A NOTE ON THE COHOMOLOGY OF FINITE
DIMENSIONAL COCOMMUTATIVE HOPF ALGEBRAS
JOHN H. PALMIERI
Abstract.In the context of finite dimensional cocommutative Hopf alge-
bras, we prove versions of various group cohomology results: the Quille*
*n-
Venkov theorem on detecting nilpotence in group cohomology, Chouinard's
theorem on determining whether a kG-module is projective by restricting *
*to
elementary abelian p-subgroups of G, and Quillen's theorem which identif*
*ies
the cohomology of G, "modulo nilpotent elements." This last result is on*
*ly
proved for graded connected Hopf algebras.
1.Introduction
When one studies the mod p cohomology of finite groups, one sees that element*
*ary
abelian p-subgroups play an important role. For example, one can detect nilpote*
*nce
in group cohomology by restricting to elementary abelian subgroups, one can det*
*ect
projectivity of modules, and up to F -isomorphism one can describe group cohomo*
*l-
ogy. In this paper we try to generalize all of this to finite dimensional cocom*
*mutative
Hopf algebras; Wilkerson laid the foundations for this work in his paper [15]. *
*Since
the setting is more general, the analogues of elementary abelian subgroups are *
*not
as pleasant.
Unless otherwise indicated, every Hopf algebra in this paper will be finite d*
*imen-
sional, graded, cocommutative, and defined over a field k of characteristic p >*
* 0. We
require cocommutativity for two reasons: so that the cohomology is a commutative
k-algebra with an action of the Steenrod algebra [8], and so that the Hopf alge*
*bra
is free as a left module over every sub-Hopf algebra [13].
Definition 1.1.A Hopf algebra C over k is elementary if it is bicommutative and
has xp = 0 for all x 2 IC, the augmentation ideal of C (we place no condition on
the comultiplication, other than cocommutativity). We will use C(x) to denote a
monogenic elementary Hopf algebra, generated by x; i.e., C(x) is isomorphic as *
*an
algebra to k[x]=(xn), where n is p or 2. For a Hopf algebra E, we call a nonzero
element v 2 Ext2E(k; k) a Serre element if there is a Hopf algebra extension
E0,! E!!C(x)
so that under the induced map in Ext, v is the image of a nonzero element of
Ext2C(x)(k; k). A Hopf algebra E is quasi-elementary if no product of Serre ele*
*ments
____________
Date. 27 September 1994.
1991 Mathematics Subject Classification. Primary 16W30,18G15, 57T05; Seconda*
*ry 20J06.
Research partially supported by NSF grant DMS-9407459.
1
2 JOHN H. PALMIERI
is nilpotent. (See the appendix for an alternate description of Serre elements *
*and
quasi-elementary Hopf algebras.)
In the p-group setting, "elementary" and "quasi-elementary" are the same no-
tion by a theorem of Serre [14]_they both correspond to the group algebras of
elementary abelian p-groups_but they differ for more general Hopf algebras. Wil*
*k-
erson [15] has shown that for every p, there is a (graded connected) non-elemen*
*tary
quasi-elementary Hopf algebra E over a field of characteristic p. For cohomolog*
*ical
purposes, quasi-elementary Hopf algebras carry more information; our main theo-
rems illustrate this. In other words, quasi-elementary Hopf algebras seem to be*
* the
correct generalization of elementary abelian p-groups. As such, it would be nic*
*e to
have alternate characterizations of them; see the appendix for a few examples.
We move on to the statements of the main results. These are analogous to group
cohomology theorems of Carlson [3], Chouinard [4], and Quillen [12]. The first *
*is a
generalization of a theorem of Wilkerson [15]; see also [10] (and we've pattern*
*ed the
statement on the nilpotence theorem of [5]). For all of these, B is a finite di*
*mensional
graded cocommutative Hopf algebra over k, char(k) = p.
For a class z 2 Ext*B(L; M), we say that z is tensor-nilpotent if the element
zn 2 Ext*B(Ln ; Mn ) is zero for n sufficiently large.
Theorem 1.2. (a)Let be an associative B-algebra with unit j : k ! ; fix
z 2 Ext*B(k; ). Then z is nilpotent if and only if resB;E(z) is nilpotent *
*for
every quasi-elementary sub-Hopf algebra E of B.
(b)Let M be a B-module; fix z 2 Ext*B(M; M). Then z is nilpotent if and only
if resB;E(z) is nilpotent for every quasi-elementary sub-Hopf algebra E of*
* B.
(c)Let L and M be B-modules; fix z 2 Ext*B(L; M). Then z is tensor-nilpotent *
*if
and only if resB;E(z) is tensor-nilpotent for every quasi-elementary sub-H*
*opf
algebra E of B.
Theorem 1.3. Let M be a B-module. Then M is projective if and only if M
restricted to E is projective for every quasi-elementary sub-Hopf algebra E of *
*B.
Theorem 1.4. Suppose that B is connected. Let AB be the category with objects
the quasi-elementary sub-Hopf algebras of B and their sub-Hopf algebras, and mo*
*r-
phisms given by inclusions. Then the natural map
qB : Ext*B(k; k) ! lim-E2ABExt*E(k; k)
is an F -isomorphism: the range is finitely generated as a module over Ext*B(k;*
* k),
every element in the kernelnof qB is nilpotent, and for every z in the range, t*
*here
is an integer n so that zp is in the image of qB .
We make a few comments about Theorem 1.4. First, it is not clear that in gene*
*ral
a sub-Hopf algebra of a quasi-elementary Hopf algebra is again quasi-elementary;
hence we have to include these sub-Hopf algebras explicitly in AB . Next, as in*
* the
group setting, this theorem has a useful reformulation in terms of varieties, a*
*nd can
also be extended to a similar result about Ext*B(M; M)_see [9]. Also, note that
in the corresponding statement for a group G, the objects in the category AG are
elementary abelian p-subgroups E of G, and the morphisms are given by inclusions
and conjugations (and composites of these). In the graded connected Hopf algebra
case, we don't need conjugations. In the group case, given E an elementary abel*
*ian
subgroup of G, the conjugations are needed to reflect the action on H*(E) by the
FINITE DIMENSIONAL HOPF ALGEBRAS 3
normalizer of E_the image of the restriction map from H*(G) is contained in the
invariants. By [6, Theorem 4.12], though, in the Hopf algebra case all of H*(E)*
* is
in the image of the restriction map, up to nilpotence.
It would be nice to remove the connectedness condition in Theorem 1.4; curren*
*tly,
though, it is not even known if Ext*B(k; k) is a finitely generated algebra whe*
*n B is
not connected, so it may be unreasonable to ask for more precise information. O*
*ur
proof will certainly not generalize, because we deduce Theorem 1.4 from the work
of Hopkins and Smith, and they make heavy use of connectedness. In any case, we
have the following conjecture.
Conjecture 1.5. Let B be a finite dimensional cocommutative Hopf algebra; let
AB be the category with objects the quasi-elementary sub-Hopf algebras of B and
their sub-Hopf algebras, and morphisms given by inclusions and conjugations by
elements of B. The natural map
qB : Ext*B(k; k) ! lim-E2ABExt*E(k; k)
is an F -isomorphism.
The structure of the paper is as follows: in Section 2 we explain our notatio*
*n and
review a few facts about Hopf algebras; in Section 3 we prove Theorems 1.2 and *
*1.3;
we prove Theorem 1.4 in Section 4; and we give some examples of quasi-elementary
Hopf algebras in the appendix.
Acknowledgments: I would like to thank Haynes Miller and Clarence Wilkerson
for providing the inspiration for the above definition of quasi-elementary; ind*
*eed,
Theorem 1.2 is essentially their result, at least for M = k (private communicat*
*ion).
Example A.7 is also due to them. I had several quite profitable discussions abo*
*ut
Hopf algebras with Mark Feshbach, as well. Lastly, I would like to acknowledge
the hospitality of the Universities of Minnesota, Wisconsin, and Washington, wh*
*ere
this research was conducted.
2. Notation and conventions
Throughout the paper all Hopf algebras are graded, cocommutative and defined
over a field k of characteristic p > 0 (cocommutative is in the usual graded se*
*nse).
E[x] (respectively, D[x]) will denote any such Hopf algebra which is isomorphic*
*, as
an algebra, to k[x]=(x2) (resp., k[x]=(xp)). (Note that if p is odd, then E[x] *
*has a
Hopf algebra structure only if x is in an odd grading.) We put no restriction o*
*n the
coalgebra structure, aside from cocommutativity. We will also use C(x) to denote
either E[x] or D[x], so C(x) is an arbitrary monogenic elementary Hopf algebra.
If B is a Hopf algebra, we use H*(B) to denote Ext*B(k; k), as usual. The
augmentation ideal of B is written IB. We write A B to mean that A is a sub-
Hopf algebra of B; similarly, A / B means that A is a normal sub-Hopf algebra of
B (so IA . B = B . IA). If A B, we write resB;Afor the restriction map
resB;A: Ext*B(- ; -) ! Ext*A(- ; -):
If A / B, then B==A is the quotient Hopf algebra B A k = B=(B . IA). If O denot*
*es
the anti-automorphism of B, then we can define conjugation by b 2 B to be the m*
*ap
B -cb! B;
X
a 7! b0iaO(b00i)
i
4 JOHN H. PALMIERI
P
where b has coproduct ib0i b00i. Then a sub-Hopf algebra A of B is normal if
and only if cb(a) 2 A for all a 2 A, b 2 B (see [11]). There are a number of ot*
*her
similarities between normal sub-Hopf algebras and normal subgroups; we will need
the following: if A; A0/B, then AA0 B; furthermore, we have A\A0/A0, A/AA0,
and there is an isomorphism
A0==(A \ A0) ~=(AA0)==A:
(The proof is a straightforward verification.)
Note that for any cocommutative Hopf algebra B, the cohomology of B has an
action of the Steenrod algebra (see [8, Section 11]); we use May's notation for
indexing of Steenrod operations on Hopf algebra cohomology.
3.Proofs of Theorems 1.2 and 1.3
We prove Theorems 1.2 and 1.3 by imitating proofs for the group theory ana-
logues. We start with the following definition; this replaces the product of t*
*he
Bocksteins of the one-dimensional classes in group cohomology.
Definition 3.1.Suppose B is not quasi-elementary, so that there is some product
of Serre elements which is nilpotent. Define the fundamental class of B, uB , t*
*o be
any such product. (This is far from unique, of course.)
The following is our key lemma, analogous to [15, 5.2] and [2, Vol. II, 5.2.1*
*].
Lemma 3.2. Suppose v is a Serre element, corresponding to the extension
A ! B ! C(x):
Let be an associative B-algebra with unit j : k ! , and fix z 2 Ext*B(k; ). If
resB;A(z) = 0, then z2 = j*(v)z0 for some z0.
Proof.This is proved just as in [2]; Benson attributes the proof to Kroll [7]. *
*We
prove this for the case C(x) = E[x] and leave the other case to the reader. Not*
*e that
in the E[x] case, the Serre element v is the square of some class w 2 Ext1B(k; *
*k).
(For the other case, v = fiPe0(w) for some w 2 Ext1B(k; k).)
There is a short exact sequence of E[x]-modules, and hence of B-modules:
(1) 0 ! k ! E[x] q-!k ! 0:
This module extension corresponds to w 2 Ext1B(k; k). Note that the composite
*
Ext *B(k; ) q-!Ext*B(E[x]; ) ~=Ext*A(k; )
is the restriction map_see [2, Vol. I, Section 2.8]. Consider the long exact se*
*quence
in Ext*B(- ; ) associated to the extension (1): since q*(z) = 0 then z is in th*
*e image
of the boundary homomorphism; but the boundary homomorphism is multiplication
by j*(w). Since this element is central in Ext*B(k; ), then z = j*(w)y for_some*
*_y;
so the result follows. |__|
Proof of Theorem 1.2.First we prove part (a); the other parts will follow from *
*this.
This is proved by induction on the sub-Hopf algebras of B using Lemma 3.2, ju*
*st
as in [15]. If A B and dimk(A) = 1, then A = k and the theorem is trivially tr*
*ue
for z 2 Ext*A(k; ). This starts the induction.
Suppose the theorem holds for all proper sub-Hopf algebras A of B, and suppose
z 2 Ext*B(k; ) is nilpotent upon restriction to every quasi-elementary sub-Hopf
FINITE DIMENSIONAL HOPF ALGEBRAS 5
algebra of B. We may assume that B is not quasi-elementary, so that B has a
fundamental class uB . For each Serre element v which is a factor in uB , we ha*
*ve
an extension
A ! B ! C(x):
By induction, we know that resB;A(z) is nilpotent, and so we may replace z with*
* a
large enough power so that resB;A(z) = 0. Hence z2 = j*(v)z00for some z00. Since
this is true for every factor of uB , we see that zr = j*(uB )i for some intege*
*r r and
some i 2 Ext*B(k; ). Since j*(uB ) is nilpotent and central in Ext*B(k; ), then*
* z is
nilpotent.
Part (b) follows from (a) via the algebra isomorphism
Ext*B(M; M) ~=Ext*B(k; Hom k(M; M)):
Part (c) also follows easily: we have Ext*B(L; M) ~=Ext*B(k; Hom k(L; M)), so*
* let
be the tensor-algebra of the B-module Hom k(L; M):
M
= (Hom k(L; M))n :
n0
Then z 2 Ext*B(L; M) is tensor-nilpotent if and only if the corresponding eleme*
*nt_
"z2 Ext*B(k; ) is nilpotent. |__|
Remark 3.3. The mod 2 Steenrod algebra A is a union of finite dimensional sub-
Hopf algebras, and one can prove Theorem 1.2 for A (see [10]). (Note that there*
* are
no Serre elements in Ext*A(F2 ; F2); so in this context (graded, connected, pos*
*sibly
infinite dimensional Hopf algebras), we add to the definition of quasi-elementa*
*ry
that there exist Serre elements in Ext2E(k; k).) It may be worthwhile to exami*
*ne
detection of nilpotence in ExtB, for any cocommutative Hopf algebra B which is
the union of finite dimensional sub-Hopf algebras.
We move on to a discussion of Theorem 1.3.
Lemma 3.4. Let B, v, and A be as in Lemma 3.2. Let M and N be B-modules.
Define the function
: Ext*B(M; N) ! Ext*B(M; N)
to be composition with v 1M . If the restriction of M to A is projective, then*
* is
an isomorphism in positive degrees.
Proof.This is proved as in [4]; namely, we observe that the spectral sequence
Ext*C(x)(k; Ext*A(M; N)) ) Ext*B(M; N)
associated to the extension
A ! B ! C(x)
collapses. |___|
Proof of Theorem 1.3.First, if M is projective over B, then by a result of Radf*
*ord
[13], M is projective over every E B. The converse is proved as in [4]: we
may assume that B is not quasi-elementary; let uB be a fundamental class for B.
Fix N an arbitrary B-module. By induction on the factors of uB and Lemma 3.4,
multiplication by uB 1M is an isomorphism in ExtsB(M; N) for s > 0. But uB is
nilpotent; hence ExtsB(M; N) = 0 for s > 0. Since this holds for all B-modules_*
*N,
then Hom B(M; -) is exact, and so M is projective over B. |__|
6 JOHN H. PALMIERI
Remark 3.5. For the sub-Hopf algebras of the mod 2 Steenrod algebra A, the
notions of elementary and quasi-elementary are the same_see [15]. So Theorem 1.3
provides an almost immediate proof of a theorem of Adams and Margolis [1], that
a bounded below A-module M is free if and only if H(M; Pts) = 0 for all Pts2
A with s < t. Indeed, their theorem is basically a special case of Theorem 1.3,
combined with [15, Theorem 6.4]. Here is a related conjecture; this would provi*
*de
a replacement for the Adams-Margolis theorem when working with non-bounded
below modules.
Conjecture 3.6. Theorem 1.3 holds for arbitrary sub-Hopf algebras of the mod 2
Steenrod algebra. In particular, it holds for the full Steenrod algebra.
4. Proof of Theorem 1.4
Let B be a graded connected cocommutative Hopf algebra over a field k of char-
acteristic p. Our goal here is to prove Theorem 1.4, the analogue of the Quill*
*en
stratification theorem. It follows from Wilkerson's work on finite generation i*
*n [15],
Theorem 1.2 for M = k, and the next result.
Theorem 4.1. Let C be a poset of sub-Hopf algebras of B which is closed under
inclusions_if A 2 C and A0 A, then A02 C. Then the natural map
q : Ext*B(k; k) ! lim-A2CExt*A(k; k)
is an F -surjection: for every z in the range, there is an integer n so that zp*
*n is in
the image of q.
This in turn essentially follows from a theorem of Hopkins-Smith [6, Theoremn
4.12]: if A is a sub-Hopf algebra of B and z 2 H*(A), then for some n, zp is i*
*n the
image of resB;A.
Proof.The goal here is to show that if we have a compatible family of elements
zA 2 H*(A) for each A 2 C,nthen there is an element y 2 H*(B) and an integer n
so that resB;A(y) = (zA )p for each A.
By induction on the sub-Hopf algebras of B, it suffices to consider the case *
*where
C is the collection of all proper sub-Hopf algebras of B: let B0be a minimal su*
*b-Hopf
algebra of B not in C; then C contains all of the proper sub-Hopf algebras of B*
*0,
so there is an element z of H*(B0) compatible with powers of the other zA 's, a*
*nd
hence we can add B0to C (after replacing the zA 's with large enough powers_note
that each B0 has finitely many sub-Hopf algebras).
We have the following lemmas.
Lemma 4.2. If A is a maximal sub-Hopf algebra of B, then A is normal and
B==A ~=C[u]. Slightly more generally, if A B and A0 B are maximal with
A 6= A0, then
A0==(A \ A0) ~=B==A ~=C[u]:
Proof.For the first statement, see [6, Lemma A.11]. To identify the quotient Ho*
*pf
algebra in the second statement, note that AA0= B_since A; A0/ B, then AA0is
a sub-Hopf algebra of B, and it properly contains the maximal sub-Hopf algebra *
*A.
Now, use the isomorphism given in Section 2:
A0==(A \ A0) ~=AA0==A:
|___|
FINITE DIMENSIONAL HOPF ALGEBRAS 7
Let A1; A2; : :;:An be the (distinct) maximal elements of C. Then for each i *
*we
have a Hopf algebra extension
Ai! B ! C[ui]:
Let vi2 Ext2B(k; k) be the corresponding Serre element.
p ____
Lemma 4.3. With the above notation, we have (vi) ker(resB;Ai) (vi).
Slightly more generally, for i 6= j we have
q _________
(vi) + (vj) ker(resB;Ai\Aj) (vi) + (vj):
Proof.The statement about ker(resB;Ai) is the content of [15, Proposition 5.2],*
* or
equivalently Lemma 3.2. It immediately follows that (vi) + (vj) is contained in
ker(resB;Ai\Aj). Suppose x 2 ker(resB;Ai\Aj). Then
resB;Ai(x) 2 ker(resAi;Ai\Aj):
p ___________
If we can show that ker(resAi;Ai\Aj) (resB;Ai(vj)), we are done. We invokep*
*__
Lemma 4.2: since Ai==(Ai\Aj) ~=C[uj], then ker(resAi;Ai\Aj) is contained in (*
*v),
where v is the homomorphic image of the polynomial generator in H*(Ai==Ai\ Aj)._
This v is precisely resB;Ai(vj). |__|
For each i, we have an elementnzi2 H*(Ai); by the Hopkins-Smith theorem, we
can replace each zi by zpi so that there is an element xi 2 H*(B) which restric*
*ts
to zi. There are many choices for such an xi_by Lemma 4.3, any element in the
coset xi+ (vi) will do. We want to find a single element x which restricts to e*
*ach
zi. We may raise everything to a large power; this means that we want to show t*
*hat
for some r, the set
n" r
(xpi+ (vi))
i=1
is nonempty. We do thisTby induction: clearly, x1 + (v1) is nonempty. Assume we
have an element y 2 k-1i=1(xi+ (vi)). For some r, we want an element in
"k r k-1" r ! r
(xpi+ (vi)) = (yp + (vi)) \ (xpk+ (vk));
i=1 i=1
so it certainly suffices to find an element in
r pr
(yp + (v1v2: :v:k-1)) \ (xk + (vk)):
Since y and xk restrict compatibly to Ai\ Ak, then y - xk 2rker(resB;Ai\Ak) for
each i. By Lemma 4.3, if we replace y and xk by ypr and xpk for r large enough,
then for each i we have elements ai and bi so that
(2) y - xk - bivk = aivi:
Suppose pr is the smallest power of p at least as big as k - 1; then we multipl*
*y the
equations (2) together (with the first with multiplicity pr - k + 2) to get
r pr
yp - xk + cvk = dv1v2: :v:k-1
for some elements c and d. This equality shows that the desired intersection_is
nonempty, and so completes the induction. |__|
8 JOHN H. PALMIERI
Appendix A. Some quasi-elementary Hopf algebras
The above definitions of Serre elements and quasi-elementary Hopf algebras are
a bit on the abstract side. In this appendix, we recall from [15, Proposition *
*5.1]
an alternate characterization of Serre elements of positive internal degree, we*
* give
a different description of connected quasi-elementary Hopf algebras, and we giv*
*e a
few examples.
Lemma A.1 ([15]). Let B be a graded cocommutative Hopf algebra over a field k
of characteristic p. Fix a nonzero v 2 Ext2;nB(k; k), with n > 0.
(a)Suppose p = 2. Then v is a Serre element if and only if v = w2 for some
0
w 2 Ext1B(k; k) \ kerfSq.
(b)Suppose p is odd. Then v is a Serre element if and only if v = w2 for some
w 2 Ext1;mB(k; k) with m odd, or v = fiPe0(w) for some w 2 Ext1;mB(k; k) \
kereP0with m even.
0
(Wilkerson proves, for example, that if w 2 kerfSq, then w2 is a Serre elemen*
*t;
the converse is immediate.) The following seems like a reasonable compromise
0
between the above case and the group algebra case (in which fSqacts as the iden*
*tity),
0 1
and is inspired by the possible actions of fSq (or eP0) on ExtC(x)(k; k). We do*
*n't
have enough evidence to grace this with the label of "Conjecture."
Guess A.2. Fix v 2 Ext1;0B(k; k).
(a)For p = 2, v is a Serre element if and only if v = w2 for some w an
0
eigenvector of fSq.
(b)For p odd, v is a Serre element if and only if v = fiPe0(w) for some w an
eigenvector of eP0.
We can use Lemma A.1 to give a slightly different characterization of quasi-
elementary Hopf algebras, at least in the connected case.
Proposition A.3. Suppose E is a finite dimensional graded connected cocommu-
tative Hopf algebra. Then E is quasi-elementary if and only if no product of the
form Q
w2SQw; Q p = 2;
( u2Soddu)( v2SevenfiPe0v);p odd;
is nilpotent, where S is a subset of Ext1E(k; k)-{0}, Soddis a subset of Ext1;o*
*ddE(k; k)-
{0}, and Sevenis a subset of Ext1;evenE(k; k) - {0}.
Proof.We proveQthis for p = 2, leaving the other case for the reader. Certainly
if no product w2Sw is nilpotent,Qthen E is quasi-elementary. So assume that E
is quasi-elementary, and consider w2Sw where S Ext1E(k; k) - {0}. We may
assume that S is of the form S = {wij: 1 i m; 1 j ni}, where for each
0 0
iQ m, j < ni, we have fSqwij= wi;j+1, and fSqwi;ni= 0. WePwant to show that
w2Sw is non-nilpotent; we do this0by induction on N = (ni- 1), theQnumber
of wij's not in the kernel of fSq. If N is zero, then by Lemma A.1 w2S w is
non-nilpotent. Assume we have shown that every such product with at most N - 1
FINITE DIMENSIONAL HOPF ALGEBRAS 9
0 0
factors not in ker(fSq) is non-nilpotent, and suppose S has N factors in ker(fS*
*q),
N + m factors altogether. Then
Y Y 0 Y 1 Y Y
fSqm( wij) = fSq(wij) fSq(wi;ni) = wi;j+1 w2i;ni
im im im im
j