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., every non-nilpotent element of Ext**A(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 how to detect all non-nilpotent el-
ements in Ext **A(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 of Quillen
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 instance when 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 *Bis the restriction map on Ext.
1
Theorem 1.1 Let A be a sub-Hopf algebra of the mod 2 Steenrod algebra;
fix z 2 Ext**A(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 proving structure 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 A, then z is nilpotent.
This is the "ring spectrum" version of the conjecture; one can make a similar
conjecture about Ext **A(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, given A, can we find all
of the non-nilpotent elements in Ext **A(F2 ; F2)? One approach would be
to investigate the image of *Efor each E. Assume that E is normal; then
this image lies in the set of generators for Ext**E(F2 ; F2) as an A==E-module
(since *Eis an edge homomorphism in the spectral sequence associated 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 alrea*
*dy
for the case E = E(2) = (F2[2; 3; . .].=(4i))*, the maximal elementary sub-
Hopf algebra of A containing P21.
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.
2
There is also an appendix in which we give a brief description of Eisen's
calculation of certain localized Ext groups.
The authors would like to thank Haynes Miller and Doug Ravenel for useful
conversations regarding this subject.
2 Proof of Theorem 1.1
In this section we prove the main theorem. The proof is analogous to that
for the nilpotence theorem for spectra (see [1] or [3]).
We prove the theorem in the case where A is the mod 2 Steenrod algebra;
the proof easily generalizes to any sub-Hopf algebra. We fix some notation:
A is dual to A* = F2[1; 2; 3; . .].; we dualize with respect to the monomial
basis in A*, and set Pts= (2st)*. The maximal exterior sub-Hopf algebras of
A are E(n) = E[Pts: t n; 0 s n - 1], for n 1 (see [5], for example).
For n 1 let Y (n) be the sub-Hopf algebra dual to F2[n; n+1; . .].(so we
have A = Y (1) Y (2) Y (3) . .)..
Let z 2 Ext**A(F2 ; F2); we will also use z to denote the restriction *Y (n)(z)*
* 2
Ext**Y (n)(F2 ; F2). Assume that z is "not detected" by any exterior algebra
E A (i.e., the restriction *E(z) = 0 for all E). We will show that z 2
Ext**Y (n)(F2 ; F2) is nilpotent by downward induction on n.
First, since Exts;tY((n)F2; F2) = 0 if (2n - 1)s > t, then for n 0, z restricts
to 0 over Y (n); this starts the induction. The inductive step is somewhat
more involved.
Assume that z restricts to zero in Ext**Y (n+1)(F2 ; F2). We want to show that
z is nilpotent when restricted to Ext**Y (n)(F2 ; F2).
Note that Y (n)==Y (n + 1) ~= E[Pns: s 0]. Define a module Gk over
this exterior algebra by Gk = E[Pns: k - 1 s 0]; let G0 = F2. Note
also that for each s, Pnsis indecomposable in Y (n), so that the polynomial
generators of Ext**Y (n)==Y (n+1)(F2 ; F2) = F2[hns : s 0] map nontrivially to
Ext**Y (n)(F2 ; F2). We also use hns to denote their images in Ext**Y (n)(F2 ; *
*F2).
We will show the following:
Lemma 2.1 For each s, there exist integers i and j so that h2inszj = 0.
3
Lemma 2.2 For some k > 0, there is an integer N so that zN 1Gk = 0
in Ext**Y (n)(Gk ; Gk).
Lemma 2.3 If for some k > 0 we have z 1Gk = 0, then there is an integer
N0 so that zN0 1Gk-1 = 0 in Ext**Y (n)(Gk-1 ; Gk-1).
Lemmas 2.2 and 2.3 give us a downward induction on m to show that z
1Gm is nilpotent in Ext**Y (n)(Gm ; Gm ); since G0 = F2, this is good enough.
Lemma 2.1 is used to prove 2.3.
Proof of Lemma 2.1: This is in two parts: if s n, then hns is nilpo-
tent in Ext **Y((n)F2; F2) (see [5], [7]). Otherwise, z restricts to zero in
Ext**E(n)(F2 ; F2); so z goes to zero in h-1n0Ext**E(n)(F2 ; F2). But by Eisen*
*'s
calculation (see [2], or Theorem A.1 in the appendix), h-1n0Ext**Y((n)F2; F2)
embeds in h-1n0Ext**E(n)(F2 ; F2), so z is zero in h-1n0Ext**Y((n)F2; F2). Henc*
*e in
Ext**Y (n)(F2 ; F2) we have h2in0z = 0 for some i. Let |z| = m, and choose i so
that 2i> 2n-1m. Then applying Sq0 s times to the previous equation gives
h2insz2s = 0 for all s n - 1. 2
Proof of Lemma 2.2: Fix a finite module M. We will show by induction on
the dimension of M that for k 0 and for any ff 2 Ext**Y (n)(Gk ; M), some
power of z 1Gk annihilates ff. We will apply this to M = Gk and ff = 1Gk .
We start with M = F2. We have a normal algebra extension
Y (n + 1) ! Y (n) ! Y (n)==Y (n + 1):
Let D = Y (n)==Y (n + 1); as noted above, D ~=E[Pns: s 0]. Note that for
any k, Gk has a D-resolution
Gk D F2[hns; s k];
where |hns| has bidegree (1; 2s(2n - 1)). Let c = 2n - 1. Then for any
bounded above D-module N, Ext **D(Gk ; N) has a vanishing line of slope
2kc.
We use a Cartan-Eilenberg spectral sequence associated to this extension:
E2 ~=Extp;*D(Gk ; Extq;-*Y (n+1)(F2);)F2)Extp+q;*Y((n)Gk; F2):
4
Ext**Y (n+1)(F2 ; F2) has a vanishing line of slope 2c - 1, so the E2-term has
a vanishing plane: Ep;q;r2= 0 if r < 2kcp + (2c - 1)q. Of course, we have
another such spectral sequence which computes Ext **Y((n)F2; F2), and the
action of Ext**Y (n)(F2 ; F2) on Ext**Y (n)(Gk ; F2) manifests itself as a pair*
*ing of
the two spectral sequences. We are interested in the z-action, so we want to
find the permanent cycle "zin the F2-spectral sequence that corresponds to
z. So assume that "z2 Ep0;q0;r02. Can p0 = 0? No, because z 7! 0 under the
restriction Ext**Y (n)(F2 ; F2) ! Ext**Y (n+1)(F2 ; F2), and this map is the ed*
*ge
homomorphism in the spectral sequence. Hence p0 > 0. This is enough:
now we choose k large enough so that 2kc > p0; then multiplication by a
high enough power of "zin E2 for Gk lands above the vanishing plane, and
hence is zero. So for each ff 2 Ext**Y (n)(Gk ; F2), some power of z kills ff.
Assume this is true for all ff 2 Ext **Y((n)Gk; N), as long as dim N < m.
Let M be any module of dimension m. We can always find a short exact
sequence of Y (n)-modules (up to suspension)
'
0 ! F2---! M---! N ! 0;
with dim N = m - 1. Applying Ext**Y (n)(Gk ; -) gives a long exact sequence
'* *
. .!.Ext**Y (n)(Gk ; F2)---! Ext**Y (n)(Gk ; M)---! Ext**Y (n)(Gk ; N) ! . .*
*:.
Given any ff 2 Ext **Y((n)Gk; M), we can find i so that *(ziff) = 0, by
induction. Then ziff 2 im '*, say '*(fi) = ziff. But we can find j so that
zjfi = 0, so 0 = '*(zkfi) = zi+jff. 2
Proof of Lemma 2.3: For each k there is a short exact sequence
kc
0 ! 2 Gk-1 ! Gk ! Gk-1 ! 0
(where, as above, c = 2n - 1), which gives y 2 Ext**Y (n)(Gk-1 ; Gk-1). One
can check that this element is the image of hnk under the map
-Gk-1
Ext**Y (n)(F2 ; F2)----! Ext**Y (n)(Gk-1 ; Gk-1);
i.e., y = hnk 1Gk-1. For brevity, let Ext(M) denote Ext**Y (n)(M; F2). The
short exact sequence above gives a long exact sequence in Ext:
hnk1
. .!.Ext(Gk-1 )---! Ext(Gk-1 ) ! Ext(Gk ) ! . .:.
5
We may assume (by taking powers) that z1Gk = 0; we have a commutative
diagram
hnk1
. . .! Ext(Gk-1?) ---! Ext (Gk-1?) ! Ext?(Gk ) ! . . .
?yz1 _ ?z1 ? z1=0
ss z y y
. . .! Ext(Gk-1 ) ---! Ext (Gk-1 ) ! Ext (Gk ) ! . . .
hnk1
Since z 1Gk : Ext (Gk ) ! Ext (Gk ) is zero, we have a factorization z
1Gk-1 = (hnk 1) O __z: Ext(Gk-1 ) ! Ext(Gk-1 ). A simple diagram chase
then shows that (z 1Gk-1)j = (hjnk 1) O __zjfor all j. Thus for any i,
(z 1)i+j = (hi+jnkzi 1) O __zj; by choosing i and j large enough, we have (by
Lemma 2.1) hi+jnkzi = 0. Hence zi+j 1Gk-1 = 0, as desired. 2
This completes the proof of Theorem 1.1. 2
Remark 2.4 There are (at least) two obstacles to applying the method in
this section to study non-nilpotence in Ext**A(C; F2), for C a bounded below
coalgebra: the first is that we don't have a calculation like Eisen's for the
appropriate localized Ext groups. In the proof of Theorem A.1, we can still
embed the E2-term of the Y (n) spectral sequence in the E2-term for E(n),
but in this case there is no reason for either spectral sequence to collapse.
The second problem is that if C is not co-commutative, then we don't have
Steenrod operations acting on Ext**Y (n)(C; F2), so knowing that some power
of hn0 kills z doesn't necessarily tell us anything about hn1 acting on z2.
A Appendix: Eisen's calculation
In his thesis, Eisen proves the following result (with notation as above):
Theorem A.1
( )
h-1n0Ext**Y((n)F2; F2) ~=F2[hn0; h-1n0; hts: ifisf=n0,>thens2n >1t,>tnhen]t:*
* n
Since his work has never been published, we outline a proof.
6
First of all, for any Y (n)-module M, there is a spectral sequence, called the
Margolis Adams spectral sequence (see [10] or [9]), with
E2 = Ext**Y (n)0n(H(M; Pn0) ; F2) F2[hn0; h-1n0] ) h-1n0Ext**Y((n)M; F2);
where Y (n)0nis the algebra of operations for Pn0-homology. This spec-
tral sequence is formed by making a "resolution" of M by direct sums of
Y (n)=Y (n)Pn0and Y (n) satisfying certain properties with respect to Pn0-
homology. For our purposes, we only need to know that Y (n)0nis given
by Y (n)0n= H(Y (n)=Y (n)Pn0; Pn0), and that one can calculate without too
much trouble that
Y (n)0n~=E[Pts: s and t as in A.1]:
So when M = F2 we have a spectral sequence with
E2 ~=F2[hn0; h-1n0; hts: s and t as in A.1];
we want to show that this spectral sequence collapses. To do this, we embed
it in another Margolis Adams spectral sequence, this time for E(n). For this
one we have
E(n)0n = H(E(n)=E(n)Pn0; Pn0)
= E(n)=E(n)Pn0;
so
E2 ~=F2[h-1n0; hts: t n; n > s 0]:
Also, since E(n) is an exterior algebra, we can see that the spectral sequence
collapses. Lastly, we observe that the map E(n) ! Y (n) induces an embed-
ding of the E2-term for the Y (n)-spectral sequence into that for E(n), and
hence the Y (n) spectral sequence collapses as well. 2
References
[1]E. S. Devinatz, M. J. Hopkins, and J. H. Smith: Nilpotence and
stable homotopy theory I, Ann. of Math. (2) 128 (1988), 207-241.
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[2]D. K. Eisen, Localized Ext groups over the Steenrod algebra, Ph.D. the-
sis, Princeton University, 1987.
[3]M. J. Hopkins: Global methods in homotopy theory, Proceedings
of the Durham Symposium on Homotopy Theory (J. D. S. Jones and
E. Rees, eds.), 1987, LMS Lecture Note Series 117, pp. 73-96.
[4]M. J. Hopkins and J. H. Smith: Nilpotence and stable homotopy
theory II, to appear.
[5]W. H. Lin: Cohomology of sub-Hopf algebras of the Steenrod algebra
I and II, J. Pure Appl. Algebra 10 (1977), 101-114, and 11 (1977),
105-110.
[6]H. R. Miller: private communication.
[7]H. R. Miller and C. Wilkerson: Vanishing lines for modules over
the Steenrod algebra, J. Pure Appl. Algebra 22 (1981), 293-307.
[8]G. Nishida: The nilpotency of elements of the stable homotopy groups
of spheres, J. Math. Soc. Japan 25 (1973), 707-732.
[9]J. H. Palmieri: The Margolis Adams spectral sequence, in prepara-
tion.
[10]J. H. Palmieri, A chromatic spectral sequence to study Ext over the
Steenrod algebra, Ph.D. thesis, Mass. Inst. of Tech., 1991.
[11]C. Wilkerson: The cohomology algebras of finite dimensional Hopf
algebras, Trans. Amer. Math. Soc. 264 (1981), 137-150.
Massachusetts Institute of Technology
Second author's current address: School of Mathematics, University
of Minnesota, Minneapolis, MN 55455
8