NILPOTENCE FOR MODULES OVER
THE MOD 2 STEENROD ALGEBRA I
JOHN H. PALMIERI
Abstract.Let Bbe a sub-Hopf algebra of the mod 2 Steenrod algebra,and let
M be a finite dimensional B-module.We prove that one can detect nilpoten*
*ce
of elements in ExtB(M;M) by restricting to elementary sub-Hopf algebras,
and we prove a similar result at odd primes.
1. Introduction
In [4], Devinatz, Hopkins, and Smith proved the nilpotence theorem,a remark-
able result which provides algebraic means for detecting nilpotence in the coll*
*ection
of homotopy classes of self-maps of any finite spectrum. This theorem has many
important consequences, and so has opened up new approaches to studying homo-
topy theory (see [19, 10, 7], for example).See [20] for a thorough discussion o*
*f this
material; [5] also gives an overview of the nilpotence theorem and related resu*
*lts.
Let A be the mod 2 Steenrod algebra, and let M be a finite A-module. In
this paper we show that there is an analogous result, Theorem 1.1, for detecting
nilpotence in ExtA (M;M). We hope that this leads to structure theorems for the
category of finite A-modules, comparable to those for finite spectra in [8]. We
begin to develop this material in a sequel [17]. Also,the nilpotence theorem fo*
*r A-
modules extends the strong parallel between results in stable homotopy theory a*
*nd
results for A-modules, as described in [11] and [16]. Some of these earlier A-m*
*odule
results have been used to prove results in homotopy theory via the Adams spectr*
*al
sequence, as in [10] and [18]; we hope that one can do likewise with Theorem 1.*
*1.
In order to state our main theorem, we need a few definitions. Given a Hopf
algebra B over a field kof characteristic p, an elementary sub-Hopf algebra E B
is a bicommutative sub-Hopf algebra such that ep= 0 for all e 2 IE. (Here, IE
denotes the augmentation ideal of E, IE =ker(" : E! k).) Of course, given any
Hopf algebra inclusion C : C ! Band B-modules L and M, we have a restriction
map C : ExtB (L; M) ! ExtC (L;M). Note that if is a (coassociative) B-
coalgebra (so that there are B-module maps : ! and " : ! k making
certain diagrams commute), then ExtB(; k) is an (associative) algebra, via the
map . We prove the following result.
Theorem 1.1. Let B be a sub-Hopf algebra of the mod 2 Steenrod algebra A.
(a)Let be a bounded belowcoassociative B-coalgebra; fix z2 ExtB (;F2).
Then z is nilpotent if and only ifE (z) is nilpotent foral l elementary *
*sub-
Hopf algebras E of B.
2 JOHN H. PALMIERI
(b)Let M be a finite dimensional B-module;fix z 2 ExtB(M; M). Then zis
nilpotent under Yoneda composition if and only if E(z) is nilpotent for *
*all
elementary sub-Hopf algebras E of B.
(c)Let L and M be B-modules with L boundedbelow and M finite dimen-
sional; fix z 2 ExtB (L;M). Then zn 2 ExtB (Ln ;Mn ) is zero for n
sufficiently large if and only if (E(z))nE is zero for nE sufficiently *
*large,
for all elementary sub-Hopf algebras E of B.
We alsoprove a similar result about detecting nilpotence for sub-Hopf algebr*
*as
of the Steenrod reduced powers at an odd prime; see Section 4.
In [21] Wilkerson proved the special case of (b) where Bis finite dimensiona*
*l and
M = F2;Lin [9] proved this for a moregeneral collection of sub-Hopf algebras (t*
*hose
with finite "profile" function_see Section 2),again only for M = F2. Hopkins and
the author [6] generalized this to arbitrary sub-Hopf algebras of A,but still f*
*or
M = F2.These are analogous to proving Nishida's theorem [15] for various B's.
We viewTheorem 1.1 as analogous to the nilpotence theorem of [4];it is also
directly analogous to a result of Carlson [3] regarding detection of nilpotent *
*elements
in ExtkG(M; M) for G a finite group, k a field of characteristic p > 0, andM a *
*kG-
module. This opens up other lines of questions;for example, what other results *
*for
group cohomology also hold for the Steenrod algebra (or for other Hopf algebras*
*)?
What sorts of structure theorems for modules over the Steenrod algebra carryover
to give interesting results in group theory? Are there interesting parallels be*
*tween
kG-modules and (finite) spectra? Wilkerson [21] has discussed some asp ects of *
*the
first question.
Our other main results are Corollary 2.7 and Theorem 3.1; we would haveto
introduce too much notation to state them precisely here,so we only give rough
descriptions. Let B be a finite dimensional graded connected cocommutative Hopf
algebra over a finite field k. Corollary 2.7 is the generalization of Wilkerso*
*n's
result [21, Corollary 5.6] about detection of nilpotent elements in ExtB(k; k) *
*to the
case ExtB (M;M) for M any B-module. As far as Theorem 3.1 goes,if one views
restriction to the elementary sub-Hopf algebras of Bin Theorem 1.1 as analogous*
* to
applying Morava K-theories, then Theorem 3.1 gives an analog of the BP -version
(or the M U-version) of the nilpotence theorem; namely,it gives a single sub-Ho*
*pf
algebra D of B so that restriction to D detects nilpotence.
Our proof is a combination of the work about Hopf algebra cohomology in [9]
and [21] with the proof of Carlson's theorem as presented in [2];we also need a
variant on an argument in [6]. In Section 2 we give some background about the
Steenrod algebra, its sub-Hopf algebras, and their cohomology;we also prove our
main lemma, Lemma 2.4. In Section 3 we prove the main theorem. In Section 4
we discuss the odd prime case; we summarize which ofthe results still hold, and
we indicate the changes necessary in the proofs.
I would like to thank the topologists at the University of Minnesota for the*
*ir
hospitality while this research was being conducted. Also, the notion of "quasi-
elementary Hopf algebra" in Section 4 is due to Haynes Miller and Clarence Wilk-