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 “detection” property; i.e., every non-nilpotent element of Ext_{A}^{∗∗}(F_{2}, F_{2}) 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 elements in Ext_{A}^{∗∗}(F_{2}, F_{2}). One can view this result in two ways: it is a generalization of results of Lin [5] and Wilkerson [11] about Ext over certain 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 F_{p} Hopf algebra A. An elementary sub-Hopf algebra B of A is a bicommutative sub-Hopf algebra with b^{p} = 0 for all b ∈ 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 ι_{B} : B ↪ A denote the inclusion, so ι_{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 ∈ Ext_{A}^{∗∗}(F_{2}, F_{2}). If ι_{E}^{∗}(z) = 0 for every elementary sub-Hopf algebra ι_{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 Steenrod 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 ∈ Ext_{A}^{∗∗}(C, F_{2}), if ι_{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}^{∗∗}(F_{2}, F_{2})? One approach would be to investigate the image of ι_{E}^{∗} for each E. Assume that E is normal; then this image lies in the set of generators for Ext_{E}^{∗∗}(F_{2}, F_{2}) as an A//E-module (since ι_{E}^{∗} is 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 already for the case E = E(2) = (F_{2}[ξ_{2}, ξ_{3},…]/(ξ_{i}^{4}))^{∗}, the maximal elementary sub-Hopf algebra of A containing P_{2}^{1}.

At odd primes, Wilkerson found a finite sub-Hopf algebra of the Steenrod 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}^{∗∗}(F_{p}, F_{p}) are detected by restricting to two-stage extensions of elementary 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.

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_{∗} = F_{2}[ξ_{1}, ξ_{2}, ξ_{3},…]; we dualize with respect to the monomial basis in A_{∗}, and set P_{t}^{s} = (ξ_{t}^{2^{s}})^{∗}. The maximal exterior sub-Hopf algebras of A are E(n) = E[P_{t}^{s} : 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 F_{2}[ξ_{n}, ξ_{n+1},…] (so we have A = Y (1) ⊃ Y (2) ⊃ Y (3) ⊃ ⋯).

Let z ∈ Ext_{A}^{∗∗}(F_{2}, F_{2}); we will also use z to denote the restriction ι_{Y (n)}^{∗}(z) ∈ Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}). Assume that z is “not detected” by any exterior algebra E ⊂ A (i.e., the restriction ι_{E}^{∗}(z) = 0 for all E). We will show that z ∈ Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}) is nilpotent by downward induction on n.

First, since Ext_{Y (n)}^{s,t}(F_{2}, F_{2}) = 0 if (2^{n} − 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)}^{∗∗}(F_{2}, F_{2}). We want to show that z is nilpotent when restricted to Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}).

Note that Y (n)//Y (n + 1) ≅ E[P_{n}^{s} : s ≥ 0]. Define a module G_{k} over this exterior algebra by G_{k} = E[P_{n}^{s} : k − 1 ≥ s ≥ 0]; let G_{0} = F_{2}. Note also that for each s, P_{n}^{s} is indecomposable in Y (n), so that the polynomial generators of Ext_{Y (n)//Y (n+1)}^{∗∗}(F_{2}, F_{2}) = F_{2}[h_{ns} : s ≥ 0] map nontrivially to Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}). We also use h_{ns} to denote their images in Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}).

We will show the following:

Lemma 2.1 For each s, there exist integers i and j so that h_{ns}^{2^{i}}z^{j} = 0.

Lemma 2.2 For some k > 0, there is an integer N so that z^{N} ⊗ 1_{G_{k}} = 0 in Ext_{Y (n)}^{∗∗}(G_{k}, G_{k}).

Lemma 2.3 If for some k > 0 we have z ⊗1_{G_{k}} = 0, then there is an integer N′ so that z^{N′} ⊗ 1_{G_{k−1}} = 0 in Ext_{Y (n)}^{∗∗}(G_{k−1}, G_{k−1}).

Lemmas 2.2 and 2.3 give us a downward induction on m to show that z ⊗ 1_{G_{m}} is nilpotent in Ext_{Y (n)}^{∗∗}(G_{m}, G_{m}); since G_{0} = F_{2}, 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 h_{ns} is nilpotent in Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}) (see [5], [7]). Otherwise, z restricts to zero in Ext_{E(n)}^{∗∗}(F_{2}, F_{2}); so z goes to zero in h_{n0}^{−1}Ext_{E(n)}^{∗∗}(F_{2}, F_{2}). But by Eisen’s calculation (see [2], or Theorem A.1 in the appendix), h_{n0}^{−1}Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}) embeds in h_{n0}^{−1}Ext_{E(n)}^{∗∗}(F_{2}, F_{2}), so z is zero in h_{n0}^{−1}Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}). Hence in Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}) we have h_{n0}^{2^{i}}z = 0 for some i. Let |z| = m, and choose i so that 2^{i} > 2^{n−1}m. Then applying Sq^{0} s times to the previous equation gives h_{ns}^{2^{i}}z^{2^{s}} = 0 for all s ≤ n − 1. □

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 α ∈ Ext_{Y (n)}^{∗∗}(G_{k}, M), some power of z ⊗ 1_{G_{k}} annihilates α. We will apply this to M = G_{k} and α = 1_{G_{k}}.

We start with M = F_{2}. 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[P_{n}^{s} : s ≥ 0]. Note that for any k, G_{k} has a D-resolution

G_{k} ← D ⊗ F_{2}[h_{ns}, s ≥ k],

where |h_{ns}| has bidegree (1, 2^{s}(2^{n} − 1)). Let c = 2^{n} − 1. Then for any bounded above D-module N, Ext_{D}^{∗∗}(G_{k}, N) has a vanishing line of slope 2^{k}c.

We use a Cartan-Eilenberg spectral sequence associated to this extension:

E_{2} ≅ Ext_{D}^{p,∗}(G_{k}, Ext_{Y (n+1)}^{q,−∗}(F_{2}, F_{2})) ⇒ Ext_{Y (n)}^{p+q,∗}(G_{k}, F_{2}).

Ext_{Y (n+1)}^{∗∗}(F_{2}, F_{2}) has a vanishing line of slope 2c − 1, so the E_{2}-term has a vanishing plane: E_{2}^{p,q,r} = 0 if r < 2^{k}cp + (2c − 1)q. Of course, we have another such spectral sequence which computes Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}), and the action of Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}) on Ext_{Y (n)}^{∗∗}(G_{k}, F_{2}) manifests itself as a pairing of the two spectral sequences. We are interested in the z-action, so we want to find the permanent cycle z̃ in the F_{2}-spectral sequence that corresponds to z. So assume that z̃ ∈ E_{2}^{p_{0},q_{0},r_{0}}. Can p_{0} = 0? No, because z ↦ 0 under the restriction Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}) → Ext_{Y (n+1)}^{∗∗}(F_{2}, F_{2}), and this map is the edge homomorphism in the spectral sequence. Hence p_{0} > 0. This is enough: now we choose k large enough so that 2^{k}c > p_{0}; then multiplication by a high enough power of z̃ in E_{2} for G_{k} lands above the vanishing plane, and hence is zero. So for each α ∈ Ext_{Y (n)}^{∗∗}(G_{k}, F_{2}), some power of z kills α.

Assume this is true for all α ∈ Ext_{Y (n)}^{∗∗}(G_{k}, 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 → F_{2}⟶^{φ}M⟶^{ψ}N → 0,

with dim N = m − 1. Applying Ext_{Y (n)}^{∗∗}(G_{k}, −) gives a long exact sequence

⋯ → Ext_{Y (n)}^{∗∗}(G_{k}, F_{2})⟶^{φ_{∗}}Ext_{Y (n)}^{∗∗}(G_{k}, M)⟶^{ψ_{∗}}Ext_{Y (n)}^{∗∗}(G_{k}, N) → · · · .

Given any α ∈ Ext_{Y (n)}^{∗∗}(G_{k}, M), we can find i so that ψ_{∗}(z^{i}α) = 0, by induction. Then z^{i}α ∈ im φ_{∗}, say φ_{∗}(β) = z^{i}α. But we can find j so that z^{j}β = 0, so 0 = φ_{∗}(z^{k}β) = z^{i+j}α. □

Proof of Lemma 2.3: For each k there is a short exact sequence

0 → Σ^{2^{k}c}G_{k−1} → G_{k} → G_{k−1} → 0

(where, as above, c = 2^{n} − 1), which gives y ∈ Ext_{Y (n)}^{∗∗}(G_{k−1}, G_{k−1}). One can check that this element is the image of h_{nk} under the map

Ext_{Y (n)}^{∗∗}(F_{2}, F_{2})⟶^{−⊗G_{k−1}}Ext_{Y (n)}^{∗∗}(G_{k−1}, G_{k−1});

i.e., y = h_{nk} ⊗ 1_{G_{k−1}}. For brevity, let Ext(M) denote Ext_{Y (n)}^{∗∗}(M, F_{2}). The short exact sequence above gives a long exact sequence in Ext:

⋯ → Ext(G_{k−1})⟶^{h_{nk}⊗1}Ext(G_{k−1}) → Ext(G_{k}) → · · · .

We may assume (by taking powers) that z⊗1_{G_{k}} = 0; we have a commutative diagram

⋯ → Ext(G_{k−1}) ⟶^{h_{nk}⊗1} Ext(G_{k−1}) → Ext(G_{k}) → ⋯
↓^{z⊗1} [figure: \overline{z}]  ↓^{z⊗1}  ↓^{z⊗1=0}
⋯ → Ext(G_{k−1}) ⟶_{h_{nk}⊗1} Ext(G_{k−1}) → Ext(G_{k}) → ⋯

Since z ⊗ 1_{G_{k}} : Ext(G_{k}) → Ext(G_{k}) is zero, we have a factorization z ⊗ 1_{G_{k−1}} = (h_{nk} ⊗ 1) ∘ \overline{z} : Ext(G_{k−1}) → Ext(G_{k−1}). A simple diagram chase then shows that (z ⊗ 1_{G_{k−1}})^{j} = (h_{nk}^{j} ⊗ 1) ∘ \overline{z}^{j} for all j. Thus for any i, (z ⊗ 1)^{i+j} = (h_{nk}^{i+j}z^{i} ⊗ 1) ∘ \overline{z}^{j}; by choosing i and j large enough, we have (by Lemma 2.1) h_{nk}^{i+j}z^{i} = 0. Hence z^{i+j} ⊗ 1_{G_{k−1}} = 0, as desired. □

This completes the proof of Theorem 1.1. □

Remark 2.4 There are (at least) two obstacles to applying the method in this section to study non-nilpotence in Ext_{A}^{∗∗}(C, F_{2}), 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 E_{2}-term of the Y (n) spectral sequence in the E_{2}-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, F_{2}), so knowing that some power of h_{n0} kills z doesn’t necessarily tell us anything about h_{n1} acting on z^{2}.

A Appendix: Eisen’s calculation

In his thesis, Eisen proves the following result (with notation as above):

Theorem A.1

h_{n0}^{−1}Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}) ≅ F_{2}[h_{n0}, h_{n0}^{−1}, h_{ts} : \{ \begin{matrix}if s = 0, then 2n > t > n \\ if n > s ≥ 1, then t ≥ n \end{matrix} \}].

Since his work has never been published, we outline a proof.

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

E_{2} = Ext_{Y (n)_{n}^{0}}^{∗∗}(H(M, P_{n}^{0}), F_{2}) ⊗ F_{2}[h_{n0}, h_{n0}^{−1}] ⇒ h_{n0}^{−1}Ext_{Y (n)}^{∗∗}(M, F_{2}),

where Y (n)_{n}^{0} is the algebra of operations for P_{n}^{0}-homology. This spectral sequence is formed by making a “resolution” of M by direct sums of Y (n)/Y (n)P_{n}^{0} and Y (n) satisfying certain properties with respect to P_{n}^{0}- homology. For our purposes, we only need to know that Y (n)_{n}^{0} is given by Y (n)_{n}^{0} = H(Y (n)/Y (n)P_{n}^{0}, P_{n}^{0}), and that one can calculate without too much trouble that

Y (n)_{n}^{0} ≅ E[P_{t}^{s} : s and t as in A.1].

So when M = F_{2} we have a spectral sequence with

E_{2} ≅ F_{2}[h_{n0}, h_{n0}^{−1}, h_{ts} : 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)_{n}^{0} = H(E(n)/E(n)P_{n}^{0}, P_{n}^{0})
= E(n)/E(n)P_{n}^{0},

so

E_{2} ≅ F_{2}[h_{n0}^{−1}, h_{ts} : 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 embedding of the E_{2}-term for the Y (n)-spectral sequence into that for E(n), and hence the Y (n) spectral sequence collapses as well. □

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.

[2] D. K. Eisen, Localized Ext groups over the Steenrod algebra, Ph.D. thesis, 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 preparation.

[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