hom

HA

+

(F

p

[x

s

];H)

�

=

(PH)

n

, the result follows from the next lemma.

Lemma 3.13. For all H 2 HA

+

and all m � 0, there are natural isomorphisms

Ext

m

HA

+

(F

p

[x

s

];H)

�

=

!(R

m

PH)

n

:

19

Proof: Let F

p

! H

0

! H

1

! H

2

! F

p

be a short exact of Hopf algebras. Then one

gets a long exact sequence in Ext

�

and in R

�

P (�), the latter by Theorem 3.6 of [4]. Since

HA

+

has enough injectives and R

m

PH = 0 for m � 1 for all injectives H, by Proposition

3.10, and

Hom

HA

+

(F

p

[x

s

];H)

�

=

(PH)

n

the result follows from standard facts about derived functors in abelian categories. See [6],

p. 206, for example. �

Note that Lemma 3.13 and the two term projective resolution of 3.12.1 immediately

imply

Corollary 3.14. For all H 2 HA

+

, R

s

PH = 0 for s � 2.

Also, Proposition 3.12 immediately implies

Corollary 3.15. A Hopf algebra H 2 HA

+

is injective as a coalgebra if and only if

V : D

�

H ! �D

�

H is surjective.

In the future we may say V is surjective on M if V :M ! �M is surjective.

x4. Derived functors of limits for abelian Hopf algebras.

We apply the technology of the previous section to calculate R

�

CA

lim

I

H for various

diagrams of abelian Hopf algebras. Note that just as the forgetful functor from abelian

groups to sets makes all limits of abelian groups, so too does the forgetful functor from

HA

+

to CA

+

. Thus if H : I ! HA

+

is a diagram of Hopf algebras, the expression lim

I

H

is unambiguous.

Recall that R

�

CA

lim

I

C = �

�

diag�

�

�

S

�

C is a bigraded coalgebra. The elements of

R

s

CA

lim

I

C will be said to have cosimplicial degree s. An element in [R

�

CA

lim

I

C]

t

will be

said to have internal degree t.

The �rst result concerns products; that is, the case when I is discrete.

Proposition 4.1. Let fH

�

g be an arbitrary set of abelian Hopf algebras. Then

R

s

CA

(

�

�

)H

�

= 0 for s > 0 and

R

0

CA

(

�

�

)H

�

=

�

�

H

�

:

The proof is below in 4.4 after some preliminaries. Note that the statement about R

0

is automatic (See 1.9.3.), so the heart of the matter is the vanishing of the higher derived

functors.

20

To set the stage for the proof, let D

�

H

�

2 D be the Dieudonn�e module of H

�

. In D,

choose an inclusion D

�

H

�

! J

0

�

where J

0

�

is injective. This is possible by Corollary 3.7.

Let J

1

�

be the cokernel of the inclusion, so there is an exact sequence in D

(4.2) 0! D

�

H

�

! J

0

�

! J

1

�

! 0:

Since J

0

�

is injective, Lemma 3.11 and Corollary 3.15 imply V : J

0

�

! �J

0

�

is surjective.

Since � is exact, V : J

1

�

! �J

1

�

is surjective. Thus (4.2) corresponds to a short exact

sequence of Hopf algebras

(4.3) F

p

! H

�

! K

0

�

! K

1

�

! F

p

with D

�

K

0

�

�

=

J

0

�

and D

�

K

1

�

�

=

J

1

�

and, by Corollary 3.15, K

0

�

and K

1

�

are injective as

coalgebras.

4.4 Proof of Proposition 4.1: The central observation is that because the sequence

of Dieudonn�e modules

0!

�

�

D

�

H

�

!

�

�

J

0

�

!

�

�

J

1

�

! 0

is exact, the sequence of Hopf algebras

F

p

!

�

�

H

�

!

�

�

K

0

�

!

�

�

K

1

�

! F

p

is exact. (Compare Remark 4.8 below.) We will argue that the " = 0; 1

(4.5) R

�

CA

(

�

�

K

"

�

)

�

=

�

�

K

"

�

concentrated in cosimplicial degree 0 and then, using an argument of Bous�eld's, that the

result follows.

To begin, note that if we have a set of coalgebras of the form S(V

�

) then

R

�

CA

(

�

�

S(V

�

))

�

=

�

�

S(V

�

) concentrated in cosimplicial degree zero. To see this, note that

the canonical resolution

S(V

�

)!

�

S

�

S(V

�

)

has a cosimplicial contraction, hence

�

S(V

�

)!

�

�

�

S

�

S(V

�

)

has a cosimplicial contraction. Now use Example 1.11.

From this observation and Proposition 3.10, equation 4.5 follows.

21

To complete the argument, we adapt an argument of Bous�eld's (pp. 477{479 of [4]).

The reader who doesn't like the language of closed model categories can use Bous�eld's

explicit constructions.

Consider, for each �, the canonical resolution

K

1

�

!

�

S

�

K

1

�

:

Regarding K

1

�

as a constant simplicial coalgebra, this is a weak equivalence from K

1

�

to a

�brant cosimplicial coalgebra. In cCA

+

, factor the composite K

0

�

! K

1

�

!

�

S

�

K

1

�

as

K

0

�

j

!W

�

�

q

�

!

�

S

�

K

1

�

where j is a weak equivalence and q

�

is a �bration. This can be done explicitly using the

cobar construction of [4]. Since compositions of �brations are �brations,W

�

�

is �brant, so

j : K

0

�

!W

�

�

is a resolution of K

0

�

. Now let

�

W

�

�

= F

p

�

�

S

�

K

1

�

W

�

�

and

(4.6) H

�

�

=

F

p

�

K

1

�

K

0

�

!

�

W

�

�

the induced map. Since pull-backs of �brations are �brations,

�

W

�

�

is �brant. Also, the

pull-back diagram of cosimplicial coalgebras

�

W

�

�

w

u

W

�

�

u

q

�

F

p

w

�

S

�

K

1

�

yields a spectral sequence, because q

�

is a �bration:

Cotor

s

�

��

S

�

K

1

�

(F

p

; �

�

W

�

�

)

t

) �

s+t

�

W

�

�

:

(To see this, use [11, xII.6] where the simplicial case is considered, or adapt [4].) The degree

t comes from the cosimplicial degree. Since �

�

�

S

�

K

1

�

�

=

K

1

�

, �

�

W

�

�

�

=

K

0

�

in degree 0 and

K

0

�

! K

1

�

is a surjective map of Hopf algebras, Cotor

s

= 0 for s > 0 and Cotor

0

�

=

H

�

, so

(4.6) is a resolution of H

�

.

22

Now there are isomorphisms

R

�

CA

(

�

�

)K

0

�

�

=

�

�

(

�

�

W

�

�

)

�

=

�

�

K

0

�

(4.7.1)

R

�

CA

(

�

�

)H

�

�

=

�

�

(

�

�

�

W

�

�

):(4.7.2)

To see this, for example, in the second case, note there is a weak equivalence

�

S

�

H

�

!

�

W

�

�

in

cCA

+

. Since every object of cCA

+

is co�brant, this is a cosimplicial homotopy equivalence.

Hence

�

�

�

S

�

H

�

!

�

�

W

�

�

is a homotopy equivalence. The second isomorphism of (4.7.1) is

(4.5).

There is a pull-back diagram in cCA

+

�

�

�

W

�

�

u

w

�

�

W

�

�

u

�

q

�

F

p

w

�

�

�

S

�

K

1

�

:

Since products of �brations are �brations we get, as above, a spectral sequence

Cotor

s

�

�

K

1

�

(F

p

;

�

�

K

0

�

)

t

) �

s+t

(

�

�

�

W

�

�

):

Here we use (4.5). Since

�

�

K

0

�

!

�

�

K

1

�

is a surjective map of Hopf algebras (this is the observation at the beginning of the proof)

Cotor

s

= 0 for s > 0 and, in degree 0,

F

p

�

�

�

K

1

�

�

�

K

0

�

�

=

�

�

(F

p

�

K

1

�

K

0

�

)

�

=

�

�

H

�

as required. �

Remark 4.8: This argument, although long, is basically formal except for one point. To

elucidate this, consider the situation: Bous�eld de�nes a coalgebra to be nice if there is a

sequence of coalgebras

F

p

! C

j

!C

0

q

!C

1

! F

p

so that qj is trivial, q is surjective, C

0

is an injective C

1

comodule, there are vector spaces

W

0

and W

1

and isomorphisms C

0

�

=

SW

0

, C

1

�

=

SW

1

, and C

�

=

F

p

�

C

1C

0

. Given a set

of nice coalgebras C

�

one has such sequences

F

p

! C

�

! C

0

�

q

�

!C

1

�

! F

p

23

and can form

F

p

! �C

�

! �C

0

�

�q

�

��!�C

1

�

but it isn't clear that �q

�

is surjective, because of the way the product is de�ned. (See

1.2.1 and 1.2.2.) So the �nal step of the argument breaks down. What we have done is

arrange a situation where the product map is surjective.

Proposition 4.1 has the following immediate consequence.

Corollary 4.9. Let H : I ! HA

+

be a diagram of abelian Hopf algebras. Then the

augmentation

�

�

H ! diag(�

�

�

S

�

H)

induces an isomorphism

�

�

(�

�

H)

�

=

!R

�

CA

lim

I

H:

Proof: The argument is exactly that of Example 1.10, once one applies Proposition 4.1

to prove

R

�

CA

(�

p

)H

�

=

�

�

�

p

�

S

�

H

�

=

�

P

H

concentrated in cosimplicial degree zero. �

We use this to make a sequence of calculations. We'll need the following lemma, whose

proof is postponed to the next section because the argument uses more model category

technology.

Lemma 4.10. Let H

�

! K

�

be a morphism of cosimplicial abelian Hopf algebras. If

�

�

D

�

H

�

! �

�

D

�

K

�

is an isomorphism, then

�

�

H

�

! �

�

K

�

is an isomorphism.

Corollary 4.11. Let H : I ! HA

+

be a diagram of abelian Hopf algebras. If lim

I

s

D

�

H

= 0 for s > 0, then R

s

CA

lim

I

H = 0 for s > 0 and

R

0

CA

lim

I

H

�

=

lim

I

H:

Note: In this statement, lim

I

s

D

�

H are the usual derived functors of graded abelian groups.

24

Proof: Again the statement about R

0

is formal. For the higher derived functors, consider

the cosimplicial Dieudonn�e module

D

�

(�

�

H)

�

=

�

�

D

�

H:

Then �

s

�

�

D

�

H

�

=

lim

I

s

D

�

H = 0 if s > 0. Hence the augmentation

lim

I

D

�

H ! �

�

D

�

H

induces a �

�

isomorphism from the constant cosimplicial Dieudonn�e module of lim

I

D

�

H

to �

�

D

�

H. Thus, by Lemma 4.10, the induced map

lim

I

H ! �

�

H

from the constant cosimplicial object is a �

�

isomorphism and

lim

I

H

�

=

�

�

(�

�

H)

�

=

R

�

CA

lim

I

C

(using 4.9) concentrated in degree zero. �

Example 4.12: Suppose we have a tower over the natural numbers in HA

+

H

1

H

2

H

3

� � � :

Then under any of the following conditions

R

�

CA

limH

i

�

=

lim

i

H

i

concentrated in cosimplicial degree zero,

1) Each of the Hopf algebras is of �nite type;

2) The tower fH

i

g is pro-isomorphic, in the category of Hopf algebras, to a constant

tower;

3) The tower fH

i

g is Mittag-Le�er.

Proof: In each case lim

s

D

�

H

i

= 0 for s > 0. For s > 1, this is always true. So one need

only argue lim

1

D

�

H

i

= 0. Case by case we have:

1) If H 2 HA

+

is of �nite type then D

�

H is of �nite type. This is because D

n

H

�

=

Hom

HA

+

(H(n);H) � Hom

algebras

(H(n);H) and H(n) is a free algebra on �nitely many

generators. The result now follows because lim

1

vanishes on graded abelian groups of �nite

type.

25

2) If fH

i

g is pro-isomorphic to a constant tower, so is fD

�

H

i

g. Hence lim

1

D

�

H

i

= 0.

3) Mittag-Le�er is the following condition: for �xed i, let

H

(k)

i

= ImfH

i+k

! H

i

g:

Then H

(k+1)

i

� H

(k)

i

and one demands that this descending chain stabilize: for each i,

there is an N so that H

(N)

i

= \H

(k)

i

. The claim is that

D

�

(H

(k)

i

)

�

=

ImfD

�

H

i+k

! D

�

H

i

g = (D

�

H

i

)

(k)

:

If so, the tower of abelian groups fD

�

H

i

g is Mittag-Le�er and has vanishing lim

1

. (See

[5, xIX.3].)

To see the claim, note that since D

n

H = Hom

HA

+

(H(n);H),

(D

�

H

i

)

(k)

� D

�

(H

(k)

i

):

To get the other inclusion, given a map H(n) ! H

(k)

i

one gets a diagram

H

i+k

u

H

i+k

u

H(n)

i

i

i

ij

w H

(k)

i

w

�

H

i

:

The dotted arrow exists since H

i+k

! H

(k)

i

is surjective and H(n) is projective. �

We next turn to the case where lim

I

s

D

�

H = 0 for s � 2; that is

�

s

�

�

D

�

H = 0

for s � 2. We indicate how to calculate R

�

CA

lim

I

H.

First, suppose D

�

is a cosimplicial Dieudonn�e module, and H

�

a cosimplicial abelian

Hopf algebra so that D

�

H

�

�

=

D

�

: Suppose

�

s

D

�

= 0

for s � 2. We now state the calculation of �

�

H

�

.

Note that �

0

D

�

and �

1

D

�

are Dieudonn�e modules and that D

�

�

0

H

�

�

=

�

0

D

�

. De�ne

graded vector spaces P�

1

and R

1

P�

1

by the exact sequence

0! P�

1

! �

1

D

�

V

!��

1

D

�

! R

1

P�

1

! 0:

If k is a �eld and V a graded vector space over k which is concentrated in even degrees if

the characteristic of k is not 2, let k[V ] be the \polynomial algebra" on V ; that is, k[V ] is

the symmetric algebra on V , which becomes isomorphic to a polynomial algebra after one

chooses a basis.

26

Lemma 4.13. Suppose p > 2 and �

1

D

�

is trivial in odd degrees. Then there is an isomor-

phism of bigraded Hopf algebras

�

�

H

�

�

=

�

0

H

�

�(P�

1

) F

p

[R

1

P�

1

]

where P�

1

and R

1

P�

1

are in cosimplicial degree 1 and 2 respectively. Furthermore the

elements of P�

1

and R

1

P�

1

are primitive.

This will be proved in the next section. If �

1

D

�

is not trivial in odd degrees one can

still write down the answer, but it is more complicated. The techniques of the next section

su�ce for the computation.

If p = 2, the conclusion of 4.13 holds coalgebras without restrictions on �

1

D

�

, however

there is the possibility of an algebra extension.

Lemma 4.14. If p = 2, there is a natural homomorphism

� : P�

1

! R

1

P�

1

doubling internal degree and an isomorphism of Hopf algebras

�

�

H

�

�

=

�

0

H

�

F

2

[P�

1

] F

2

[R

1

P�

1

]=(�x + x

2

)

where P�

1

and R

1

P�

1

are in cosimplicial degree 1 and 2 respectively. Furthermore P�

1

and R

1

P�

1

are primitive.

This also will be proved in the next section.

Example 4.15: Consider a tower of abelian Hopf algebras over the natural numbers

H

1

H

2

� � �

where eachH

i

is injective as a coalgebra. Then we must calculate with lim

1

D

�

H

i

. However

there is an exact sequence

0! limD

�

H

i

! �D

�

H

i

! �D

�

H

i

! lim

1

D

�

H

i

! 0:

Since V is surjective on D

�

H

i

for all i (by Corollary 3.15) it is surjective on lim

1

D

�

H

i

.

Hence R

1

P�

1

= 0 and

R

�

CA

limH

i

= limH

i

�(P lim

1

D

�

H

i

):

27

Furthermore, treating �D

�

H

i

! �D

�

H

i

as a two term cochain complex, one gets a short

exact sequence

0! R

1

P (limH

i

)! lim

1

PH

i

! P lim

1

D

�

H

i

! 0:

Hence if R

1

P (limH

i

) = 0, or equivalently, if limH

i

is an injective coalgebra, lim

1

PH

i

�

=

P lim

1

D

�

H

i

.

To be speci�c, let H

i

= �

p

(x

i

; x

i+1

; : : : ) be the divided algebra on in�nitely many

generators of degree 2t for some t, and let H

i+1

! H

i

be the inclusion. Then limH

i

= F

p

and

lim

1

PH

i

=

1

�

i=1

F

p

=

1

M

i=1

F

p

:

Hence

R

�

CA

limH

i

�

=

�(

1

�

i=1

F

p

=

1

M

i=1

F

p

)

with the generators in cosimplicial degree 1 and internal degree 2t. Note that R

s

CA

limH

i

6=

0 for all s � 1.

Example 4.16: Let p > 2 and H

i

= F

p

[x

i

; x

i+1

; : : : ]=(x

p

i

; x

p

i+1

; : : : ) be the truncated

polynomial algebra on primitive generators of degree 2t and H

i+1

! H

i

the inclusion.

Then if W =

1

�

i=1

F

p

=

1

L

i=1

F

p

in degree 2t, then

R

�

CA

limH

i

= �(W ) F

p

[�W ]

with W and �W in cosimplicial degrees 1 and 2 respectively. The same example at p = 2

yields, with W in cosimplicial degree 1,

R

�

CA

limH

i

= F

2

[W ]:

Note in this case PR

�

CA

limH

i

is non-zero in in�nitely many cosimplicial degrees.

Example 4.17: Let G be a free group on �nitely many generators and H 2 HA

+

a G-

Hopf algebra; that is G acts on H through morphisms in HA

+

. Then D

�

H is a G-module

and

limD

�

H = H

0

(G;D

�

H) = (D

�

H)

G

lim

s

D

�

H = H

s

(G;D

�

H)and

28

which is zero for s > 1. Finally H

�

(G;D

�

H) can be calculated by an exact sequence

(4.18) 0! H

0

(G;D

�

H) ! D

�

H

@

!

n

�

i=1

D

�

H ! H

1

(G;D

�

H) ! 0

where, if �

i

2 G, 1 � i � n, are the generators

@(x) = (x � �

1

x; : : : ; x � �

n

x):

Let's confuse G with category with one object and G as morphisms. Note that in CA

+

the inclusion lim

G

H � H

G

is usually strict.

Assume p > 2 and that H is concentrated in even degrees. Then

R

�

CA

lim

G

H

�

=

lim

G

H �(PH

1

(G;D

�

H)) F

p

[R

1

PH

1

(G;D

�

H)]:

If H is injective as a coalgebra (e.g., H

�

BU), then V is surjective on D

�

H, so on

H

1

(G;D

�

H) by 4.18 and

R

�

CA

lim

G

H

�

=

lim

G

H �(PH

1

(G;D

�

H)):

The latter formula holds at p = 2 if H is injective as a coalgebra, but not necessarily in

even degrees.

We �nish this example by showing that R

�

CA

lim

G

H can be identi�ed as a Cotor. Fix

generators �

1

; : : : ; �

n

of G and de�ne two maps

H ! H � � � H

| {z }

n+1

= H

n+1

as follows. The �rst is �

n+1

: H ! H

n+1

, the (n + 1)

st

iterated diagonal. The second is

the composite

H

�

n+1

���!H

n+1

1�

1

����

n

������������!H

n+1

:

Both of these maps make H into an H

n+1

comodule. Denote the �rst comodule structure

by H, the second by H

�

.

Proposition 4.19. Let H be a connected G-abelian Hopf algebra. Then there is a natural

isomorphism of bigraded Hopf algebras

R

�

CA

lim

G

H

�

=

Cotor

H

n+1(H;H

�

):

29

Proof: Let B

�

(H) = B

�

(H;H

n+1

;H

�

) be the cobar construction. we will show there is

a natural map

(4.20.1) �

�

D

�

H = D

�

�

�

H ! D

�

B

�

(H)

giving an isomorphism on cohomotopy. The result will then follow from Corollary 4.9 and

Lemma 4.10.

If M is any G-module, then one de�nes

H

s

(G;M) = lim

G

s

M = �

s

�

�

M:

However, one can calculate H

�

(G;M) as the cohomology of the two stage cochain complex

T

�

(M) = fM �M

@

!M � � � � �M

| {z }

n+1

g

where @(x; y) = (x�y; x��

1

y; : : : ; x��

n

y). Indeed, H

0

T

�

(M)

G

�

=

M

G

andH

s

T

�

(M) = 0

for s > 0 and M an injective G-module, so there must be a natural map of cochain

complexes

(4.20.2) N�

�

M ! T

�

(M)

inducing an isomorphism on cohomology. (This is a standard fact about derived functors;

see [6] x3.1). Now one easily checks that ND

�

B

�

(H)

�

=

T

�

D

�

H. Hence the normalized

cochain equivalence of (4.20.2) gives an (unnormalized) cohomotopy equivalence in (4.20.1)

and the result follows. �

x5. From cosimplicial Dieudonn�e modules to cosimplicial Hopf algebras.

In this section we prove those technical results needed in Section 4 on passing from

�

�

D

�

H

�

to �

�

H

�

where H

�

is a cosimplicial abelian Hopf algebra. We end with some

generalities.

The �rst result we used was Lemma 4.10, which we record now as:

Proposition 5.1. Let H

�

! K

�

be a morphism of cosimplicial abelian Hopf algebras. If

�

�

D

�

H

�

! �

�

D

�

K

�

is an isomorphism, then �

�

H

�

! �

�

K

�

is an isomorphism.

30

This is proved by combining Lemmas 5.3, 5.4 and 5.6 below with some model category

techniques. The argument is given in 5.7.

Because D has enough injectives there is a simplicial model category structure on the

category cD. Indeed, the simplicial structure is the obvious analog of that given for cCA

+

in (2.1) and a morphism f : A

�

! B

�

in cD is

5.2.1) a weak equivalence if it is a �

�

isomorphism;

5.2.2) a co�bration if the normalized chain complex Nf : NA

�

! NB

�

is an injective in

positive degrees; and

5.2.3) a �bration if it is a surjection and K

�

= kerff : A

�

! B

�

g is level-wise an injective

object.

The proof is entirely standard; see [2, Proposition 6.5] or [11, xII.4] and use the fact

that the opposite category of cD is the category of simplicial objects in D

op

and D

op

is

an abelian category with enough projectives.

Lemma 5.3. Suppose H

�

! K

�

is a morphism of cosimplicial abelian Hopf algebras. If

D

�

H

�

and D

�

K

�

are �brant in cD and �

�

D

�

H

�

! �

�

D

�

K

�

is an isomorphism, then

�

�

H

�

! �

�

K

�

is an isomorphism.

Proof: Every object of cD is co�brant so a weak equivalence of �brant objects is a

homotopy equivalence. Hence H

�

! K

�

is a cosimplicial homotopy equivalence. �

Lemma 5.4. Let H

�1

! H

�

be an augmented cosimplicial abelian Hopf algebra and

suppose �

�

D

�

H

�

�

=

D

�

H

�1

concentrated in cosimplicial degree 0. Then �

�

H

�

�

=

H

�1

.

Proof: We will de�ne sub-cosimplicial Hopf algebras

H

�1

= H(�1) � H(0) � H(1) � � � � � H

�

so that �

�

H(n)

�

=

�

�

H(n + 1) for all n and H(n)

s

= H(n + 1)

s

for all s � n. The result

follows.

To de�ne H(n) consider the normalized augmented cochain complex of Dieudonn�e

modules

0! D

�

H

�1

! ND

�

H

0

@

!ND

�

H

1

@

!ND

�

H

2

! � � � :

Let K

n

= ker(@ : ND

�

H

n

! ND

�

H

n+1

)

�

=

Im(@ : ND

�

H

n�1

! ND

�

H

n

) and let

H(n) 2 cHA

+

be the augmented object with augmented normalized Dieudonn�e module

0! DH

�1

�

! ND

�

H

0

! ND

�

H

1

! � � � ! ND

�

H

n

! K

n+1

! 0! � � �

31

Then there is a pull-back diagram in cHA

+

, n � 0,

H(n � 1) w

u

H(n)

u

F

p

w K(n)

where K(n) 2 cHA

+

is the object with normalized Dieudonn�e module

(5.5) � � � 0! K

n+1

! K

n+1

! 0 � � �

in degrees n and n+ 1. Thus H

�1

= H(�1), and there is a spectral sequence

Cotor

�

�

�

K(n)

(F

p

; �

�

H(n)) ) �

�

H(n � 1):

Now the chain complex of (5.5) has a contraction, so K(n) has a cosimplicial contraction

and the �

�

K(n)

�

=

F

p

. �

Lemma 5.6. Let H

�

2 cHA

+

be a cosimplicial abelian Hopf algebra. Then there is a

natural map " : H

�

! H

�

1

in cHA

+

so that �

�

" and �

�

D

�

" are isomorphisms and D

�

H

�

1

is �brant as an object in cD.

Proof: For M 2 D, let (with J(n) as in 3.8)

J(M) =

�

n

�

J(n)

x2Hom(M

n

;Q =Z )

and let M ! J(M) be the evident inclusion. Then for M 2 D there is a natural injective

resolution

M ! J

0

(M) ! J

1

(M) ! � � �

with J

0

(M) = J(M) and J

n+1

(M) = J(cokerJ

n�1

(M) ! J

n

(M)). Thus in H 2 HA

+

one gets a natural cosimplicial resolution in cHA

+

H ! J(H)

�

where the normalized Dieudonn�e module of J(H)

�

is J

�

(D

�

H). By Lemma 5.4, H

�

=

�

�

J(H). If H

�

2 cHA

+

let J(H

�

)

�

be the obvious bicosimplicial object and set H

�

1

=

diagJ(H

�

)

�

. A bicomplex argument shows that " : H

�

! H

�

1

is an �

�

and �

�

D

�

iso-

morphism. Since D

�

H

�

1

is a product of injectives at each level, D

�

H

�

1

is �brant by the

de�nition 5.2.3. �

32

5.7 Proof of Proposition 5.1: Use Lemma 5.6 to produce a diagram

H

�

u

"

H

w

f

K

�

u

"

K

H

�

1

w

f

1

K

�

1

so that the conclusions of Lemma 5.6 hold. Now D

�

f , D

�

"

H

, and D

�

"

K

are all �

�

isomor-

phisms, so D

�

f

1

is a �

�

isomorphism. Hence, by Lemma 5.3 and Lemma 5.6, �

�

f

1

is an

isomorphism. Since �

�

"

H

and �

�

"

K

are isomorphisms, so is �

�

f . �

The next project is to calculate �

�

H

�

where �

s

D

�

H

�

= 0 for s > 1. By Lemma 5.1

we may assume ND

�

H

s

= 0 for s > 1, simply by letting

H(1)

�

� H

�

be the sub-cosimplicial Hopf algebra with

ND

�

H(1)

s

=

8

<

:

ND

�

H

0

s = 0

ker(ND

�

H

1

! ND

�

H

2

) s = 1

0 s � 2:

Then �

�

D

�

H(1)

�

�

=

�

�

D

�

H

�

, so �

�

H(1)

�

�

=

�

�

H

�

. For short, let us write D

s

for ND

�

H

s

.

Then there is a pull-back diagram of cochain complexes

ND

�

H

�

u

w fD

1

! D

1

! 0! � � � g

u

fD

0

! 0 � � � g w fD

1

! 0! � � � g

:

Hence, there is an associated pull-back diagram of cosimplicial Hopf algebras

H

�

u

w A

0

u

H

0

w A

1

:

Therefore we get a spectral sequence

E

s;t

2

= Cotor

s

A

1

(F

p

;H

0

)

t

) �

s+t

H

�

33

where A

1

is the Hopf algebra so that D

�

N

1

�

=

D

1

. The grading t arises from the cosimpli-

cial degree on these cosimplicial Hopf algebra. Since A

1

and H

0

are both in cosimplicial

degree 0, the spectral sequence collapses. Furthermore the obvious change of rings implies

(5.8) �

�

H

�

�

=

Cotor

�

A

1

(F

p

;H

0

)

�

=

�

0

H

�

Cotor

�

H

1

(F

p

;F

p

)

where H

1

is the Hopf algebra, so that D

�

H

1

�

=

�

1

D

�

H

�

.

We now show how to calculate Cotor

�

H

1

(F

p

;F

o

) as a functor of

PH

1

�

=

P�

1

D

�

H

�

(5.9)

R

1

PH

1

�

=

R

1

P�

1

D

�

H

�

:and

Choose an exact sequence of Dieudonn�e modules

0! D

�

H

1

! J

0

! J

1

! 0

where V is surjective on J

0

and J

1

. This can be done because D has enough injectives and

V is surjective on injectives. This corresponds to a short exact sequence of Hopf algebras

F

p

! H

1

! K

0

! K

1

! F

p

where K

i

, i = 0; 1, are injective as coalgebras. Let B

�

(�) denote the cobar construction,

so that

�

�

B

�

(K) = Cotor

�

K

(F

p

;F

p

):

Then we have a pull-back diagram of cosimplicial Hopf algebras

B

�

(H

1

) w

u

B

�

(K

0

)

u

F

p

w B

�

(K

1

)

and hence a spectral sequence

(5.10) E

s;t

2

= Cotor

s

�

�

B

�

(K

1

)

(F

p

; �

�

B

�

(K

0

))

t

) Cotor

s+t

H

1

(F

p

;F

p

)

If K 2 HA

+

is injective as a coalgebra, K

�

=

S(W ) for some graded vector space W and

(where if p > 2, W is concentrated is even degrees)

�

�

B

�

(K)

�

=

Cotor

�

K

(F

p

;F

p

)

�

=

�(PK):

34

This isomorphism is natural in maps of injective coalgebras. Thus we can write

E

2

�

=

Cotor

�

�(PK

1

)

(F

p

;�(PK

0

)):

By a change of rings since is isomorphic to

�(PH

1

) Cotor

�

�(R

1

PH

1

)

(F

p

;F

p

)(5.11)

�

=

�(PH

1

) F

p

[R

1

PH

1

]

since there is an exact sequence

0! PH

1

! PK

0

! PK

1

! R

1

PH

1

! 0:

Since (5.9) is a spectral sequence of Hopf algebras it must collapse, and there can be no

coalgebra extensions. Thus we can give

5.12: The proof of 4.13: Combine 5.8, 5.9, and 5.11 with the observations that if p > 2

there can be no algebra extensions. �

If p = 2 there can be an algebra extension in the spectral sequence of 5.10. For

example if H

1

is an exterior algebra on a single generator, then Cotor

H

1

(F

2

;F

2

)

�

=

F

2

[x].

This example is generic, and is detected by an operation � : PH

1

! R

1

PH

2

.

To de�ne � consider a short exact sequence of Dieudonn�e modules

0! D

�

H

1

"

!J

0

q

!J

1

! 0

where V is surjective on J

0

. Let x 2 PH

1

= kerfV : D

�

H

1

! �D

�

H

1

g. Then there is a

y 2 J

0

so that V y = x. The element

q(y) 2 PH

1

= kerfV : J

1

! �J

1

g:

Then �(x) is the image of q(y) in R

1

PH

1

in the exact sequence

0! PH

1

! PH

0

! PJ

1

! R

1

PH

1

! 0:

It is an easy exercise that this is independent of the choices and �(x) = 0 if and only if x

is in the image of V in D

�

H

1

.

5.13: The proof of 4.14: Because the spectral sequence 5.10 collapses and there can

be no coalgebra extensions, the result follows from 5.8, 5.9, and 5.11 once we determine

35

the algebra extensions. Since Cotor

H

1

(F

p

;F

p

) commutes with �ltered colimits, we may

assume H

1

is of �nite type. Thus

Cotor

H

1

(F

2

;F

2

)

�

=

Ext

H

�

1

(F

2

;F

2

):

Use Borel's structure theorem to write an algebra decomposition

H

�

1

�

=

O

i

F

2

[x

i

]=(x

2

n

i

i

);

where 1 � n

i

�1. Since an algebra decomposition of this sort yields a tensor product on

Ext, we may assume

H

�

1

�

=

F

2

[x]=(x

2

n

)

with 1 � n �1. But, if deg(x) = t, we have

(D

�

H)

k