A nilpotence theorem for modules over
the mod 2 Steenrod algebra
Michael J. Hopkins
John H. Palmieri
4 September 1992
Abstract
We prove that the mod 2 Steenrod algebra A satisfies the "detec-
tion" property; i.e., everynon-nilpotent element of ExtA (F2;F2) can
be detected by restricting to an exterior sub-Hopf algebra of A.
1 Introduction and results
Let A be the mod 2 Steenrod algebra. In this paper we prove Theorem 1.1,
a conjecture of Adams, which describes howto detect all non-nilpotent el-
ements in ExtA (F2 ;F2). One can view this result in two ways: it is a
generalization of results of Lin [5] and Wilkerson [11] about Ext over cer-
tain sub-Hopf algebras of A (and hence is analogous to results ofQuillen
and others on group cohomology); and it is a Steenrod algebra version of
Nishida's theorem [8], a special case of the nilpotence theorem of Devinatz,
Hopkins, and Smith [1].
We need one definition in order to state our result: fix a prime p and a
cocommutative Fp Hopf algebra A. An elementary sub-Hopf algebra B of
A is a bicommutative sub-Hopf algebra with bp= 0 for all b 2 IB (IB is
the augmentation ideal). For instancewhen p = 2, then the elementary sub-
Hopf algebras are the sub-Hopf algebras which are exterior algebras. Let
B : B ,! A denote the inclusion, so B is the restriction map on Ext.
Theorem 1.1 Let A be a sub-Hopf algebra of the mod 2 Steenrod algebra;
fix z 2 ExtA (F2; F2). If E (z) = 0 for every elementary sub-Hopf algebra
E : E ,! A,then z is nilpotent.
Theorem 1.1 was first conjectured by Adams, as reported by Lin in [5].
We view Theorem 1.1 as a first step in provingstructure theorems for S-
teenrod algebra modules analogous to those for spectra given in [3] and [4];
for instance, one has the following conjecture (analogous to the nilpotence
theorem):
Conjecture 1.2 Let A be a sub-Hopf algebra of the mod 2 Steenrod algebra;
let C be a bounded below coalgebra over A. Given z 2 Ext A(C; F2), if
B (z) = 0 for every elementary sub-Hopf algebra B aeA, then z is nilpotent.
This is the "ring spectrum" version of the conjecture;one can make a similar
conjecture about ExtA (M; M) for any finite A-module M. If one could
prove this, then one should be able to work as in [3] to determine the thick
subcategories of the category of finite A-modules, and hence to prove an
appropriate "periodicity" theorem.
Theorem 1.1 raises other questions; for instance, givenA, can we find all
of the non-nilpotent elements in ExtA (F2; F2)? One approach would be
to investigate the image of E for each E. Assume that E is normal; then
this image lies in the set of generators for ExtE(F2 ;F2 ) asan A==E -module
(since E is an edge homomorphism in the spectral sequence asso ciated to the
extension E ! A ! A==E);hence, the first step should be determining this
set of generators. When A is the full Steenrod algebra,this is difficult already
for the case E = E(2) = (F2[2; 3; . .].=(i4)) , the maximal elementary sub-
Hopf algebra of A containing P12.
At odd primes, Wilkerson found a finite sub-Hopf algebra of the Steen-
rod algebra for which the odd primary version of Theorem 1.1 fails. A
weakened version could still be true_perhaps all non-nilpotent elements in
Ext A(Fp ;Fp ) are detected by restricting to two-stage extensions of elemen-
tary sub-Hopf algebras [6].
In Section 2 we prove Theorem 1.1,and at the end of that section we discuss
some reasons that our proof doesn't work for an arbitrary coalgebra C.