NILPOTENCE FOR MODULES OVER THE MOD 2 STEENROD ALGEBRA II

JOHN H. PALMIERI

Abstract. We imitate much of the work of Hopkins and Smith in [7], in the context of modules over the mod 2 Steenrod algebra A. Our main result is an analog of the periodicity theorem of [7], implying, for example, the existence of infinitely many non-nilpotent elements in Ext_{A}^{∗∗}(F_{2}, F_{2}).

1. Introduction

The periodicity theorem of Hopkins and Smith [7] (see also [16]) is an almost formal consequence of the nilpotence theorem of Devinatz, Hopkins, and Smith [4]; given a finite spectrum X, the former result classifies many of the non-nilpotent elements in [X, X], the homotopy classes of self-maps of X. In [11], we prove an analog of the nilpotence theorem for modules over the mod 2 Steenrod algebra A, describing functors which detect nilpotence in Ext_{A}^{∗∗}(M, M) for any finite A-module M—see Theorem 2.8 below. In this paper we imitate as much as we can of [7]; our main result is the existence of many non-nilpotent elements in Ext_{A}^{∗∗}(M, M) for any finite A-module M. Note that we describe what is probably only a small subset of the collection of non-nilpotent elements in Ext_{A}^{∗∗}(M, M), so our result is not as complete as the geometric result in [7]; however, we do succeed in establishing much of the framework used in the geometric setting there.

To detect nilpotence in Ext_{A}^{∗∗}(M, M), one restricts to elementary sub-Hopf algebras. In this paper we use these sub-Hopf algebras to define certain kinds of self-maps, analogous to v_{n}-maps, and we work out some of their properties. We use these properties to prove the existence of certain non-nilpotent elements in Ext_{A}^{∗∗}(M, M) for M a finite A-module. Restricting to elementary sub-Hopf algebras is not as nice, algebraically, as computing Morava K-theories, so we are not able to develop things as fully as in [7]. In particular, we do not prove an analog of the thick subcategory theorem.

Our main results are the following. Recall that P_{t}^{s} is the Milnor basis element dual to ξ_{t}^{2^{s}}, and that (P_{t}^{s})^{2} = 0 if s < t; hence for any A-module M, if s < t then P_{t}^{s} acts as a differential on M. Roughly speaking, for non-negative integers s < t, an h_{t,s}-map y in Ext_{A}^{∗∗}(F_{2}, F_{2}) is an element represented by some power of h_{t,s} = [ξ_{t}^{2^{s}}] in the cobar complex; alternatively, it is an element that restricts to a power of the generator in Ext_{E[P_{t}^{s}]}^{∗∗}(F_{2}, F_{2}) ≅ F_{2}[h_{t,s}]—see Definition 3.1 for a precise definition. Note that such a y will have bidegree (k, 2^{s}(2^{t} − 1)k) for some k. For example, the element usually written as h_{0} or h_{10} in Ext_{A}^{1,1}(F_{2}, F_{2}) is an h_{1,0}-map; g ∈ Ext_{A}^{4,24}(F_{2}, F_{2}) is an h_{2,1}-map. These (and their powers) are the only known examples of h_{t,s}-maps in Ext_{A}^{∗∗}(F_{2}, F_{2}), but we assert that there are many others.

Theorem 1.1. For each integer n ≥ 0, there is a non-nilpotent h_{n+1,n}-map z in Ext_{A}^{∗∗}(F_{2}, F_{2}). If y is another h_{n+1,n}-map, then there are integers i and j so that z^{i} = y^{j}.

One can also define h_{t,s}-maps in Ext_{A}^{∗∗}(M, M)—see Definition 3.1.

Theorem 1.2. Let M be a finite A-module. If H(M, P_{t}^{s}) ≠ 0 but H(M, P_{v}^{u}) = 0 for all u and v with u < v, u + v ≤ s + t, u ≥ s, and v < t, then there is a non-nilpotent h_{t,s}-map z in Ext_{A}^{∗∗}(M, M). As above, if y is another h_{t,s}-map, then there are integers i and j so that z^{i} = y^{j}.

These are the only non-nilpotent h_{t,s}-maps in Ext_{A}^{∗∗}(M, M) (see Remark 4.2); however, these are not the only non-nilpotent elements. Determining all of them seems hard; we discuss this further in Section 6.

The collection of P_{v}^{u}’s in this theorem is the smallest collection S of P_{v}^{u}’s with u < v so that S′ = S ∪ {P_{t}^{s}} is closed under the following operations; see Theorem A.1.

• If P_{v}^{u} ∈ S′ and u < v − 1, then P_{v−1}^{u} ∈ S′.

• If P_{v}^{u} ∈ S′ and u + 1 < v − 1, then P_{v−1}^{u+1} ∈ S′. This whole picture is somewhat less clear at odd primes. For one thing, we

don’t have a nilpotence theorem for Ext_{A}, we only have one for Ext_{P}, where P is the algebra of reduced powers. For another, this nilpotence theorem for odd primes is weaker than the result when p = 2—elementary sub-Hopf algebras are not sufficient to detect nilpotence, we need “quasi-elementary” sub-Hopf algebras in addition. This is good enough, though, to prove the analog of Theorem 1.2; we discuss this in Section 5.

The structure of the paper is as follows. In order to imitate some of the homotopy theoretic constructions in [7], we will work with chain complexes of modules; we present the categories we will use in Section 2. In Section 3, we define h_{t,s}-maps and we describe properties of these elements of Ext; we also discuss thick subcategories of the category of finite chain complexes of A-modules. In Section 4, we prove our main theorems. We have a brief discussion of periodicity at odd primes in Section 5. In Section 6, we discuss some questions related to this material; for example, we point out several non-nilpotent elements in Ext_{A}^{∗∗}(F_{2}, F_{2}), the existence of which is not predicted by Theorem 1.1, and ask how to describe Ext_{A}^{∗∗}(F_{2}, F_{2}) “mod nilpotents.” We also include an appendix in which we prove a result about the P_{t}^{s}-homology groups of finite (chain complexes of) A-modules.

I would like to thank Doug Ravenel for providing the main idea of the proof of Theorem A.1, and Mike Hopkins for insisting that all of this would be much easier with chain complexes of modules.

2. Chain complexes of modules

In this section we do some preliminary work with chain complexes of modules. Let A be the mod 2 Steenrod algebra; we assume the reader is familiar with P_{t}^{s}- homology groups and related machinery. See [8] for a reference.

Let M be the category of bounded below A-modules and A-module maps. We discuss two other categories: first, the category C(M) of bounded below chain complexes of A-modules: the objects are bounded-below chain complexes of finitetype bounded-below modules, and the morphisms are chain maps. Given a chain complex C, we define ΩC to be the chain complex with nth term C_{n+1} (so ΩC is the “desuspension” of C). A chain map from C to D is a quasi-isomorphism if it induces an isomorphism on homology; with these playing the role of weak equivalences and surjections acting as fibrations, C(M) is a closed model category. Our second category is Ho C(M), the homotopy category of C(M): the objects are chain complexes, and the morphisms are given by Ext_{A}^{∗∗}(−, −)—i.e., the morphisms of degree i from C to D are chain homotopy classes of maps from Ω^{i}P to D, where P is a cofibrant object (a chain complex of projective modules) quasi-isomorphic to C. (See [14] for closed model categories and their associated homotopy categories;

[17] discusses these categories of chain complexes from a slightly different point of view.)

Given any A-module M, we can form a chain complex with M in degree zero, and 0 elsewhere. We denote this chain complex by M.

Definition 2.1. Call a property P on C(M) generic if the following conditions are satisfied:

(a) If C is weakly equivalent to D, then C satisfies P if and only if D satisfies P.

(b) If C ⊕ D satisfies P, then so do C and D.

(c) If C → D → E is an exact triangle of chain complexes (a fibration, in the language of [14]), and two of C, D, and E satisfy P, then so does the third.

(d) If C satisfies P, then Ω^{k}C satisfies P for all k ∈ Z.

A full subcategory D of C(M) is thick if “M ∈ ob D” is generic. Finally, we define F to be the thick subcategory of C(M) generated by the finite modules concentrated in degree 0; we call the objects of F finite chain complexes.

Note that since F is thick, then Ho F makes sense. This will be our analog of the category of finite p-local spectra, which plays such an important role in the work of Hopkins and Smith. We have two goals for the remainder of this section: to generalize various results about A-modules to chain complexes, and to describe the appropriate analog of Spanier-Whitehead duality. We need a simple observation about finite chain complexes, first.

Lemma 2.2. Assume C is a chain complex. The following are equivalent.

(a) C is quasi-isomorphic to a complex which has finite total dimension.

(b) C is finite.

(c) H_{∗}(C) has finite total dimension.

Proof. The implications (a) ⇒ (b) ⇒ (c) are evident. So assume that H_{∗}(C) is finite-dimensional. Then there exists a k so that H_{n}(C) is zero for all n ≥ k, and so we have a quasi-isomorphism from C to a complex E, where E is given by

⋯ → 0 → 0 → ker(∂_{k+1}) → C_{k+1} ⟶^{∂_{k+1}} C_{k} → C_{k−1} → ….

Since C is bounded below, then this new complex is nonzero in only finitely many degrees. Now, H_{n}(C) is a graded vector space for each n; since H_{∗}(C) is finitedimensional, then there is an r so that H_{n}(C) = 0 above grading r for every n. We form a chain complex F by truncating each term in E above dimension r + 1; then F is finite-dimensional (taken as a whole), and is quasi-isomorphic to C.

Corollary 2.3. If C and D are finite, then so is C ⊗ D.

Remark 2.4. We note that by definition, any generic property which holds for finite A-modules will also hold for finite chain complexes. Indeed, most cohomological properties of A-modules extend easily. As an example, we have the following theorem; we will need this in Section 4. The proof is left as an exercise (hint: imitate the work in [12]).

Definition 2.5. The differentials in A are the elements P_{t}^{s} with s < t (recall that these satisfy (P_{t}^{s})^{2} = 0). Let h_{t,s} denote the polynomial generator of Ext_{E[P_{t}^{s}]}^{∗∗}(F_{2}, F_{2}). For a finite complex C, we write H(C, P_{t}^{s}) = 0 if h_{t,s}^{−1} Ext_{E[P_{t}^{s}]}^{∗∗}(C, F_{2}) = 0. Of course, if M is an A-module, then H(M, P_{t}^{s}) = 0 in this sense if and only if M has no P_{t}^{s}-homology in the usual sense.

Theorem 2.6 (cf. Theorem 2.2 in [12]). Let C be a finite chain complex. Let P_{t}^{s} be the differential so that H(C, P_{t}^{s}) ≠ 0, while H(C, P_{v}^{u}) = 0 if |P_{v}^{u}| < |P_{t}^{s}|. Then for some k, there is a non-nilpotent element v in Ext_{A}^{k,k|P_{t}^{s}|}(C, F_{2}). Furthermore, v is an h_{t,s}-map (see Definition 3.1).

As a corollary, we have the following construction (see [12, Lemma 3.2]). Let x_{0}, x_{1}, x_{2}, … be the differentials in A, ordered by degree.

Corollary 2.7. For each n, there is a finite complex C_{n} with H(C_{n}, x_{j}) ≠ 0 for j ≥ n and H(C_{n}, x_{j}) = 0 for j < n. Furthermore, we have an exact triangle (up to weak equivalences and Ω’s)

C_{n} ⟶ C_{n+1} ⟶ C_{n}.

Sketch of proof. These are constructed by induction: start with C_{0} = F_{2}. Suppose we have C_{n}; let x_{n} = P_{t}^{s}, and let v be the h_{t,s}-map given in Theorem 2.6. Define C_{n+1} by the exact triangle

C_{n+1} ⟶ Ω^{k}C̃_{n} ⟶^{v} C_{n}.

(Here, C̃_{n} is a cofibrant object weakly equivalent to C_{n}.)

The nilpotence theorem of [11] holds for (finite) chain complexes of modules in place of (finite) A-modules; see [11, Remark 3.4]. We record it here for later use. Given a sub-Hopf algebra B ⊆ A, let ι_{B}^{∗} denote the restriction map on Ext. A sub-Hopf algebra E of A is elementary if it is bicommutative and isomorphic, as an algebra, to an exterior algebra.

Theorem 2.8 ([11]). (a) Let Γ be a bounded below A-coalgebra chain complex (coassociative, with counit). Then z ∈ Ext_{A}^{∗∗}(Γ, F_{2}) is nilpotent if and only if ι_{E}^{∗}(z) is nilpotent for all elementary sub-Hopf algebras E of A.

(b) Let M be a finite chain complex. Then z ∈ Ext_{A}^{∗∗}(M, M) is nilpotent if and only if ι_{E}^{∗}(z) is nilpotent for all elementary sub-Hopf algebras E of A.

We move on to Spanier-Whitehead duality.

Definition 2.9. Suppose F is a chain complex of finite total dimension. Define the dual of F, written DF, to be the chain complex with nth term Hom_{F_{2}}(F_{−n}, F_{2}). This is a chain complex of A-modules in the usual way (see [8], for instance).

Lemma 2.10. D extends to give a contravariant functor D : Ho F → Ho F.

Proof. We extend D in the obvious way: given a finite complex C, we replace C by an equivalent complex F of finite total dimension, and then we set DC = DF. If C_{1} is weakly equivalent to C_{2}, then the same is true for DC_{1} and DC_{2}.

Given a morphism C_{1} → C_{2}, represent the morphism as a chain map P_{1} → F_{2}, where P_{1} is a complex of projectives and F_{2} has finite total dimension. For some k, the modules (F_{2})_{n} are zero for n > k; the same is true for the homology of P_{1} (with the same k, we may assume). We have a similar situation for “internal” degree—there is an r so that (F_{2})_{n} is 0 above degree r, for all n, and similarly for H_{n}(P_{1}). So truncate P_{1} as in the proof of Lemma 2.2, above homological degree k +2 and internal degree r +2, and then dualize. If we have two morphisms f and g (which both may be represented as chain maps P_{1} → F_{2}) and a chain homotopy H between them, then clearly DH will be a chain homotopy between Df and Dg.

Lemma 2.11. Let C, E, and F be chain complexes with E finite.

(a) The complexes E and D(DE) weakly equivalent.

(b) There is a natural isomorphism

Ext_{A}^{∗∗}(C ⊗ E, F) ≅ Ext_{A}^{∗∗}(C, DE ⊗ F).

Proof. Part (a) is immediate. For (b), imitate the proofs for modules; see [2], for instance.

We will most often apply part (a) to part (b) to conclude that when E is finite, there is an isomorphism

Ext_{A}^{∗∗}(C ⊗ DE, F) ≅ Ext_{A}^{∗∗}(C, E ⊗ F).

3. Properties of h_{t,s}-maps

In this section we define h_{t,s}-maps, and we describe some of their basic properties. We also discuss thick subcategories briefly.

As above, let A be the mod 2 Steenrod algebra. Given an elementary sub-Hopf algebra E = E[P_{t_{1}}^{s_{1}}, P_{t_{2}}^{s_{2}}, …] of A, let

Ext_{E}^{∗∗}(F_{2}, F_{2}) = F_{2}[h_{t_{1},s_{1}}, h_{t_{2},s_{2}}, …],

so that h_{t_{i},s_{i}} is in bidegree (1, |P_{t_{i}}^{s_{i}}|).

Definition 3.1. Let M be an A-module (or more generally, a chain complex of A-modules). An h_{t,s}-map v ∈ Ext_{A}^{∗∗}(M, M) is an element such that if E is an elementary sub-Hopf algebra containing P_{t}^{s}, then for some j, we have ι_{E}^{∗}(v) = h_{t,s}^{j} ⊗ 1_{M} ∈ Ext_{E}^{∗∗}(M, M); and if E is elementary with P_{t}^{s} ∉ E, then ι_{E}^{∗}(v) is nilpotent. We also say that such a v is an h_{t,s}-map “on M.” Similarly, if Γ is a coalgebra (in the category either of modules or of chain complexes), then we have a notion of an h_{t,s}-map in Ext_{A}^{∗∗}(Γ, F_{2}).

Remark 3.2. On one hand, this definition bears some resemblance to the notion of v_{n}-map in [7, 3.3]. On the other hand, this definition is not very practical, computationally. We do not know how to improve it, though—if we were to use an alternative condition like “ι_{E[P_{t}^{s}]}^{∗}(v) = h_{t,s}^{j} ⊗ 1_{M} and ι_{E}^{∗}(v) = 0 if P_{t}^{s} ∉ E,” then many of our proofs in this section would fail.

We have a number of results about h_{t,s}-maps analogous to results about v_{n}-maps; the h_{t,s} results are proved just as the v_{n} ones are in [6] and [7]. The following is our main goal.

Proposition 3.3. The property of having an h_{t,s}-map is generic for finite chain complexes of A-modules.

Lemma 3.4. Let M be a finite chain complex, and let v ∈ Ext_{A}^{∗∗}(M, M) be an h_{t,s}-map. For some k > 0, for every elementary sub-Hopf algebra E with P_{t}^{s} ∉ E, we have ι_{E}^{∗}(v^{k}) = 0.

Proof. It suffices to prove this for the exterior sub-Hopf algebras E of A which are maximal among those that do not contain P_{t}^{s}; then using a vanishing line argument, e.g. [11, 2.9 and 2.10], we only need to prove this for finitely many different such E’s.

Lemma 3.5. Let M be a finite chain complex, and let v ∈ Ext_{A}^{∗∗}(M, M) be an h_{t,s}-map. For some k > 0, v^{k} is central in Ext_{A}^{∗∗}(M, M).

Proof. This is proved as in [16] and [7]: since M ⊗ DM is a finite coalgebra chain complex and there is an algebra isomorphism

Ext_{A}^{∗∗}(M, M) ≅ Ext_{A}^{∗∗}(M ⊗ DM, F_{2}),

we may prove this instead for the case of an h_{t,s}-map v in Ext_{A}^{n}(Γ, F_{2}) for Γ a finite coalgebra. Define ad(v) ∈ Ext_{A}^{∗∗}(Γ, Γ) as the composite

Γ̃ ⟶^{Δ} Γ̃ ⊗ Γ̃ ⟶^{1−T} Γ̃ ⊗ Γ̃ ⟶^{v⊗ι} Ω^{−n}Γ,

where Γ̃ is a projective complex, ι : Γ̃ → Γ is a quasi-isomorphism, and T is the map that interchanges the two factors. Then for x ∈ Ext_{A}^{∗∗}(Γ, F_{2}), we have (ad(v))^{∗}(x) = [v, x]. By definition, for each elementary E, the element ι_{E}^{∗}(v) is in the center of Ext_{E}^{∗∗}(Γ, F_{2}), so all of the restrictions of ad(v) are zero. Hence ad(v) is nilpotent, and so by the argument in the proof of [16, Lemma 6.1.2], some power v^{2^{k}} is central in Ext_{A}^{∗∗}(Γ, F_{2}).

Lemma 3.6. Let M be a finite chain complex; let v and w be two h_{t,s}-maps on M. Then for some i and j we have v^{i} = w^{j}.

Proof. By replacing v and w by powers if necessary, we may assume that ι_{E}^{∗}(v) = ι_{E}^{∗}(w) for all E; hence v − w is nilpotent. By Lemma 3.5, v and w commute, so v^{2^{k}} = w^{2^{k}} for some k.

Lemma 3.7. Let L and M be finite. Assume that v is an h_{t,s}-map on L and w is an h_{t,s}-map on M. Given f ∈ Ext_{A}^{∗∗}(L, M), there exist integers i and j so that w^{j} ∘ f = f ∘ v^{i}. (Here, ∘ denotes Yoneda composition.)

Proof. We have an isomorphism

Ext_{A}^{∗∗}(L, M) ≅ Ext_{A}^{∗∗}(L ⊗ DM, F_{2}).

There exist two h_{t,s}-maps in Ext_{A}^{∗∗}(L ⊗ DM, L ⊗ DM), namely ṽ = v ⊗ 1 and w̃ = 1 ⊗ Dw. So by Lemma 3.6, we have ṽ^{i} = w̃^{j} for some i and j. Now argue as in [7] (see also [16]).

Proof of Proposition 3.3. We are trying to prove that the property of possessing an h_{t,s}-map is generic on F, the category of finite chain complexes. The only nontrivial aspect of genericity is part (c) of Definition 2.1, and just as in [7], this follows from Lemma 3.7.

Now we describe some thick subcategories of the category of finite chain complexes of A-modules. The following definition of “admissible” is motivated by Theorem A.1.

Definition 3.8. Recall that the differentials in A are the P_{t}^{s}’s with s < t. A collection S of differentials is admissible if and only if P_{t}^{s} ∈ S implies that P_{t−1}^{s} ∈ S (if s < t − 1) and that P_{t−1}^{s+1} ∈ S (if s + 1 < t − 1). Given a differential P_{t}^{s}, let S(P_{t}^{s}) be the smallest admissible collection containing P_{t}^{s}. Given a chain complex M, the support of M is the collection of differentials supp(M) = {P_{t}^{s} : H(M, P_{t}^{s}) = 0} (perhaps this should be called the “anti-support,” instead). By Theorem A.1, if M is finite, then the support of M is admissible. Given a collection S of P_{t}^{s}’s with s < t, define C(S) to be the full subcategory of the category F of finite chain complexes with objects

ob C(S) = {M ∈ F : S ⊆ supp(M)}.

For example, suppose we put the differentials in order of degree as x_{0}, x_{1}, x_{2}, …, and given n we define k(n) so that x_{k(n)} = P_{n}^{0}. Then the complex C_{k(n)+1} in Corollary 2.7 has support S(P_{n}^{0}). If Φ denotes the doubling functor—see Section 4— then

supp(Φ^{s}C_{k(n)+1}) = S(P_{n}^{s}).

Question 3.9. C(S) is thick for any admissible S; are there other thick subcategories?

We also have the following.

Proposition 3.10. For each finite admissible set S, there is a finite complex L(S) with supp(L(S)) = S. Hence the categories C(S), for S finite admissible, are distinct.

Proof. If S is a finite admissible set of differentials, then S is the union of finitely many sets of the form S(P_{t}^{s}). If S = ⋃_{i} S(P_{t_{i}}^{s_{i}}), then we set

L(S) = ⨂_{i} Φ^{s_{i}}C_{k(t_{i})+1}.

One can easily verify that this module does the trick.

Remark 3.11. Let A(n) be the sub-Hopf algebra of A generated by the elements {P_{t}^{s} : s + t ≤ n + 1}. Mitchell [10] has shown that there is an A-module structure on A(n) extending its A(n)-module structure; with this structure, one can show that supp(A(n)) = S(P_{n+1}^{0}). Hence given a finite admissible set of differentials S, one can use A(n) and its doubles in place of the complexes C_{k} to construct a finite A-module with support exactly S. This could be helpful if for some odd reason one wanted to avoid working with chain complexes.

4. h_{t,s}-maps in Ext_{A}^{∗∗}(M, M)

In this section we prove this “periodicity theorem” for finite A-modules.

Theorem 4.1. Let M be a finite chain complex of A-modules; let S be the support of M, S = {P_{v}^{u} : H(M, P_{v}^{u}) = 0}. Suppose that for some s < t, P_{t}^{s} ∉ S and S ∪ {P_{t}^{s}} is admissible. Then there is a non-nilpotent h_{t,s}-map in Ext_{A}^{∗∗}(M, M).

Remark 4.2. (a) We proved a special case of this in [12]; namely, given S, we prove this for P_{t}^{s} equal to the lowest dimensional differential not in S. See Theorem 2.6.

(b) Theorem 4.1 implies Theorems 1.1 and 1.2 immediately; for the uniqueness claims in these results, see Lemma 3.6.

(c) If S ∪ {P_{t}^{s}} is not admissible, then M does not have an h_{t,s}-map. Any element f ∈ Ext_{A}^{q}(M, M) can be represented by a map f̃: Ω^{q}P → M, where P is a complex of projectives quasi-isomorphic to M. Let K be the fiber of this map; then K is a finite chain complex, but it violates Theorem A.1: since f is an h_{t,s}-map, then h_{t,s}^{−1} Ext_{E[P_{t}^{s}]}^{∗∗}(f, F_{2}) is an isomorphism, while for P_{v}^{u} ≠ P_{t}^{s}, the map h_{v,u}^{−1} Ext_{E[P_{v}^{u}]}^{∗∗}(f, F_{2}) is zero.

(d) We expect that this theorem would follow rather easily if one had an analog of the thick subcategory theorem. After all, the property of having an h_{t,s}- map is generic, so the subcategory of all finite complexes with such a map is thick; presumably, from a classification of thick subcategories, one could then conclude that this thick subcategory contained C(S).

We outline the proof of a special case of the theorem, to demonstrate the ideas involved. First we need a definition.

Definition 4.3. Let ΦA denote the doubled Steenrod algebra: when one ignores grading, it is isomorphic as a Hopf algebra to A; one grades it by putting ΦSq^{n} in degree 2n. Note that ΦA is also the quotient Hopf algebra of A obtained by dividing by the ideal

A(P_{1}^{0}, P_{2}^{0}, P_{3}^{0}, …).

See [8], for example. We denote the quotient map by φ : A → ΦA, so that if L and M are ΦA-modules, then we have a map φ^{∗} : Ext_{ΦA}^{∗∗}(L, M) → Ext_{A}^{∗∗}(L, M).

Let Φ also denote the doubling functor from the category of A-modules to the category of ΦA-modules, defined by doubling degrees, as above. For any A-module M, we can view ΦM as an A-module via the map φ; the action is given by

Sq^{n}(Φx) = \begin{cases}0 & if n is odd, \\ Φ(Sq^{n/2}x) & if n is even. \end{cases}

Of course, by doubling each module in a given chain complex, we also have a functor from chain complexes of A-modules to chain complexes of ΦA-modules.

Example 4.4. There is an h_{2,1}-map in Ext_{A}^{∗∗}(F_{2}, F_{2}).

Outline of proof. We have a short exact sequence of A-modules

0 → ΣF_{2} → M → F_{2} → 0, (4.1)

where M is the cohomology of the mod 2 Moore spectrum. This corresponds to the element h_{1,0} ∈ Ext_{A}^{1,1}(F_{2}, F_{2}). Applying Φ gives the short exact sequence

0 → Σ^{2}F_{2} → ΦM → F_{2} → 0,

representing h_{1,1} ∈ Ext_{A}^{1,2}(F_{2}, F_{2}); it is well known that this element is nilpotent (or one can prove it with Theorem 2.8—see Lemma 4.6).

We claim that ΦM has an h_{2,1}-map. First of all, there is an h_{2,0}-map in Ext_{A}^{∗∗}(M, M)—this is well known, and it can also be proven using Theorem 2.6. Applying Φ yields an h_{2,1}-map in Ext_{ΦA}^{∗∗}(ΦM, ΦM), and hence also in Ext_{A}^{∗∗}(ΦM, ΦM).

Since having an h_{2,1}-map is generic, the nilpotence of the element h_{1,1} shows that F_{2} must also have an h_{2,1}-map: for some n, we have h_{1,1}^{n} = 0, and so the extension

0 → Σ^{2n}F_{2} → Σ^{2n−2}ΦM → Σ^{2n−4}ΦM → ⋯ → ΦM → F_{2} → 0

is trivial. Using a little homological algebra, one can show that any generic property that holds for ΦM holds for F_{2} (see Lemma 4.8).

The general idea behind the proof of Theorem 4.1 is to use the finite complexes from Corollary 2.7

F_{2} = C_{0}, C_{1}, C_{2}, …, C_{k(t)},

where C_{k(t)} has an h_{t,0}-map, and there are exact triangles

C_{i} → C_{i+1} → C_{i}. (4.2)

We apply Φ^{s} to the C_{i}’s, tensor them with M, and show that the elements corresponding to the sequences (4.2) in Ext_{A}^{1}(M ⊗ Φ^{s}C_{i}, M ⊗ Φ^{s}C_{i}) are nilpotent. We use this to show that the h_{t,s}-map on M ⊗ Φ^{s}C_{i+1} induces one on M ⊗ Φ^{s}C_{i}; the result follows by induction.

Here are the tools that we need to prove Theorem 4.1.

Definition 4.5. Let W be a chain complex. Given an h_{t,s}-map y ∈ Ext_{A}^{∗∗}(W, W), we have an h_{t,s+n}-map Φ^{n}y ∈ Ext_{Φ^{n}A}^{∗∗}(Φ^{n}W , Φ^{n}W). The algebra map φ_{n} : A → → Φ^{n}A induces the map φ_{n}^{∗} on Ext; we refer to

φ_{n}^{∗}(Φ^{n}y) ∈ Ext_{A}^{∗∗}(Φ^{n}W , Φ^{n}W)

as a Φ^{n}h_{t,s}-map.

Lemma 4.6. Let C be a finite chain complex, suppose that s + n ≥ t, and let z ∈ Ext_{A}^{∗∗}(Φ^{n}C, Φ^{n}C) be a Φ^{n}h_{t,s}-map. Then z is nilpotent.

Proof. To prove this, we use the nilpotence theorem, 2.8. Let B_{n} be the Hopf algebra kernel of φ_{n} : A → → Φ^{n}A (so that B_{n} is dual to F_{2}[ξ_{1}, ξ_{2}, ξ_{3}, …]/(ξ_{i}^{2^{n}})— see [8]). Let E be an elementary sub-Hopf algebra of A. If we have an inclusion ι_{B_{n},E} : E ↪ B_{n}, then the following diagram commutes.

Ext_{Φ^{n}A}^{∗∗}(Φ^{n}C, Φ^{n}C) ⟶^{φ_{n}^{∗}} Ext_{A}^{∗∗}(Φ^{n}C, Φ^{n}C)
ι_{F_{2}}^{∗} ↓  ↓^{ι_{Bn}^{∗}}
Ext_{F_{2}}^{∗∗}(Φ^{n}C, Φ^{n}C) ⟶ Ext_{B_{n}}^{∗∗}(Φ^{n}C, Φ^{n}C)
↓^{ι_{Bn,E}^{∗}}
Ext_{E}^{∗∗}(Φ^{n}C, Φ^{n}C).

Since z is in the image of the map φ_{n}^{∗}, then clearly ι_{E}^{∗}(z) = ι_{B_{n},E}^{∗} ∘ ι_{B_{n}}^{∗}(z) = 0. On the other hand, if E ⊈ B_{n}, then Ẽ = E//(E ∩ B_{n}) is an elementary sub-Hopf algebra of Φ^{n}A which does not contain P_{t}^{s+n}. If we let z = φ_{n}^{∗}(Φ^{n}y), where Φ^{n}y is an h_{t,s+n}-map in Ext_{Φ^{n}A}^{∗∗}(Φ^{n}C, Φ^{n}C), then ι∗ ; Ẽ(Φ^{n}y) is nilpotent, and hence the same is true of ι_{E}^{∗}(z). Hence by Theorem 2.8, z is nilpotent.

Lemma 4.7. Let M and C be finite chain complexes; assume that C has an h_{t,s}- map y ∈ Ext_{A}^{∗∗}(C, C). Assume that s + n < t. If H(M, P_{t}^{s+n}) = 0, then

1_{M} ⊗ φ_{n}^{∗}(Φ^{n}y) ∈ Ext_{A}^{∗∗}(M ⊗ Φ^{n}C, M ⊗ Φ^{n}C)

is nilpotent.

Proof. M ⊗ Φ^{n}C is finite, so we use Theorem 2.8. Let z = 1_{M} ⊗ φ_{n}^{∗}(Φ^{n}y); z is clearly an h_{t,s+n}-map, so if E is elementary and P_{t}^{s+n} ∉ E, then ι_{E}^{∗}(z) is nilpotent. On the other hand, if E is elementary and P_{t}^{s+n} ∈ E, then Lemma A.2 tells us that ι_{E}^{∗}(z) is nilpotent. (Lemma A.2 implies that z^{−1} Ext_{E}^{∗∗}(M ⊗ Φ^{n}C, L) = 0 for any finite chain complex L.)

Lemma 4.8. Let L be a finite chain complex. Suppose that y ∈ Ext_{A}^{1}(L, L) is an element represented by the extension

0 → L → W → L → 0,

and suppose that y is nilpotent. If W has an h_{t,s}-map, then so does L.

Proof. Suppose y^{n} = 0; then the n-extension

0 → L → \underbrace{W → W → ⋯ → W → W}_{n} → L → 0

is trivial. Equivalently, we have a trivial extension

0 → L → W (n) → L → 0,

where W (n) is the total complex of the double complex \overbrace{W → ⋯ → W}^{n}. Clearly W (n) is finite, and by genericity and induction on n, it has an h_{t,s}-map. Since this extension is trivial, then L is a summand of W (n), and hence has an h_{t,s}-map.

Recall that x_{0}, x_{1}, … are the differentials in A, in order of degree. Recall further that given t, k(t) is the number so that x_{k(t)} = P_{t}^{0}. By Corollary 2.7, for each t we have a finite chain complex C_{k(t)} with an h_{t,0}-map.

Proof of Theorem 4.1. Assume that M is as in the statement of the theorem. We want to show that M has an h_{t,s}-map. If P_{t}^{s} = P_{1}^{0}, then Theorem 2.6 proves the result.

Otherwise, we have t > 1; for each j ≤ k(t), set W_{j} = Φ^{s}C_{j}. Then W_{k(t)} has an h_{t,s}-map, and hence so does M ⊗ W_{k(t)}. We have a series of exact triangles

M ⊗ W_{i} → M ⊗ W_{i+1} → M ⊗ W_{i}.

By Lemmas 4.6 and 4.7, the corresponding elements in Ext are nilpotent. So by Lemma 4.8 and downward induction on i, M ⊗W_{0} = M has an h_{t,s}-map. This map is non-nilpotent because it induces an isomorphism on h_{t,s}^{−1} Ext_{E[P_{t}^{s}]}^{∗∗}(−, F_{2}).

5. The odd prime case

Fix an odd prime p, let A be the mod p Steenrod algebra, and let P = A/AβA be the algebra of reduced powers. Now, Ext_{A} has more topological relevance than Ext_{P}; unfortunately, we only have a nilpotence theorem for the latter ([11, 4.3 and 4.5]). Because of this, we only present an outline of the results on periodicity in Ext_{P}.

There is one main difference between the p = 2 case and the p odd case: at odd primes one detects nilpotence by restricting to quasi-elementary sub-Hopf algebras (see [11, Section 4]). The maximal ones of these are called the Q(r)’s: for r ≥ 1, Q(r) is dual to

Q(r)_{∗} = F_{p}[ξ_{r}, ξ_{r+1}, ξ_{r+2}, …]/(ξ_{r}^{p}, ξ_{r+i}^{p^{r+1}}).

(Here and in what follows, i runs through all positive integers.) There is a central extension of Hopf algebras, dual to

F_{p}[ξ_{r}, ξ_{r+1}^{p^{r}}, ξ_{r+2}^{p^{r}}, …]/(ξ_{r}^{p}, ξ_{r+i}^{p^{r+1}}) → Q(r)_{∗} → F_{p}[ξ_{r+1}, ξ_{r+2}, …]/(ξ_{r+i}^{p^{r}}).

We can argue as in [18, 6.3] to show that, if h_{t,s} denotes the cobar element [ξ_{t}^{p^{s}}] and if β is the Bockstein acting on Hopf algebra cohomology, then we have a polynomial subalgebra of Ext_{Q(r)}^{∗∗}(F_{p}, F_{p}),

R = F_{p}[β(h_{r,0})] ⊗ F_{p}[β(h_{t,s}) : t ≥ r + 1, s < r + 1],

and furthermore Ext_{Q(r)}^{∗∗}(F_{p}, F_{p})//R consists of nilpotent elements. (Sketch of proof: look at the Cartan-Eilenberg spectral sequence associated to the extension above. The E_{2}-term is the tensor product of R with an exterior algebra,

E[h_{r,0}, h_{t,s} : t ≥ r + 1, s < r + 1].

One can compute that for n ≥ 2r + 1 we have d_{2}(h_{n,0}) = h_{r,0}h_{n−r,r}, and that d_{2} is zero on all the other generators. Then using Steenrod operations, one can show that each β(h_{t,s}) is a permanent cycle.)

Using this (and similar computations for sub-Hopf algebras of Q(r)), we can define β(h_{t,s})-maps in Ext_{P}^{∗∗}(M, M). Working as in Sections 2 and 3, we can prove the following.

Proposition 5.1. The property of having a β(h_{t,s})-map is generic for finite chain complexes of P -modules.

We do not have a proof for the odd prime version of Theorem A.1; nonetheless, we define admissible the same way. Corollary 2.7 still holds—all of the work in [12] immediately carries over to the context of modules over P —and we can replace doubling by the pth power map; hence we get an odd prime version of our main theorem.

Theorem 5.2. Let M be a finite chain complex of P -modules; let S be the support of M, S = {P_{v}^{u} : H(M, P_{v}^{u}) = 0}. Suppose that for some s < t, P_{t}^{s} ∉ S and S∪{P_{t}^{s}} is admissible. Then there is a non-nilpotent β(h_{t,s})-map z in Ext_{A}^{∗∗}(M, M). If y is another β(h_{t,s})-map, then there are integers i and j so that z^{i} = y^{j}.

Remark 5.3. (a) We do not have the odd prime version of Theorem A.1, so it is possible for there to be other β(h_{v,u})-maps in Ext_{B}^{∗∗}(M, M). A proof of Conjecture A.8 would eliminate this possibility.

(b) When working with the mod p Steenrod algebra A for p odd, the analog of doubling is not as pleasant as for the mod p reduced powers or for the mod 2 Steenrod algebra. If there were a nilpotence theorem for A and one wanted to prove a periodicity theorem, this difference could conceivably cause problems. (One may be able to use Theorem 5.2 to avoid these problems, though.)

6. Questions and comments

The work of Hopkins and Smith [7] identifies, among other things, all of the central non-nilpotent self-maps of a finite p-local spectrum (up to “N-isomorphism,” anyway). Theorem 1.1 tells us about certain non-nilpotent elements in Ext_{A}^{∗∗}(F_{2}, F_{2}), but it does not identify all of the non-nilpotent elements (and note that Ext_{A}^{∗∗}(F_{2}, F_{2}) is commutative). Neither does our more general periodicity theorem, Theorem 4.1; this is one of its flaws. For instance, d_{0} ∈ Ext_{A}^{4,18}(F_{2}, F_{2}), e_{0} ∈ Ext_{A}^{4,21}(F_{2}, F_{2}), and B_{21} ∈ Ext_{A}^{10,69}(F_{2}, F_{2}) are non-nilpotent, and none of them is an h_{t,s}-map for any (s, t)—one can see this immediately, because they are each in the wrong bidegree. (Of course, these elements are all detected by restricting to elementary sub-Hopf algebras—if E is the sub-Hopf algebra dual to F_{2}[ξ_{2}, ξ_{3}, …]/(ξ_{i}^{4}), then ι_{E}^{∗}(d_{0}) = h_{2,0}^{2}h_{2,1}^{2}, ι_{E}^{∗}(e_{0}) = h_{2,0}h_{2,1}^{3}, and

ι_{E}^{∗}(B_{21}) = h_{2,0}^{3}h_{2,1}^{5}h_{4,0}^{2} + h_{2,0}^{5}h_{2,1}^{4}h_{4,1} + h_{2,0}^{3}h_{2,1}^{3}h_{3,0}^{2}h_{3,1}^{2} + h_{2,0}^{5}h_{2,1}^{2}h_{3,1}^{3}.)

So a natural question to ask is: what are all of the non-nilpotent elements in Ext_{A}^{∗∗}(F_{2}, F_{2})? Let E be a maximal elementary sub-Hopf algebra; then E is normal (see [8], for instance). One has a Hopf algebra extension

E → A → A//E

which gives a Cartan-Eilenberg spectral sequence:

Ext_{A//E}^{∗∗}(F_{2}, Ext_{E}^{∗∗}(F_{2}, F_{2})) ⇒ Ext_{A}^{∗∗}(F_{2}, F_{2}).

The restriction ι_{E}^{∗} is an edge homomorphism in this spectral sequence, so one has an inclusion of algebras im(ι_{E}^{∗}) ↪ (Ext_{E}^{∗∗}(F_{2}, F_{2}))^{A//E}. What is this ring of invariants? How close is this inclusion to an equality? When E = E(0) = E[P_{1}^{0}, P_{2}^{0}, …], then the image of ι_{E(0)}^{∗} and the ring of invariants are both equal to F_{2}[h_{10}]. When E = E(1) = E[P_{t}^{s} : s ≤ 1, t ≥ 2], then the ring of invariants is strictly larger than im(ι_{E(1)}^{∗})—the former contains F_{2}[h_{20}] as a subalgebra, for instance.

Remark 6.1. If G is a finite group and k a field of nonzero characteristic, then one has a “nilpotence theorem” precisely analogous to Theorem 2.8, due to Carlson [3]; as a corollary of a theorem of Evens [5], we also see that if M is a kG-module, given any non-nilpotent element z of Ext_{kG}^{∗}(M, M), then some power of z is in the image of

Ext_{kG}^{∗}(k, k) ⟶^{−⊗M} Ext_{kG}^{∗}(M, M).

(See [2, Volume II, Corollary 4.2.4].) This fails in our case; for example, let M denote the cohomology of the mod 2 Moore spectrum. Then by [12, Theorem 3.1] there is an h_{2,0}-map in Ext_{A}^{k,3k}(M, M) for some k, but Ext_{A}^{k,3k}(F_{2}, F_{2}) = 0 for all k, so no power of this element is in the image of

Ext_{A}^{∗∗}(F_{2}, F_{2}) ⟶^{−⊗M} Ext_{A}^{∗∗}(M, M).

Also, in the group theory case, a theorem of Avrunin-Scott [1] identifies essentially all of the non-nilpotent elements in Ext_{kG}^{∗}(M, M) for M a finitely generated kGmodule; this is the sort of result we would like for A-modules, but we have not found yet.

Another question is, what information does Theorem 1.1 give about stable homotopy groups of spheres? Nishida’s theorem tells us that none of the h_{n+1,n}-maps can survive the Adams spectral sequence to give non-nilpotent elements in π_{∗}(S^{0}) (except of course for h_{1,0}), so some power of each of these elements in Ext must be killed by a differential. It is not clear whether one can get any useful information about the Adams spectral sequence from this, but it is worth investigating.

We make one further comment about self maps of spectra. Let X be a finite spectrum. The nilpotence theorem of [4] tells us that h_{t,s}-maps in Ext_{A}^{∗∗}(H^{∗}X, H^{∗}X) can survive to give non-nilpotent homotopy classes in [X, X] only if s = 0. If s > 0, then h_{t,s}-maps can survive to the E_{∞}-term, but the resulting homotopy classes will be nilpotent. Particular ones of these geometric h_{t,s}-maps have been used to construct type n spectra with v_{n}-maps, giving an alternate proof of the periodicity theorem of Hopkins and Smith—see [13]. So it is reasonable to hope for applications of these newly discovered non-nilpotent self-maps to stable homotopy theory.

Appendix A. P_{t}^{s}-homology of finite complexes

In this appendix we prove the following result, analogous to Ravenel’s in [15] that if X is a finite spectrum and K(n)_{∗}(X) = 0, then K(n − 1)_{∗}(X) = 0. Recall that for M a chain complex of A-modules, we say that H(M, P_{t}^{s}) = 0 if h_{t,s}^{−1} Ext_{E[P_{t}^{s}]}^{∗∗}(M, F_{2}) = 0.

Theorem A.1. If M is a finite chain complex of A-modules and H(M, P_{t}^{s}) = 0,

(a) if s < t − 1 then H(M, P_{t−1}^{s}) = 0;
(b) if s + 1 < t − 1 then H(M, P_{t−1}^{s+1}) = 0.

Of course, this theorem applies to finite A-modules, as a special case. Also, note that these are the only restrictions on the vanishing of the homology groups; see Proposition 3.10 and Remark 3.11.

Proof. First we prove part (a), so we assume that s < t − 1. Let

Q = E[P_{v}^{u} : 0 ≤ u ≤ s, v ≥ t − 1];

this is a normal elementary sub-Hopf algebra of A (see [8], for example). Its cohomology Ext_{Q}^{∗∗}(F_{2}, F_{2}) is a polynomial algebra, generated by the h_{v,u}’s with 0 ≤ u ≤ s and v ≥ t − 1.

It suffices to prove the following lemmas.

Lemma A.2. Let M be a finite chain complex of Q-modules. If H(M, P_{t}^{s}) = 0, then h_{t,s}^{−1} Ext_{Q}^{∗∗}(M, F_{2}) = 0.

Lemma A.3. Let M be a finite chain complex of A-modules. If h_{t,s}^{−1} Ext_{Q}^{∗∗}(M, F_{2}) = 0, then h_{t−1,s}^{−1} Ext_{Q}^{∗∗}(M, F_{2}) = 0.

Lemma A.4. Let M be a finite complex of Q-modules. If h_{t−1,s}^{−1} Ext_{Q}^{∗∗}(M, F_{2}) = 0, then H(M, P_{t−1}^{s}) = 0.

Proof of Lemma A.2. For M a finite chain complex, this follows from [9, 2.2 and 2.3]: by definition, H(M, P_{t}^{s}) = 0 means that h_{t,s}^{−1} Ext_{E[P_{t}^{s}]}^{∗∗}(M, F_{2}) = 0. Therefore the E_{2}-term of the algebraic Eilenberg-Moore spectral sequence is zero, and hence the abutment, h_{t,s}^{−1} Ext_{Q}^{∗∗}(M, F_{2}), is zero as well.

Proof of Lemma A.3. Since Q is normal in A, then A//Q acts on Ext_{Q}^{∗∗}(M, F_{2}). When M = F_{2} this action can be computed as follows: first, note that

Ext_{Q}^{∗∗}(F_{2}, F_{2}) ≅ Ext_{A}^{∗∗}(A//Q, F_{2});

the right action of A//Q on itself induces a left action on Ext_{Q}^{∗∗}(F_{2}, F_{2}). So we let P_{∗} be a projective resolution of A//Q as an A-module; then an element θ ∈ A//Q gives an A-linear map from A//Q to itself: 1 ↦ θ. This induces a map of the resolution to itself, so we compose this map with the map P_{1} → F_{2} representing h_{t,s}. The net result is the following:

Sq^{2^{k}}(h_{t,s}) = \begin{cases}0 & if k < s + t − 1, \\ h_{t−1,s} & if k = s + t − 1. \end{cases}

So the Cartan formula gives the result: given x ∈ Ext_{Q}^{∗∗}(M, F_{2}) with h_{t,s}^{2^{i}}x = 0, we have

0 = Sq^{2^{s+t−1+i}}(h_{t,s}^{2^{i}}x) = h_{t−1,s}^{2^{i}}x + h_{t,s}^{2^{i}}Sq^{2^{s+t−1+i}}(x).

The A//Q-action lowers degrees. Since M is a bounded below chain complex of bounded below modules, then Ext_{Q}^{∗∗}(M, F_{2}) is bounded below as an A//Q-module; thus for i ≫ 0, we have Sq^{2^{s+t−1+i}}(x) = 0 and hence h_{t−1,s}^{2^{i}}x = 0.

Proof of Lemma A.4. Since M is finite and h_{t−1,s}^{−1} Ext_{Q}^{∗∗}(M, F_{2}) = 0, then

h_{t−1,s}^{−1} Ext_{Q}^{∗∗}(M, M) = 0.

We have a commutative diagram

h_{t−1,s}^{−1} Ext_{Q}^{∗∗}(M, M) ⟶ h_{t−1,s}^{−1} Ext_{E[P_{t−1}^{s}]}^{∗∗}(M, M)
↑  ↑^{i}
h_{t−1,s}^{−1} Ext_{Q}^{∗∗}(F_{2}, F_{2}) ⟶^{ρ} h_{t−1,s}^{−1} Ext_{E[P_{t−1}^{s}]}^{∗∗}(F_{2}, F_{2})

The map ρ is a surjection. If H(M, P_{t−1}^{s}) ≠ 0, then i is nonzero; but we know that h_{t−1,s}^{−1} Ext_{Q}^{∗∗}(M, M) = 0.

This finishes the proof of part (a) of the theorem.

To prove part (b), we work over the algebra

Q′ = E[P_{v}^{u} : 0 ≤ u ≤ s + 1, v ≥ t − 1].

Lemma A.2 still holds, as does the analogue of Lemma A.4. Instead of A.3, though, we use the following.

Lemma A.5. Let M be a finite chain complex of A-modules. If h_{t,s}^{−1} Ext_{Q′}^{∗∗}(M, F_{2}) = 0, then h_{t−1,s+1}^{−1} Ext_{Q′}^{∗∗}(M, F_{2}) = 0.

Proof of Lemma A.5. We imitate the proof of Lemma A.3; as above, A//Q′ acts on Ext_{Q′}^{∗∗}(M, F_{2}), and on Ext_{Q′}^{∗∗}(F_{2}, F_{2}) we have

Sq^{2^{k}}(h_{t,s}) = \begin{cases}0 & if k < s, \\ h_{t−1,s+1} & if k = s. \end{cases}

Now use the Cartan formula, as above.

This finishes the proof of Theorem A.1.

We have the following corollary.

Corollary A.6. Let M be a finite A-module. If H(M, P_{n+1}^{0}) = 0, then M is free over A(n), the sub-Hopf algebra of A generated by Sq^{k} for all k ≤ 2^{n}.

Remark A.7. One might ask about generalizing Theorem A.1. Here are some limitations: first of all, there are bounded below A-modules for which the theorem fails; for example, A/AP_{2}^{0} has zero P_{3}^{0}-homology but nonzero P_{2}^{0}-homology [8, 19.24]. It is also important for the module to be an A-module, not just a module over some sub-Hopf algebra; for instance, M = E[P_{2}^{0}] is a finite dimensional module over E[P_{t}^{0} : t ≥ 1] which has no P_{2}^{0}-homology, but H(M, P_{1}^{0}) ≠ 0.

One might also ask if there is an odd prime analog. The following seems reasonable.

Conjecture A.8. Let p be an odd prime, and let A be the mod p Steenrod algebra. If M is a finite chain complex of A-modules,

(a) if H(M, Q_{t}) = 0, then H(M, Q_{t−1}) = 0;

(b) if H(M, P_{t}^{s}) = 0 and s < t − 1, then H(M, P_{t−1}^{s}) = 0; and
(c) if H(M, P_{t}^{s}) = 0 and s + 1 < t − 1, then H(M, P_{t−1}^{s+1}) = 0.

In fact, part (a) can be proved exactly as Theorem A.1(a). Imitating the above proof for parts (b) and (c) depends on having an analog of [9, Theorem 2.2] for a truncated polynomial algebra instead of an exterior algebra, and there may be difficulties in proving such a thing.

References

1. G. S. Avrunin and L. L. Scott, Quillen stratification for modules, Invent. Math. 66 (1982), 277–286.

2. D. J. Benson, Representations and cohomology, Volumes I and II, Cambridge University Press, 1991.

3. J. F. Carlson, Complexity and Krull dimension, Representations of Algebras (M. Auslander and E. Lluis, eds.), 1981, Springer Lecture Notes in Mathematics 903, pp. 62–67.

4. E. S. Devinatz, M. J. Hopkins, and J. H. Smith, Nilpotence and stable homotopy theory, Ann. of Math. (2) 128 (1988), 207–241.

5. L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224– 239.

6. 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.

7. M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory II, preprint.

8. H. R. Margolis, Spectra and the Steenrod algebra, North-Holland, 1983.

9. H. R. Miller, A localization theorem in homological algebra, Math. Proc. Cambridge Philos. Soc. 84 (1978), 73–84.

10. S. Mitchell, Finite complexes with A(n)-free cohomology, Topology 24 (1985), 227–246.

11. J. H. Palmieri, Nilpotence for modules over the mod 2 Steenrod algebra I, this volume.

12. , Self-maps of modules over the Steenrod algebra, J. Pure Appl. Algebra 79 (1992), 281–291.

13. J. H. Palmieri and H. Sadofsky, Self maps of spectra, a theorem of J. Smith, and Margolis’ killing construction, to appear in Math. Zeit.

14. D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. 43, Springer-Verlag, 1967.

15. D. C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. of Math. 106 (1) (1984), 351–414.

16. , Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, vol. 128, Princeton University Press, 1992.

17. J.-L. Verdier, Catégories dérivées, Cohomologie Etale (SGA 4 \frac{1}{2} ) (P. Deligne, ed.), 1977, Springer Lecture Notes in Mathematics 569, pp. 262–311.

18. C. Wilkerson, The cohomology algebras of finite dimensional Hopf algebras, Trans. Amer. Math. Soc. 264 (1981), 137–150.

University of Minnesota Current address: Department of Mathematics, University of Wisconsin, Madison, WI 53706 E-mail: palmieri@math.umn.edu