NILPOTENCE FOR MODULES OVER THE MOD 2 STEENROD ALGEBRA I

JOHN H. PALMIERI

Abstract. Let B be 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 nilpotence of elements in Ext_{B}^{∗∗}(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 remarkable result which provides algebraic means for detecting nilpotence in the collection 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 homotopy theory (see [19, 10, 7], for example). See [20] for a thorough discussion of this material; [5] also gives an overview of the nilpotence theorem and related results.

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 Ext_{A}^{∗∗}(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 for Amodules extends the strong parallel between results in stable homotopy theory and results for A-modules, as described in [11] and [16]. Some of these earlier A-module results have been used to prove results in homotopy theory via the Adams spectral 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 k of characteristic p, an elementary sub-Hopf algebra E ⊆ B is a bicommutative sub-Hopf algebra such that e^{p} = 0 for all e ∈ IE. (Here, IE denotes the augmentation ideal of E, IE = ker(ε : E → k).) Of course, given any Hopf algebra inclusion ι_{C} : C → B and B-modules L and M, we have a restriction map ι_{C}^{∗} : Ext_{B}^{∗∗}(L, M) → Ext_{C}^{∗∗}(L, M). Note that if Γ is a (coassociative) Bcoalgebra (so that there are B-module maps Δ : Γ → Γ ⊗ Γ and ε : Γ → k making certain diagrams commute), then Ext_{B}^{∗∗}(Γ, k) is an (associative) algebra, via the map Δ^{∗}. We prove the following result.

Theorem 1.1. Let B be a sub-Hopf algebra of the mod 2 Steenrod algebra A.

(a) Let Γ be a bounded below coassociative B-coalgebra; fix z ∈ Ext_{B}^{∗∗}(Γ, F_{2}). Then z is nilpotent if and only if ι_{E}^{∗}(z) is nilpotent for all elementary sub-Hopf algebras E of B.

(b) Let M be a finite dimensional B-module; fix z ∈ Ext_{B}^{∗∗}(M, M). Then z is nilpotent under Yoneda composition if and only if ι_{E}^{∗}(z) is nilpotent for all elementary sub-Hopf algebras E of B.

(c) Let L and M be B-modules with L bounded below and M finite dimensional; fix z ∈ Ext_{B}^{∗∗}(L, M). Then z^{⊗n} ∈ Ext_{B}^{∗∗}(L^{⊗n}, M^{⊗n}) is zero for n sufficiently large if and only if (ι_{E}^{∗}(z))^{⊗n_{E}} is zero for n_{E} sufficiently large, for all elementary sub-Hopf algebras E of B.

We also prove a similar result about detecting nilpotence for sub-Hopf algebras of the Steenrod reduced powers at an odd prime; see Section 4.

In [21] Wilkerson proved the special case of (b) where B is finite dimensional and M = F_{2}; Lin [9] proved this for a more general collection of sub-Hopf algebras (those with finite “profile” function—see Section 2), again only for M = F_{2}. Hopkins and the author [6] generalized this to arbitrary sub-Hopf algebras of A, but still for M = F_{2}. These are analogous to proving Nishida’s theorem [15] for various B’s.

We view Theorem 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 Ext_{kG}^{∗}(M, M) for G a finite group, k a field of characteristic p > 0, and M a kGmodule. 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 carry over to give interesting results in group theory? Are there interesting parallels between kG-modules and (finite) spectra? Wilkerson [21] has discussed some aspects of the first question.

Our other main results are Corollary 2.7 and Theorem 3.1; we would have to 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 Wilkerson’s result [21, Corollary 5.6] about detection of nilpotent elements in Ext_{B}^{∗∗}(k, k) to the case Ext_{B}^{∗∗}(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 B in Theorem 1.1 as analogous to applying Morava K-theories, then Theorem 3.1 gives an analog of the BP -version (or the MU-version) of the nilpotence theorem; namely, it gives a single sub-Hopf 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 of the 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 their hospitality while this research was being conducted. Also, the notion of “quasielementary Hopf algebra” in Section 4 is due to Haynes Miller and Clarence Wilkerson; I would like to thank them for pointing it out to me.

2. Preliminaries and the main lemma

In this section we recall some notions about the mod 2 Steenrod algebra, and we prove our main lemma, Lemma 2.4. [13] and [11] are basic references for results on the Steenrod algebra.

Recall that the mod 2 Steenrod algebra A is a Hopf algebra; its dual is isomorphic, as an algebra, to F_{2}[ξ_{1}, ξ_{2}, …]. Let the Milnor basis be the dual to the monomial basis; as usual, we set P_{t}^{s} = (ξ_{t}^{2^{s}})^{∗}. Given a sequence (n_{1}, n_{2}, …) where each n_{i} is either a non-negative integer or ∞, consider the subspace (F_{2}[ξ_{1}, ξ_{2}, …]/(ξ_{i}^{2^{n_{i}}}))^{∗} of A. Under certain conditions on the sequence of n_{i}’s, this is a sub-Hopf algebra of the Steenrod algebra—see [1] or [11], for example. Call the corresponding sub-Hopf algebra A(n_{1}, n_{2}, …); the sequence (n_{1}, n_{2}, …) is the profile for this sub-Hopf algebra. This is the main result we need:

(a) If (n_{1}, n_{2}, …) is a non-decreasing sequence of elements of {0, 1, 2, … } ∪ {∞}, then it is the profile function of a normal sub-Hopf algebra of A—see [11].

Here are some examples:

(b) Let D = A(1, 2, 3, …) (Lin [9] called this A_{d}).

(c) Fix r ≥ 1. Let D′ = A(\underbrace{0, 0, …, 0}_{r−1}, r, r + 1, …).

(d) Fix m ≥ 0. Let E(m) = A(\underbrace{0, 0, …, 0}_{m}, m+1, m+1, m+1, …). Then E(m)

is a sub-Hopf algebra of A, and (as an algebra) it is an exterior algebra (see [9], for example). Indeed, the E(m)’s are the maximal elementary sub-Hopf algebras of A.

(e) Fix m ≥ 0 and r > m. Let

E(m)′ = A(\underbrace{0, 0, …, 0}_{m}, \underbrace{m + 1, …, m + 1}_{r−m−1}, r, r + 1, r + 2, …).

Finally, we have one remark.

(f) By convention, we let n_{0} = 0. This is important in order to have the full generality for Lemmas 2.2, 2.3 and 2.4.

Definition 2.1. Denote by h_{j,i} the element [ξ_{j}^{2^{i}}] in the cobar construction for A (and similarly for any sub-Hopf algebra B of A with profile function satisfying n_{j} > i). If this element is a permanent cycle, then let h_{j,i} also denote the corresponding element in Ext_{A}^{∗∗}(F_{2}, F_{2}) (or Ext_{B}^{∗∗}(F_{2}, F_{2})).

For example, if E = A(n_{1}, n_{2}, …) is an elementary sub-Hopf algebra of A (so that there is some t so that n_{i} = 0 for all i ≤ t, and n_{i} ≤ t + 1 for all i > t), then

Ext_{E}^{∗∗}(F_{2}, F_{2}) ≅ F_{2}[h_{j,i} : j ≥ 1, n_{j} > i ≥ 0].

Note that h_{j,i} is in bidegree (1, 2^{i}(2^{j} − 1)).

Lemma 2.2. Let B = A(n_{1}, n_{2}, …) be a sub-Hopf algebra of the Steenrod algebra with n_{1} ≤ n_{2} ≤ · · ·. Let j ≥ 1 be an integer for which n_{j−1} < n_{j}; then h_{j,q} is a nonzero element in Ext_{B}^{∗∗}(F_{2}, F_{2}), for all q with n_{j−1} ≤ q < n_{j} if n_{j} is finite, and for all q with n_{j−1} ≤ q if n_{j} = ∞.

Proof. This is a simple computation with the cobar construction.

Lemma 2.3. If B and j are as in Lemma 2.2 with n_{j} < ∞, then there is a Hopf algebra extension

B′ → B → E[P_{j}^{n_{j}−1}],

where B′ has the same profile sequence as B except that its jth coordinate is n_{j} −1, rather than n_{j}.

Proof. This is also straightforward; one only needs to know that B′ is a normal sub-Hopf algebra of B.

The following is our key lemma, analogous to [21, 5.2] and [2, Vol. II, 5.2.1].

Lemma 2.4. Let B = A(n_{1}, n_{2}, …) be a sub-Hopf algebra of the Steenrod algebra with n_{1} ≤ n_{2} ≤ ⋯, and assume that there is an integer j ≥ 1 so that n_{j−1} < n_{j} < ∞. Let B′ = A(m_{1}, m_{2}, …) be the sub-Hopf algebra defined by setting m_{i} = n_{i} if i ≠ j, and by setting m_{j} = n_{j} − 1.

(a) Let Γ be a coassociative B-coalgebra, with structure maps Δ : Γ → Γ⊗Γ and ε : Γ → F_{2}. Assume that z ∈ Ext_{B}^{∗∗}(Γ, F_{2}) is an element of positive homological degree which restricts to zero in Ext_{B′}^{∗∗}(Γ, F_{2}); then z = ε^{∗}(h_{j,n_{j}−1})z_{0} for some z_{0} ∈ Ext_{B}^{∗∗}(Γ, F_{2}).

(b) Let M be a B-module. Assume that z ∈ Ext_{B}^{∗∗}(M, M) is an element of positive homological degree which restricts to zero in Ext_{B′}^{∗∗}(M, M); then z = (h_{j,n_{j}−1} · 1_{M})z_{0} for some z_{0} ∈ Ext_{B}^{∗∗}(M, M).

Proof. We prove this just as in [2]. We prove part (b); (a) is done similarly. Set x = P_{j}^{n_{j}−1}. We have a Hopf algebra extension

B′ → B → E[x]. (1)

Consider the following short exact sequence of E[x]-modules (and hence of Bmodules):

0 → Σ^{|x|}F_{2} → E[x] → F_{2} → 0.

Tensoring with M gives a short exact sequence

0 → Σ^{|x|}M → E[x] ⊗ M ⟶^{η′} M → 0. (2)

Corresponding to these we have h_{j,n_{j}−1} ∈ Ext_{B}^{1,|x|}(F_{2}, F_{2}) and h_{j,n_{j}−1} · 1_{M} ∈ Ext_{B}^{1,|x|}(M, M).

Note that the composite

Ext_{B}^{∗∗}(M, M) ⟶^{(η′)^{∗}} Ext_{B}^{∗∗}(E[x] ⊗ M, M) ≅ Ext_{B′}^{∗∗}(M, M)

is just the restriction map—see Section 2.8, Volume I, of [2], for instance. So since the restriction of z is zero, then we have (η′)^{∗}(z) = 0.

From the short exact sequence (2) we get a long exact sequence in Ext:

⋯ → Ext_{B}^{n−1}(M, M) ⟶^{δ} Ext_{B}^{n}(M, M) ⟶^{(η′)^{∗}} Ext_{B}^{n}(E[x] ⊗ M, M) → · · · .

Here δ is Yoneda composition with h_{j,n_{j}−1} · 1_{M}. So since (η′)^{∗}(z) = 0, we see that z = z_{0} ∘ (h_{j,n_{j}−1} · 1_{M}) for some z_{0} ∈ Ext_{B}^{∗∗}(M, M). The element h_{j,n_{j}−1} · 1_{M} is central in Ext_{B}^{∗∗}(M, M), so this finishes the proof.

Remark 2.5. Lemma 2.4 holds if we work with chain complexes of modules instead of modules. For example, if C is a finite chain complex of finite A-modules and if z ∈ Ext_{B}^{∗∗}(C, C) restricts to zero in Ext_{B′}^{∗∗}(C, C), then z = (h_{j,n_{j}−1} · 1_{C})z_{0} for some z_{0} ∈ Ext_{B}^{∗∗}(C, C). See also Remarks 2.10 and 3.4.

We want to generalize Wilkerson’s work in [21] on detecting nilpotence; we refer the reader to that paper for more information. In particular, we use Wilkerson’s notation for Steenrod operations acting on Hopf algebra cohomology. Let B be a finite dimensional graded connected cocommutative Hopf algebra over a finite field k; let p be the characteristic of k. We make the following definition; we give the version for p = 2, since Wilkerson gives the odd prime version.

Definition 2.6 ([21], Definition 5.3). The fundamental class of B is u_{B} = ∏ v, where the product ranges over all nonzero v ∈ ker(S̃q^{0}) ∩ Ext_{B}^{1}(k, k).

Note that Lemma 2.4 can be stated more generally for finite dimensional Hopf algebras over k, as in [2] or [21]; one replaces the Steenrod algebra conditions by conditions that ensure the existence of a Hopf algebra extension as in (1). Namely, given a nonzero v ∈ ker(S̃q^{0})∩Ext_{B}^{1}(k, k), there is a normal Hopf algebra extension (1), where the polynomial generator of Ext_{E[x]}^{∗}(k, k) hits v under the map in Ext induced by B → E[x]. There is also an odd prime version. See [21, Section 5] for more details.

This is a generalization of [21, Corollary 5.6].

Corollary 2.7. Assume that every sub-Hopf algebra C of B is either elementary or has u_{C} nilpotent.

(a) Let Γ be a coassociative B-coalgebra; fix z ∈ Ext_{B}^{s,t}(Γ, k). Then z is nilpotent if and only if ι_{E}^{∗}(z) is nilpotent for all elementary sub-Hopf algebras E of B.

(b) Let M be a B-module; fix z ∈ Ext_{B}^{s,t}(M, M). Then z is nilpotent under Yoneda composition if and only if ι_{E}^{∗}(z) is nilpotent for all elementary sub-Hopf algebras E of B.

Proof. Use Wilkerson’s argument for the proof of [21, Corollary 5.6], replacing his Lemma 5.2 with (the appropriate restatement of) Lemma 2.4.

In the next section we will need the next lemma and its corollary. One can prove a better result, a “relative vanishing line” theorem, but we don’t need anything so precise. By the way, these are not new results; we do not know to whom they should be attributed, though.

Lemma 2.8. Let ι_{C} : C ↪ B be an inclusion of graded connected cocommutative Hopf algebras over a field k. Let n be the connectivity of the cokernel of ι_{C} (viewed as a map of k-vector spaces). Let L and M be B-modules with L bounded below and M bounded above, and let c be the connectivity of Hom_{k}^{∗}(L, M). Then the map

ι_{C}^{∗} : Ext_{B}^{s,t}(L, M) → Ext_{C}^{s,t}(L, M)

is an isomorphism for all (s, t) with t − s < c + n.

Proof. Assume that L is zero below dimension ℓ and M is zero above dimension m, so that Hom_{k}^{t}(L, M) is zero for t < ℓ − m. Since B is connected, then there is a projective resolution

⋯ → P_{2} → P_{1} → P_{0} → L → 0

for L over B so that the bottom dimension of P_{s} is at least s + ℓ, for each s ≥ 0. B is free over C by a result in [14], so we can view P_{∗} → L as a projective resolution for L over C, as well. Under the conditions on C ↪ B, we see that the map

Hom_{B}^{t}(P_{s}, M) → Hom_{C}^{t}(P_{s}, M)

is an isomorphism for all t with t < s + n + ℓ − m. This proves the result.

Corollary 2.9. Let C ⊂ B be sub-Hopf algebras of A with non-decreasing profile functions (m_{1}, m_{2}, …) and (n_{1}, n_{2}, …), respectively. Suppose that for some integer r ≥ 1, we have m_{i} = n_{i} for all i < r, and m_{r} < n_{r}; let L and M be B-modules with L bounded below and M bounded above. Given a bidegree (s, t), if either r is sufficiently large or m_{r} is sufficiently large, then the restriction map

Ext_{B}^{s,t}(L, M) → Ext_{C}^{s,t}(L, M)

is an isomorphism. (Here, “sufficiently large” depends only on (s, t) and the connectivity of Hom_{F_{2}}^{∗}(L, M).)

Remark 2.10. As with Lemma 2.4, both Lemma 2.8 and Corollary 2.9 hold for chain complexes of modules, in place of modules. More precisely, if D and F are finite chain complexes of finite B-modules, then there is a number c depending on D and F, so that if C ↪ B is an inclusion of Hopf algebras with cokernel that is “(n − 1)-connected,” then the restriction map

Ext_{B}^{s,t}(D, F) → Ext_{C}^{s,t}(D, F)

is an isomorphism for all (s, t) with t−s < c+n. To see this, note that the statement, “There is a number c … ” is generic with respect to D and F, and since the statement holds for all finite A-modules, it must hold for all finite chain complexes of finite A-modules. (See [17] for a discussion of chain complexes, genericity, and related topics. See also Remarks 2.5 and 3.4.)

Finally, we will need the following results of Lin (see also [21, Theorem 6.4]).

Theorem 2.11 (Theorem 3.1 in Part I of [9]). Let B = A(n_{1}, n_{2}, …) be a sub-Hopf algebra of A so that for some k, n_{i} is finite for all i < k. Suppose h_{j,λ} and h_{k,μ} are elements in Ext_{B}^{∗∗}(F_{2}, F_{2}), with j ≤ k and j ≤ μ. Then the class h_{j,λ}h_{k,μ} is nilpotent in Ext_{B}^{∗∗}(F_{2}, F_{2}).

Corollary 2.12 (Corollary 3.2 in Part I of [9]). Let B = A(n_{1}, n_{2}, …) be a sub-Hopf algebra of A so that for some k, n_{i} is finite for all i < k. If h_{k,μ} is an element in Ext_{B}^{∗∗}(F_{2}, F_{2}) with k ≤ μ, then h_{k,μ} is nilpotent.

3. Proof of Theorem 1.1

We prove Theorem 1.1 as well as the following result. For a Hopf algebra B and B-modules L and M, given an element z ∈ Ext_{B}^{∗∗}(L, M), we say that z is tensor-nilpotent if z^{⊗n} = 0 in Ext_{B}^{∗∗}(L^{⊗n}, M^{⊗n}), for all n sufficiently large.

Theorem 3.1. Let B be a sub-Hopf algebra of the mod 2 Steenrod algebra A; let D be the sub-Hopf algebra of A with profile function (1, 2, 3, …).

(a) Let Γ be a bounded below coassociative B-coalgebra; fix z ∈ Ext_{B}^{∗∗}(Γ, F_{2}). Then z is nilpotent if and only if ι_{D∩B}^{∗}(z) is nilpotent.

(b) Let M be a finite dimensional B-module; fix z ∈ Ext_{B}^{∗∗}(M, M). Then z is nilpotent if and only if ι_{D∩B}^{∗}(z) is nilpotent.

(c) Let L and M be B-modules with L bounded below and M finite dimensional; fix z ∈ Ext_{B}^{∗∗}(L, M). Then z is tensor-nilpotent if and only if ι_{D∩B}^{∗}(z) is tensor-nilpotent.

We view Theorem 1.1 as being analogous to the Morava K-theory version of the nilpotence theorem, [8, Theorem 3]; so Theorem 3.1 corresponds to the BP -version or the MU-version, [4, Theorem 1]. (See also [5] or [20].) By the way, Lin [9] conjectured part (b) of Theorem 3.1 for M = F_{2}. We will prove Theorem 3.1 as one of the steps in proving Theorem 1.1.

Proofs of Theorems 1.1 and 3.1. Suppose z is in the appropriate Ext group (either Ext_{B}^{∗∗}(Γ, F_{2}), Ext_{B}^{∗∗}(M, M), or Ext_{B}^{∗∗}(L, M)). It is clear that if z is nilpotent in the appropriate sense, then so is ι_{E}^{∗}(z) for any sub-Hopf algebra E of B. (For the first two cases, for instance, ι_{E}^{∗} is an algebra map.)

So assume that ι_{E}^{∗}(z) is nilpotent for every elementary sub-Hopf algebra E of B. In particular, the restriction of z to every maximal elementary sub-Hopf algebra is nilpotent. By Corollary 2.9, the restriction of z is zero for all but finitely many of these; so by raising z to a large enough power, we may assume that ι_{E}^{∗}(z) = 0 for all E.

If z is in homological degree zero, then the restriction maps are inclusions; hence if some restriction of z is nilpotent, then so is z. Most of the remaining work is in proving Theorem 1.1(a) for z of positive homological degree. We do this for the full Steenrod algebra A; the case for a sub-Hopf algebra follows similarly. There are two steps in the proof of (a): first we prove that if z ∈ Ext_{A}^{s,t}(Γ, F_{2}) is a class with s > 0 which restricts to zero over all elementary sub-Hopf algebras of A, then z is nilpotent when restricted to D. Then we prove Theorem 3.1(a), and this finishes the proof. (The first step is a generalization of one of Lin’s main theorems in [9].) Following that, we prove parts (b) and (c) of both theorems.

3.1. z is nilpotent over D. Let Γ be a bounded below coassociative A-coalgebra; let z ∈ Ext_{A}^{s,t}(Γ, F_{2}) be an element with s > 0. Let D_{R} = A(\underbrace{0, 0, …, 0}_{R−1}, R, R+1, R+

2, …). Now we apply Corollary 2.9 to F_{2} ↪ D_{R}—since Γ is bounded below, then for R large enough, we can make sure that Ext_{D_{R}}^{s,t}(Γ, F_{2}) → Ext_{F_{2}}^{s,t}(Γ, F_{2}) is an isomorphism. But since s > 0, then Ext_{F_{2}}^{s,t}(Γ, F_{2}) = 0; hence for R large enough, z restricts to zero over D_{R}.

We assume that z restricts to zero over every elementary sub-Hopf algebra; in particular, z restricts to zero over E(0), E(1), E(2), …. Using Corollary 2.9 again, we see that if r_{m} is a large enough integer for each m, then z restricts to zero over

the sub-Hopf algebra A(\underbrace{0, 0, …, 0}_{m}, \underbrace{m + 1, …, m + 1}_{r_{m}−m−1}, r_{m}, r_{m} + 1, r_{m} + 2, …). In

fact, since the isomorphism in Corollary 2.9 depends only on the connectivity of the cokernel of the Hopf algebra inclusion, we may choose r_{m} to be the same for all m, say all equal to r_{0}.

Let r = max(R, r_{0}). Define sub-Hopf algebras

D_{r} = A(\underbrace{0, 0, …, 0}_{r−1}, r, r + 1, r + 2, …),

E(m)′ = A(\underbrace{0, 0, …, 0}_{m}, \underbrace{m + 1, …, m + 1}_{r−m−1}, r, r + 1, r + 2, …),

for m < r. Then z restricts to zero over D_{r} and all the (E(m)′)’s; we want to show that ι_{D}^{∗}(z) is nilpotent.

We use induction to build D out of D_{r} and the (E(m)′)’s. Here is the precise statement: we claim that if B = A(n_{1}, n_{2}, …, n_{r−1}, r, r + 1, …) is a sub-Hopf algebra of D with n_{1} ≤ n_{2} ≤ ⋯ ≤ n_{r−1}, then ι_{B}^{∗}(z) is nilpotent. We prove this by induction on n_{1} + n_{2} + ⋯ + n_{r−1}. Let N_{B} denote this sum. (Note that since B ⊆ D, then n_{i} ≤ i for each i < r.)

If N_{B} = 0, then we have B = D_{r} and so ι_{B}^{∗}(z) = 0. For the inductive step, we need the following observation.

Sublemma 3.2. Suppose B = A(n_{1}, n_{2}, …, n_{r−1}, r, r + 1, …) is a sub-Hopf algebra of D with n_{1} ≤ n_{2} ≤ ⋯ ≤ n_{r−1}, so that B not a sub-Hopf algebra of any E(m)′. Let j = min{i : n_{i} ≠ 0}; let k = min{i : n_{i} ≥ j + 1}. Then j ≤ k < r.

Proof. This is immediate.

Assume that ι_{C}^{∗}(z) is nilpotent for all C as above with N_{C} ≤ N −1. By replacing z with some power, we may assume that ι_{C}^{∗}(z) = 0 for all such C. Let B denote a sub-Hopf algebra of D as above, with N_{B} = N. If B is a sub-Hopf algebra of E(m)′ for some m, then ι_{B}^{∗}(z) = 0 by assumption. Otherwise we have j and k as in the sublemma; by Lemma 2.2, h_{j,n_{j}−1} and h_{k,n_{k}−1} are nonzero elements of Ext_{B}^{1}(F_{2}, F_{2}), and by Theorem 2.11, the product h_{j,n_{j}−1}h_{k,n_{k}−1} is nilpotent. So apply induction and Lemma 2.4 twice to get z = h_{j,n_{j}−1}z′ = h_{k,n_{k}−1}z′′.

This completes the inductive step. When N_{B} = \frac{r(r−1)}{2}, then B = D.

3.2. z is nilpotent over A. Let z ∈ Ext_{A}^{s,t}(Γ, F_{2}) be an element with s > 0, and assume that ι_{E}^{∗}(z) = 0 for all elementary sub-Hopf algebras E of A. We know from the previous subsection that ι_{D}^{∗}(z) is nilpotent; we may assume that ι_{D}^{∗}(z) = 0, by replacing z with an appropriate power. We want to show that z is nilpotent. (In other words, we are proving Theorem 3.1(a) here.)

For each integer q ≥ 1, let B_{q} = A(1, 2, …, q − 1, q, ∞, ∞, …), and let B_{0} = A. We will show by downward induction on q that z is nilpotent when restricted to B_{q}, for each q. By Corollary 2.9, we can find an integer q large enough so that z restricts to zero over B_{q}; this starts the induction. Now we use a variant on one of the arguments used in [6]. For each q we have a normal extension of Hopf algebras

B_{q} ⟶ B_{q−1} ⟶ B_{q−1}//B_{q}, (3)

where (as algebras) we have B_{q−1}//B_{q} ≅ E[P_{q}^{u} : u ≥ q]. For k ≥ q, let G_{k} be the B_{q−1}//B_{q}-module defined by G_{k} = E[P_{q}^{u} : k ≥ u ≥ q], and let G_{q−1} = F_{2}; of course, G_{k} is also a B_{q−1}-module for each k ≥ q − 1. For any B_{q−1}-module M, Ext_{B_{q−1}}^{∗∗}(Γ ⊗ M, F_{2}) is a module over Ext_{B_{q−1}}^{∗∗}(Γ, F_{2}), via the composite

Ext_{B_{q−1}}^{∗∗}(Γ, F_{2}) ⊗ Ext_{B_{q−1}}^{∗∗}(Γ ⊗ M, F_{2}) ⟶ Ext_{B_{q−1}}^{∗∗}(Γ ⊗ Γ ⊗ M, F_{2})
↓^{(Δ⊗1_{M})^{∗}}
Ext_{B_{q−1}}^{∗∗}(Γ ⊗ M, F_{2}).

Let z̃ denote the restriction of z to Ext_{B_{q−1}}^{∗∗}(Γ, F_{2}). We have the following lemma.

Lemma 3.3. For k sufficiently large, we have z̃^{−1} Ext_{B_{q−1}}^{∗∗}(Γ ⊗ G_{k}, F_{2}) = 0.

Proof. We have the following Cartan-Eilenberg spectral sequence associated to the extension (3):

E_{2}^{p,q} ≅ Ext_{B_{q−1}//B_{q}}^{p}(G_{k}, Ext_{B_{q}}^{q}(Γ, F_{2})) ⇒ Ext_{B_{q−1}}^{p+q}(G_{k} ⊗ Γ, F_{2}).

Precisely as in the proof of [6, Lemma 2.2], we use a vanishing plane argument to show that some power of z̃ annihilates every element of the abutment.

Now we finish the proof of the theorem. For each k ≥ q, there is a short exact sequence

0 → Σ^{2^{k}(2^{q}−1)}G_{k−1} → G_{k} → G_{k−1} → 0. (4)

We tensor with Γ and apply Ext_{B_{q−1}}^{∗∗}(−, F_{2}); the result is a long exact sequence of Ext_{B_{q−1}}^{∗∗}(Γ, F_{2})-modules, in which the boundary homomorphism is composition with h_{q,k} ⊗ 1_{Γ⊗G_{k−1}}. Inverting z̃ is exact, so if we choose k sufficiently large, as in Lemma 3.3, then z̃^{−1} Ext_{B_{q−1}}^{∗∗}(Γ ⊗ G_{k}, F_{2}) = 0; and so we have an isomorphism

h_{q,k} ⊗ 1_{Γ⊗G_{k−1}} : z̃^{−1} Ext_{B_{q−1}}^{∗∗}(Γ ⊗ G_{k−1}, F_{2}) → z̃^{−1} Ext_{B_{q−1}}^{∗∗}(Γ ⊗ G_{k−1}, F_{2}).

Now we use Corollary 2.12: since k ≥ q, then h_{q,k} is nilpotent; hence

z̃^{−1} Ext_{B_{q−1}}^{∗∗}(Γ ⊗ G_{k−1}, F_{2}) = 0.

By downward induction on k, we see that z̃^{−1} Ext_{B_{q−1}}^{∗∗}(Γ, F_{2}) = 0, which means that the restriction of z to B_{q−1} is nilpotent. Replacing z with an appropriate power allows us to assume that z restricts to zero over B_{q−1}. This completes the inductive step.

3.3. Proofs of Theorems 1.1(b–c) and 3.1(b–c). These proofs are essentially the same as the corresponding arguments for spectra; both parts follow from (a) easily. Given a B-module M, let M^{∗} denote the dual of M, with B-module structure given as usual.

Proof of Theorem 1.1(b). If M is a finite dimensional B-module, then M ⊗ M^{∗} is a coassociative B-coalgebra—the coproduct map

M ⊗ M^{∗} → (M ⊗ M^{∗}) ⊗ (M ⊗ M^{∗})

is adjoint to the identity on M ⊗ M^{∗} ⊗ M, and the counit map M ⊗ M^{∗} → F_{2} is adjoint to the identity on M^{∗}. Also, as algebras we have

Ext_{B}^{∗∗}(M, M) ≅ Ext_{B}^{∗∗}(M ⊗ M^{∗}, F_{2}).

So apply part (a).

Proof of Theorem 1.1(c). We have

Ext_{B}^{∗∗}(L, M) ≅ Ext_{B}^{∗∗}(L ⊗ M^{∗}, F_{2}),
z ↔ ẑ.

Now, ι_{E}^{∗}(z) = 0 for all E if and only if ι_{E}^{∗}(ẑ) = 0 for all E, and z^{⊗n} = 0 if and only if ẑ^{⊗n} = 0. So it suffices to prove the corollary for an element z of Ext_{B}^{∗∗}(L ⊗ M^{∗}, F_{2}); i.e., we may assume that M = F_{2}. Furthermore, by suspending L we may assume that L is zero below dimension 0 (z ∈ Ext_{B}^{s,t}(L, F_{2}) is tensor-nilpotent if and only if z ∈ Ext_{B}^{s,t+n}(Σ^{n}L, F_{2}) is).

So assume that z ∈ Ext_{B}^{∗∗}(L, F_{2}). Let Γ be the tensor coalgebra on L: let

Γ = ⨁_{n≥0} L^{⊗n}

(where L^{⊗0} = F_{2}), with the evident coproduct. Then Γ is a coassociative Bcoalgebra, and since L is zero in negative dimensions, Γ is bounded below. Also, tensor powers of z correspond to powers of the corresponding element \overline{z} in the algebra Ext_{B}^{∗∗}(Γ, F_{2}), and ι_{E}^{∗}(z) = 0 if and only if ι_{E}^{∗}(\overline{z}) = 0. So apply part (a).

Proof of Theorem 3.1(b–c). Imitate the proof of Theorem 1.1(b–c).

This finishes the proofs of Theorems 1.1 and 3.1.

Remark 3.4. This proof relied on certain facts about Ext_{B}^{∗∗}(F_{2}, F_{2}), for B a sub-Hopf algebra of A—2.11 and 2.12—as well as some basic homological algebra applied to Ext_{B}^{∗∗}(L, M) for B-modules L and M—2.4 and 2.9. Hence the whole proof carries through for chain complexes of A-modules, in place of A-modules—see Remarks 2.5 and 2.10. For example, if C is a finite chain complex of finite A-modules and z is an element of Ext_{A}^{∗∗}(C, C), then z is nilpotent if and only if ι_{E}^{∗}(z) is nilpotent for every elementary sub-Hopf algebra E of A. We will need the nilpotence theorem in this generality in the sequel [17].

4. The odd prime case

In this section we discuss our results for the algebra of reduced powers. Let p be an odd prime, let A denote the mod p Steenrod algebra, and let P = A/AβA. We prove the odd prime analog of Theorem 3.1 for sub-Hopf algebras of P, and we prove a weaker version of Theorem 1.1.

P is a Hopf algebra, and its dual is isomorphic to F_{p}[ξ_{1}, ξ_{2}, …]. One has the same description of sub-Hopf algebras of P as for the mod 2 Steenrod algebra; we write P (n_{1}, n_{2}, …) for the sub-Hopf algebra with profile as indicated. Many of the results from Section 2 hold without change. There are a few exceptions: if E = P (n_{1}, n_{2}, …) is an elementary sub-Hopf algebra of P, then

Ext_{E}^{∗∗}(F_{p}, F_{p}) ≅ E[h_{j,i} : j ≥ 1, n_{j} > i ≥ 0] ⊗ F_{p}[β(h_{j,i}) : j ≥ 1, n_{j} > i ≥ 0],

where β denotes the Bockstein acting on Hopf algebra cohomology. So we have to modify Lemma 2.4 to reflect this; the conclusion of the lemma will be that z^{2} = ε^{∗}(β(h_{j,n_{j}−1}))z_{0} for some z_{0} (see [2] for the changes necessary in the proof).

Also, we would like to have analogs of Theorem 2.11 and Corollary 2.12; we have one for the latter, but not the former. Wilkerson [21, 6.3] gives an example in which the odd primary version of Theorem 2.11 fails. We have the following, though.

Proposition 4.1 (Proposition 4.1 in [12]). Let B = P (n_{1}, n_{2}, …) be a sub-Hopf algebra of P so that for some k, n_{i} is finite for all i < k. If h_{k,μ} is an element in Ext_{B}^{∗∗}(F_{p}, F_{p}) with k ≤ μ, then β(h_{k,μ}) is nilpotent.

Here is our first result for odd primes.

Theorem 4.2. Theorem 3.1 holds for sub-Hopf algebras of P.

Proof. The proof is almost identical to that given above. Let D[x] denote the algebra F_{p}[x]/(x^{p}); then with B_{q} and B_{q−1} as in Section 3, we have B_{q−1}//B_{q} ≅ D[P_{q}^{u} : u ≥ q]. We set G_{k} = D[P_{q}^{u} : k ≥ u ≥ q], so that G_{k−1} ≅ G_{k}/(P_{q}^{k}); we also set G̃_{k−1} ≅ G_{k}/((P_{q}^{k})^{p−1}). Now, instead of the short exact sequence (4), we have two short exact sequences:

0 → Σ^{|P_{q}^{k}|}G̃_{k−1} → G_{k} → G_{k−1} → 0
0 → Σ^{(p−1)|P_{q}^{k}|}G_{k−1} → G_{k} → G̃_{k−1} → 0.

We get two long exact sequences in Ext_{B_{q−1}//B_{q}}^{∗∗}(Γ ⊗ −, F_{p}), and both boundary homomorphisms are isomorphisms once z̃ is inverted. Hence so is the composite of the boundary homomorphisms, and this is just β(h_{k,q}) ⊗ 1_{Γ⊗G_{k−1}}. Now, as above, β(h_{k,q}) is nilpotent. This finishes the proof.

In light of the failure of Theorem 2.11, we cannot expect the precise analog of Theorem 1.1 to hold; indeed, the same counterexample shows that the analog does not hold. Here is an alternative. Say that a sub-Hopf algebra E of P is quasi-elementary if for some r ≥ 1 it is a sub-Hopf algebra of

Q(r) = P (\underbrace{0, 0, …, 0}_{r−1}, 1, r + 1, r + 1, r + 1, …).

Theorem 4.3. Suppose B is a sub-Hopf algebra of P and Γ is a bounded below coassociative B-coalgebra. One can detect nilpotence in Ext_{B}^{∗∗}(Γ, F_{p}) by restricting to quasi-elementary sub-Hopf algebras.

Proof. The only step we need is the following version of Theorem 2.11 (see also Sublemma 3.2). This is essentially [21, Theorem 6.4], modified to work at odd primes. As one might guess, for suitable r ≫ 0 we set

D = P (1, 2, 3, …),
Q(m)′ = P (\underbrace{0, 0, …, 0}_{m−1}, 1, \underbrace{m + 1, …, m + 1}_{r−m−1}, r, r + 1, r + 2, …).

Lemma 4.4. Suppose B = P (n_{1}, n_{2}, …, n_{r−1}, r, r + 1, …) is a sub-Hopf algebra of D with n_{1} ≤ n_{2} ≤ ⋯ ≤ n_{r−1}, so that B is not a sub-Hopf algebra of any Q(m)′. Let j = min{i : n_{i} ≠ 0}. If n_{j} = 1, then let k = min{i : n_{i} ≥ j + 2}; otherwise, let k = min{i : n_{i} ≥ j + 1}. Then i ≤ k < r, and furthermore, the class

β(h_{j,n_{j}−1})β(h_{k,n_{k}−1}) ∈ Ext_{B}^{∗∗}(F_{p}, F_{p})

is nilpotent.

Proof. Such a Hopf algebra B satisfies the conditions of [21, Lemma 6.2] with r = j and n = j + k. Hence we have a non-trivial relation h_{k,j}h_{j,0} = 0 in Ext_{B}^{∗∗}(F_{p}, F_{p}); now apply Steenrod operations. See [21, Theorem 6.4] for details. (The key here is that either h_{j,1} or h_{k,j+1} is nonzero in Ext_{B}^{∗∗}(F_{p}, F_{p}).)

This finishes the proof of Theorem 4.3.

Corollary 4.5. If L and M are B-modules with L bounded below and M finite, then one can detect nilpotence in Ext_{B}^{∗∗}(M, M) and tensor-nilpotence in Ext_{B}^{∗∗}(L, M) by restricting to quasi-elementary sub-Hopf algebras.

Remark 4.6. (a) Theorem 4.3 and Corollary 4.5 hold more generally for chain

complexes of B-modules—see Remark 3.4.

(b) The quasi-elementary sub-Hopf algebras Q of P are defined so that no product ∏ v ∏ βP̃^{0}u is zero, where v ranges over any finite subset of nonzero elements in Ext_{Q}^{1,odd}(F_{p}, F_{p}), and u ranges over any finite subset of nonzero elements in ker P̃^{0} ∩ Ext_{Q}^{1,even}(F_{p}, F_{p})—see [21, Definition 5.3]. We expect that if we used this Ext condition as a definition for quasi-elementary, then the analog of Theorem 4.3 for the full mod p Steenrod algebra A would hold. It would be nice to have a characterization of these sub-Hopf algebras in terms of profile functions.

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