QUILLEN STRATIFICATION FOR THE STEENROD ALGEBRA

JOHN H. PALMIERI

Abstract. Let A be the mod 2 Steenrod algebra, and consider the map ρ: H^{∗}(A; F_{2}) ⟶ \underleftarrow{lim} H^{∗}(E; F_{2}), where the inverse limit is taken over the exterior sub-Hopf algebras of A, ordered by inclusion. There is an action of A on this inverse limit, and ρ factors through the invariants under this action, giving a map

γ: H^{∗}(A; F_{2}) ⟶ (\underleftarrow{lim} H^{∗}(E; F_{2}))^{A} .

We show that γ is an F -isomorphism.

1. Results

In this paper we give a version of the Quillen stratification theorem [Qui71, 6.2] for the cohomology of the Steenrod algebra. This cohomology ring is the E_{2}- term of the Adams spectral sequence converging to the stable homotopy groups of spheres, so any sort of structural information is potentially applicable to stable homotopy theory. Since our result is a version of a powerful group cohomology result, it provides evidence that other structural results about group cohomology (e.g., the work of Benson, Carlson, et al., on varieties for kG-modules) may carry over to results about modules over the Steenrod algebra, and maybe to results about stable homotopy theory. Also, since A is infinite-dimensional, one might hope that the same techniques would give analogous results for the cohomology rings of other infinite Hopf algebras.

Recall that the mod 2 Steenrod algebra A is a graded connected cocommutative Hopf algebra; Milnor showed in [Mil58] that its dual A_{∗} is isomorphic, as an algebra, to

A_{∗} ≅ F_{2}[ξ_{1}, ξ_{2}, ξ_{3}, …], (∗)

which is graded by setting |ξ_{n}| = 2^{n} − 1. The product on A is dual to the diagonal map on A_{∗}, which is given by

Δ: ξ_{n} ⟼ ∑_{i=0}^{n} ξ_{n−i}^{2^{i}} ⊗ ξ_{i}.

(We take ξ_{0} to be 1.)

Date: April 29, 1997.

1991 Mathematics Subject Classification. 55S10, 55T15, 55Q45, 18G15.

Key words and phrases. Steenrod algebra, Ext, stable homotopy, F-isomorphism, Quillen stratification.

Research partially supported by National Science Foundation grant DMS-9407459.

Notation 1.1. Given a commutative ring k, a graded augmented k-algebra Γ, and a graded left Γ-module M, we write H^{∗}(Γ; M) for Ext_{Γ}^{∗∗}(k, M). Note that this is a bigraded vector space. We write M^{Γ} for Hom_{Γ}^{∗}(k, M) ⊆ M; then M^{Γ} is the set of invariants in M under the Γ-action. It consists of all x in M which support no nontrivial operations by elements of Γ.

Note that when Γ is a (graded) cocommutative Hopf algebra over a field k, then H^{∗}(Γ; k) is a (graded) commutative k-algebra—one can use the diagonal map on Γ to induce the product on H^{∗}(Γ; k), so cocommutativity of Γ implies commutativity of H^{∗}(Γ; k). This is the case when k = F_{2} and Γ = A.

Part (a) of this next definition is due to Quillen [Qui71]; part (b) is due to Wilkerson [Wil81].

Definition 1.2. Fix a field k of characteristic p.

(a) Given a map f: R ⟶ S of graded commutative k-algebras, we say that f is an F -isomorphism if

(i) every x ∈ ker f is nilpotent, and (ii) every y ∈ coker f is nilpotent, in the sense that for all y ∈ S, there is an

n so that y^{p^{n}} is in the image of f.

(b) A Hopf algebra E over k is elementary if E is commutative and cocommutative, and x^{p} = 0 for all x in IE (where IE = ker(E ⟶ k) is the augmentation ideal of E).

Remark. (a) If one can choose the same n for every y in Definition 1.2(a)(ii), then the F -isomorphism is said to be uniform. We do not expect this to be the case in our results.

(b) Note that when char(k) = 2, then a cocommutative Hopf algebra over k is elementary if and only if it is exterior, as an algebra. The elementary sub-Hopf algebras of the Steenrod algebra A were classified by Lin [Lin, 1.1]; they have also been studied in [Wil81] and [Pal96]. One should think of them as the “elementary abelian subgroups” of A.

Given a Hopf algebra Γ over a field k, for every sub-Hopf algebra Λ of Γ, there is a restriction map H^{∗}(Γ; k) ⟶ H^{∗}(Λ; k). If we let Q denote the category of elementary sub-Hopf algebras of Γ, where the morphisms are inclusions, then we can assemble the restriction maps H^{∗}(Γ; k) ⟶ H^{∗}(E; k), for E in Q, into

ρ: H^{∗}(Γ; k) ⟶ \underleftarrow{lim}_{Q}H^{∗}(E; k).

In the case k = F_{2} and Γ = A, we show in Section 2 that there is an action of A on \underleftarrow{lim}_{Q}H^{∗}(E; F_{2}), making the inverse limit an algebra over A.

Here is our main result.

Theorem 1.3. The map ρ: H^{∗}(A; F_{2}) ⟶ \underleftarrow{lim}_{Q}H^{∗}(E; F_{2}) factors through

γ: H^{∗}(A; F_{2}) ⟶ (\underleftarrow{lim}_{Q}H^{∗}(E; F_{2}))^{A} ,

and γ is an F -isomorphism.

Partial results related to this theorem can be found in [Lin], [Wil81], [HP93], and [Pal96]. Lin proves a special case of Theorem 1.3 (actually a version of Theorem 4.1, from which we deduce Theorem 1.3) for certain sub-Hopf algebras of A; in his setting, though, some power of every element is invariant, so one does not have to apply (−)^{A}. The other papers all focus on the detection of nilpotence (finding a map f that satisfies condition (i) in Definition 1.2(a)), rather than on the identification of all of the non-nilpotent elements.

The statement of Theorem 1.3 is strongly influenced by group theory; it is based, of course, on a theorem of Quillen [Qui71, 6.2] which identifies the cohomology H^{∗}(G; k) of a compact group G, up to F -isomorphism, as the inverse limit \underleftarrow{lim} H^{∗}(E; k). This inverse limit is taken over the category whose objects are elementary abelian p-subgroups of G (where p = char(k)), and whose morphisms are generated by inclusions and conjugations by elements of G. Using conjugations is technically unpleasant in our setting, so we just use inclusions as our morphisms; this is why we have to consider the invariants under the A-action.

Since A is not “compact” in any sense—A is infinite-dimensional and not profinite, and H^{∗}(A; F_{2}) is not Noetherian—one should not expect the group theoretic proof to carry over. So for the proof, we turn to homotopy theory. From the point of view of axiomatic stable homotopy theory [HPS], in the appropriate setting, H^{∗}(A; F_{2}) plays the role of the stable homotopy groups of spheres. One has Adams spectral sequences for computing this; the Lyndon-Hochschild-Serre spectral sequence associated to a Hopf algebra extension (Proposition 2.3) is an example. We use a strong property of the Adams spectral sequence in the proof: the existence of a vanishing line with fixed slope at some term of the spectral sequence is “generic” (Theorem 3.1). While this property was originally established by Hopkins and Smith [HSa] in the setting of the stable homotopy category of spectra, the formalism of [HPS] allows one to prove it in other situations, such as the present one. In this paper, we have translated the stable homotopy language into algebraic terms; see [Pal] for an untranslated version.

We should point out that Theorem 1.3 will not generalize, as stated, to other Hopf algebras, because it is known that elementary Hopf algebras do not always detect nilpotence—i.e., the map ρ: H^{∗}(Γ; k) ⟶ \underleftarrow{lim}_{Q}H^{∗}(E; k) need not satisfy condition (i) of Definition 1.2(a). This is the case, for example, with the odd primary Steenrod algebra; see [Wil81, 6.5]. One has replacements for elementary Hopf algebras, the so-called “quasi-elementary” Hopf algebras, which do a better job of detecting nilpotence. For example, Theorem 1.3 holds with A replaced by any finite-dimensional graded connected cocommutative Hopf algebra Γ, if one takes the inverse limit over the category of quasi-elementary sub-Hopf algebras of Γ [Pal97, Theorem 1.4]. See [Wil81] and [Pal97] for a further discussion of these issues.

Here is our second result: we can compute the inverse limit in Theorem 1.3, and we have a formula for the A-action.

Theorem 1.4. There is an isomorphism of A-algebras

\underleftarrow{lim}_{Q}H^{∗}(E; F_{2}) ≅ F_{2}[h_{ts} | s < t] / (h_{ts}h_{vu} | u ≥ t),

where |h_{ts}| = (1, 2^{s}(2^{t} − 1)). The action of A is given by the following on the generators:

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

and then one extends multiplicatively.

[figure: h_{10}; h_{20}; h_{21}; h_{30}; h_{31}; h_{32}; h_{40}; h_{41}; h_{42}; h_{43}; ⋯; ⋯; ⋯; ⋯; 1; 2; 3; 4; 3; 4; 5; 5; 6; 7; 0; 0; 0; 1; 1; 2]

Figure 1-1. Graphical depiction of the action of A on R. An arrow labeled by k represents an action by Sq^{2^{k}} ∈ A, or equivalently a “coaction” by ξ_{1}^{2^{k}} ∈ A_{∗}—in other words, a term of the form ξ_{1}^{2^{k}} ⊗ (target) in the coaction on the source.

The polynomial generator h_{ts} is represented by [ξ_{t}^{2^{s}}] in the cobar construction (for any elementary E for which this makes sense). By “one extends multiplicatively,” we mean that one uses the Cartan formula to determine the action of Sq^{n} on products.

Alternatively, one can rewrite the action of A as a coaction of the dual A_{∗}—that is, as a map \underleftarrow{lim} H^{∗}(E; F_{2}) ⟶ A_{∗} ⊗ \underleftarrow{lim} H^{∗}(E; F_{2}). The coaction is multiplicative, and its value on the polynomial generators is given by the following formula:

h_{ts} ⟼ ∑_{j=0}^{⌊ \frac{s+t−1}{2} ⌋} ∑_{i=j+s+1}^{t−j} ζ_{j}^{2^{s}}ξ_{t−i−j}^{2^{i+j+s}} ⊗ h_{i,j+s}.

(As usual, ζ_{j} denotes χ(ξ_{j}), where χ: A_{∗} ⟶ A_{∗} is the conjugation map.) Using this explicit description of the (co)action, one can find elements in the ring of invariants, either by hand or by computer; we give a few examples below. For these examples, we let R denote the bigraded ring F_{2}[h_{ts} | s < t] / (h_{ts}h_{vu} | u ≥ t). We give a graphical depiction of the (co)action of A on R in Figure 1-1.

Example 1.5. (a) The element h_{t,t−1} in R^{1,2^{t−1}(2^{t}−1)} is an invariant, for t ≥ 1. Indeed, we know that h_{10} lifts to H^{1,1}(A; F_{2}); also h_{21}^{4} lifts to an element in H^{4,24}(A; F_{2}) (the element known as g or \overline{κ}—see [Zac67] for the identification of this element as a lift of a power of h_{21}). We do not know which power of h_{t,t−1} survives for t ≥ 3.

(b) We have some families of invariants. Note that h_{20} is not invariant—it supports a single non-trivial operation:

Sq^{2}(h_{20}) = h_{10}.

Then since h_{21} is invariant, the element h_{20}h_{21} supports at most one possibly nontrivial operation:

Sq^{2}(h_{20}h_{21}) = h_{10}h_{21}.

But since h_{10}h_{21} = 0 in R, then h_{20}h_{21} is invariant; indeed, h_{20}^{i}h_{21}^{j} is invariant for all i ≥ 0 and j ≥ 1. It turns out that more of these lift to H^{∗}(A; F_{2}) than one might expect from Theorem 1.3: the elements in the “Mahowald-Tangora wedge” [MT68] are lifts of the elements h_{20}^{i}h_{21}^{8j} for all i ≥ 0 and j ≥ 1. (See [MPT71]; earlier, Zachariou [Zac67] verified this for elements of the form h_{20}^{2i}h_{21}^{2(i+j)} for i, j ≥ 0.) These elements are distributed over a wedge between lines of slope \frac{1}{2} and \frac{1}{5} (in the usual Adams spectral sequence picture of H^{∗}(A; F_{2})).

(c) Similarly, while h_{30} and h_{31} are not invariant, the monomials

{h_{30}^{i}h_{31}^{j}h_{32}^{k} | i ≥ 0, j ≥ 0, k ≥ 1}

are invariant elements. Hence some powers of them lift to H^{∗}(A; F_{2}); we do not know which powers, though. Continuing in this pattern, we find that for n ≥ 1, we have sets of invariant elements

{h_{n0}^{i_{0}}h_{n1}^{i_{1}} … h_{n,n−1}^{i_{n−1}} | i_{0}, …, i_{n−2} ≥ 0, i_{n−1} ≥ 1}.

Again, we do not know which powers of these elements lift. In any case, we can see that the family of elements in the Mahowald-Tangora wedge is not a unique phenomenon—we have infinitely many such families, and all but the n = 1 and n = 2 families give more than a lattice of points in the bigraded vector space H^{∗}(A; F_{2}). One can easily show, by the way, that any invariant monomial in R is of the form h_{n0}^{i_{0}}h_{n1}^{i_{1}} … h_{n,n−1}^{i_{n−1}} for some n. The invariant polynomials may be much more complicated.

(d) Margolis, Priddy, and Tangora [MPT71, p. 46] have found some other nonnilpotent elements, such as x ∈ H^{10,63}(A; F_{2}) and B_{21} ∈ H^{10,69}(A; F_{2}). These both come from the invariant

z = h_{40}^{2}h_{21}^{3} + h_{20}^{2}h_{21}^{2}h_{41} + h_{30}^{2}h_{21}h_{31}^{2} + h_{20}^{2}h_{31}^{3}

in R. While we do not know which power of z lifts to H^{∗}(A; F_{2}), we do know that B_{21} maps to the product h_{20}^{3}h_{21}^{2}z, and x maps to h_{20}^{5}z. In other words, we have at least found some elements in the ideal

(z) ⊆ R^{A}

which lift to H^{∗}(A; F_{2}).

(e) Computer calculations have led us to a few other such “sporadic” invariant elements (i.e., invariant elements that do not belong to any family—any family of which we are aware, anyway):

h_{20}^{8}h_{31}^{4} + h_{30}^{8}h_{21}^{4} + h_{21}^{11}h_{31}

in R^{12,80}, an element in R^{13,104} (a sum of 12 monomials in the variables h_{i,0} and h_{i,1}, 2 ≤ i ≤ 5), and an element in R^{9,104} (a sum of 8 monomials in the same variables). We do not know what powers of these elements lift to H^{∗}(A; F_{2}), nor are we aware of any elements in the ideal that they generate which are in the image of the restriction map from H^{∗}(A; F_{2}).

Aside from these computational results, Theorem 1.3 is potentially interesting from a structural point of view. In particular, it has led us to a similar result for Ext with non-trivial coefficients (Theorem 1.7 below), as well as a proposed classification of the thick subcategories of finite A-modules (Conjecture 1.6).

Let D be the sub-Hopf algebra of A generated by the elementary sub-Hopf algebras of A; equivalently, D is generated by the P_{t}^{s}’s with s < t; equivalently, D is dual to the following quotient Hopf algebra of A_{∗}:

D_{∗} = A_{∗}/(ξ_{1}^{2}, ξ_{2}^{4}, ξ_{3}^{8}, …, ξ_{n}^{2^{n}}, …).

One can see that there is an action of A on H^{∗}(D; F_{2}). In Theorem 4.1, we show that the map

H^{∗}(A; F_{2}) ⟶ H^{∗}(D; F_{2})^{A}

is an F -isomorphism; this is an important intermediate result in proving Theorem 1.3.

Conjecture 1.6. The thick subcategories of finite A-modules are in one-to-one correspondence with the finitely generated radical ideals of H^{∗}(D; F_{2}) which are invariant under the action of A.

Given a finite A-module M, we let I(M) be the radical of the annihilator ideal of H^{∗}(D; M) in H^{∗}(D; F_{2}), under the Yoneda composition action. The maps in the proposed bijection should go like this: given an invariant radical ideal I ⊆ H^{∗}(D; F_{2}), then we let C be the full subcategory with objects

{M | M finite, I(M) ⊇ I}.

This is clearly thick. Given a thick subcategory C, then we let I be the ideal

⋂_{M∈ob C} I(M).

With a little work, one can show that this is an invariant ideal of H^{∗}(D; F_{2}) (and we clearly have I = \sqrt{I}).

(For this conjecture, one needs to interpret “thick subcategories of finite Amodules” properly—one usually discusses thick subcategories of triangulated categories, and the category of A-modules is, of course, abelian. The intended interpretation of Conjecture 1.6 is to work in a nice triangulated category in which the category of finite A-modules sits, and to classify the finite objects there; the best option for such a category is the category of cochain complexes of injective A_{∗}-comodules. See [Pal] for details.)

This is supported by the following result, which is proved elsewhere. Part (b) is an analogue of the result of Hopkins and Smith [HSb, Theorem 11], in which they identify the center of [X, X] up to F -isomorphism, for any finite spectrum X. We write i for the following map:

i: (Ext_{D}^{∗∗}(F_{2}, F_{2})/I(M))^{A} ⟶ Ext_{D}^{∗∗}(M, M).

Theorem 1.7. Let M be a finite-dimensional A-module.

(a) [Pal96] If x ∈ Ext_{A}^{∗∗}(M, M) is in the kernel of the restriction map

Ext_{A}^{∗∗}(M, M) ⟶ Ext_{D}^{∗∗}(M, M),

then x is nilpotent.

(b) [Pal] For every

y ∈ (Ext_{D}^{∗∗}(F_{2}, F_{2})/I(M))^{A} ,

there is an n so that i(y)^{2^{n}} is in the image of the restriction map

Ext_{A}^{∗∗}(M, M) ⟶ Ext_{D}^{∗∗}(M, M).

In fact, i(y)^{2^{n}} is hit by a central element of Ext_{A}^{∗∗}(M, M).

One might in fact conjecture that there is an F -isomorphism between the center

of Ext_{A}^{∗∗}(M, M) and (Ext_{D}^{∗∗}(F_{2}, F_{2})/I(M))^{A}.

Here is the structure of this paper. In Section 2, we discuss elementary and normal sub-Hopf algebras of the Steenrod algebra; we describe the action of A on \underleftarrow{lim} H^{∗}(E; F_{2}), and we prove Theorem 1.4. In Section 3, we prove two results about the spectral sequence associated to a Hopf algebra extension: first, the property of having a vanishing plane with fixed normal vector, with some intercept and at some E_{r}-term, is generic. Second, if we are working with Hopf algebras over the field F_{2}, given a class y ∈ E_{2}^{0,t}, then y^{2^{i}} survives to the E_{2^{i}+1}-term.

In Section 4, we introduce a sub-Hopf algebra D of A, and show that Theorem 1.3 follows if we can prove a similar result—Theorem 4.1—about the restriction map H^{∗}(A; F_{2}) ⟶ H^{∗}(D; F_{2}). In Section 5, we prove Theorem 4.1.

Remark. All of the main results of this paper—Theorems 1.3, 1.4, and 4.1—have obvious analogues if we replace A by any sub-Hopf algebra B of A, and Q by the category of elementary sub-Hopf algebras of B (so the objects will be B ∩ E, for E ∈ ob Q). The proofs that we give for the B = A case carry over easily to this slightly more general setting; we omit the details.

Convention. Throughout the paper, we have a number of results about the Steenrod algebra, its sub-Hopf algebras, and modules over it. We let A denote the mod 2 Steenrod algebra, and we use letters like B, C, D, and E to denote sub-Hopf algebras of it. We write B_{∗} (etc.) for the corresponding quotient Hopf algebras of the dual of A, A_{∗}. Now, we also have a number of results about general graded cocommutative Hopf algebras over a field k. To set such results apart, we will use capital Greek letters, like Γ and Λ, to denote such Hopf algebras; typically, Λ will denote a sub-Hopf algebra of Γ. Whether we are working over A or Γ, all modules are assumed to be graded.

2. Sub-Hopf algebras of A

In this section we discuss two kinds of sub-Hopf algebras of A: elementary ones and normal ones. We also construct the action of A on \underleftarrow{lim} H^{∗}(E; F_{2}), and we prove Theorem 1.4.

First we bring the reader up to speed on elementary sub-Hopf algebras of A. We gave a description of the dual of A in Section 1 (equation (∗)); we dualize with respect to the monomial basis of A_{∗}, and we let P_{t}^{s} be the basis element dual to ξ_{t}^{2^{s}}.

Proposition 2.1. (a) As an algebra, each elementary sub-Hopf algebra E of A is isomorphic to the exterior algebra on the P_{t}^{s}’s that it contains.

(b) Hence,

H^{∗}(E; F_{2}) ≅ F_{2}[h_{ts} | P_{t}^{s} ∈ E],

where h_{ts} is represented by [ξ_{t}^{2^{s}}] in the cobar complex for E; in particular, the degree of h_{ts} is |h_{ts}| = (1, 2^{s}(2^{t} − 1)).

(c) The maximal elementary sub-Hopf algebras of A are of the form

(F_{2}[ξ_{1}, ξ_{2}, …]/(ξ_{1}, …, ξ_{n−1}, ξ_{n}^{2^{n}}, ξ_{n+1}^{2^{n}}, ξ_{n+2}^{2^{n}}, …))^{∗} ,

for n ≥ 1.

Proof. This is an accumulation of well-known facts about sub-Hopf algebras of the Steenrod algebra. Part (b) follows immediately from part (a); parts (a) and (c) can be found in [Mar83, pp. 233–235].

We also need to discuss normal sub-Hopf algebras.

Definition 2.2. Suppose that Γ is a cocommutative Hopf algebra over a field k. A sub-Hopf algebra Λ of Γ is normal if the left ideal of Γ generated by Λ is equal to the right ideal of Γ generated by Λ (i.e., we have Γ·IΛ = IΛ·Γ, where IΛ = ker(Λ ⟶ k) is the augmentation ideal of Λ). If Λ is normal, then we have Γ ⊗_{Λ} k = k ⊗_{Λ} Γ, and this vector space has the structure of a Hopf algebra; we use Γ//Λ to denote it. In this situation, we say that Λ ⟶ Γ ⟶ Γ//Λ is a Hopf algebra extension.

First we discuss properties of normal sub-Hopf algebras of an arbitrary graded cocommutative Hopf algebra Γ, and then we specialize to the case of normal sub-Hopf algebras of the Steenrod algebra A.

Proposition 2.3. Let Γ be a graded cocommutative Hopf algebra over a field k. Fix a Γ-module M and a normal sub-Hopf algebra Λ of Γ.

(a) For each integer t, there is an action of Γ//Λ on H^{t,∗}(Λ; M). This action is natural in M, and if viewed as an action of Γ on H^{t,∗}(Λ; M), then it is natural in Λ as well. Γ//Λ acts on the right, so the action lowers internal degrees: given γ ∈ Γ//Λ of degree i and x ∈ H^{t,v}(Λ; M), then γx is in H^{t,v−i}(Λ; M).

(b) There is a spectral sequence, the Lyndon-Hochschild-Serre spectral sequence, with

E_{2}^{s,t,u}(M) = H^{s,u}(Γ//Λ; H^{t,∗}(Λ; M)) ⇒ H^{s+t,u}(Γ; M),

and with differentials

d_{r}: E_{r}^{s,t,u} ⟶ E_{r}^{s+r,t−r+1,u}.

It is natural in M.

Proof. This is a simple extension of the corresponding results for group extensions. (We also outline a construction of the spectral sequence in the proof of Theorem 3.1.)

By the way, we will often be applying Proposition 2.3(a) in the case where Γ = A is the mod 2 Steenrod algebra, yielding an action of A on H^{∗}(B; F_{2}) for B a normal sub-Hopf algebra of A. Note that one also has Steenrod operations acting on the cohomology of any cocommutative Hopf algebra over F_{2}—see May’s paper [May70], for example. These two actions are quite different; in this paper, we will almost always use the A-action arising as in Proposition 2.3(a) (the only exception is in the proof of Lemma 3.5).

Proposition 2.4. Let A be the mod 2 Steenrod algebra.

(a) A sub-Hopf algebra of A is normal if and only if its dual is of the form

F_{2}[ξ_{1}, ξ_{2}, …]/(ξ_{1}^{2^{n_{1}}} , ξ_{2}^{2^{n_{2}}} , ξ_{3}^{2^{n_{3}}} , …),

with n_{1} ≤ n_{2} ≤ n_{3} ≤ ⋯ ≤ ∞. (If n_{i} = ∞ in this expression, this means that one does not divide out by any power of ξ_{i}.).

(b) Hence the maximal elementary sub-Hopf algebras of A are normal.

Proof. This is well-known; see [Mar83, Theorem 15.6], for example.

Now we construct the action of A on \underleftarrow{lim} H^{∗}(E; F_{2}). We let Q_{0} denote the full subcategory of Q consisting of the normal elementary sub-Hopf algebras of A. By Proposition 2.4(b), we see that Q_{0} is final in Q; hence we have

\underleftarrow{lim}_{Q}H^{∗}(E; F_{2}) = \underleftarrow{lim}_{Q_{0}}H^{∗}(E; F_{2}).

On the right we have an inverse limit of modules over A; we give this the induced A-module structure. So we have (since taking invariants is an inverse limit)

(\underleftarrow{lim}_{Q}H^{∗}(E; F_{2}))^{A} = (\underleftarrow{lim}_{Q_{0}}H^{∗}(E; F_{2}))^{A}
= \underleftarrow{lim}_{Q_{0}} (H^{∗}(E; F_{2})^{A})
= \underleftarrow{lim}_{Q_{0}} (H^{∗}(E; F_{2})^{A//E}) .

Theorem 1.4 follows more or less immediately:

Proof of Theorem 1.4. We want to compute \underleftarrow{lim} H^{∗}(E; F_{2}), and we want to compute the action of A on this ring. Proposition 2.1(b) tells us H^{∗}(E; F_{2}) for each E; since the maps in the category Q are inclusion maps of one exterior algebra into another exterior algebra in which one simply adds more generators, the induced maps on cohomology are surjections of one polynomial ring onto another, each polynomial generator mapping either to zero or to a polynomial generator (of the same name). So assembling the cohomology rings of the maximal elementary sub-Hopf algebras of A gives our computation of the inverse limit.

To get our formula for the action of A on \underleftarrow{lim}_{Q}H^{∗}(E; F_{2}), we have to compute the A-action on H^{∗}(E; F_{2}) for E ≤ A maximal elementary. This is done in [Pal96, Lemmas A.3 and A.5]. (Briefly: for degree reasons, we have

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

for some α, β ∈ F_{2}. So one only has to check that α = β = 1; this is a reasonably straight-forward computation.)

3. Extensions of Hopf algebras

In this section we discuss normal sub-Hopf algebras and the Lyndon-Hochschild-Serre spectral sequence that arises from a Hopf algebra extension. In particular, we show in Theorem 3.1 that the presence of a vanishing plane at some term of such a spectral sequence is a “generic” property (Definition 3.2). We also prove (Lemma 3.5) that if y is in the E_{2}^{0,t,u}-term of this spectral sequence, then y^{2^{i}} survives to the E_{2^{i}+1}-term (if we’re working over the field F_{2}).

(Recall that we defined the notions of normal sub-Hopf algebra and Hopf algebra extension in Definition 2.2, and we presented the spectral sequence of a Hopf algebra extension in Proposition 2.3.)

Theorem 3.1 is our main theorem for this section. It is originally due to Hopkins and Smith [HSa], in the setting of the Adams spectral sequence associated to a generalized homology theory. Since the Lyndon-Hochschild-Serre spectral sequence can be viewed as an Adams spectral sequence in the appropriate context, it is not surprising that their result applies here.

Remark. The “appropriate context” is the category whose objects are unbounded chain complexes of injective comodules over the dual of the Hopf algebra Γ, and whose morphisms are chain homotopy classes of maps—this is a stable homotopy category [HPS, 9.5]. We have stated our results and proofs in the language of modules, rather than chain complexes, but if one is willing to work in the chain complex setting, then some of the following can be stated and proved a bit more cleanly (see [Pal]).

Theorem 3.1. Suppose that Λ ⟶ Γ ⟶ Γ//Λ is an extension of graded cocommutative Hopf algebras over a field k. For any Γ-module M, consider the associated spectral sequence

E_{2}^{s,t,u}(M) = H^{s,u}(Γ//Λ; H^{t,∗}(Λ; M)) ⇒ H^{s+t,u}(Γ; M).

Fix integers a, b, and c, with a ≥ b ≥ 0 and c < 0, and consider the following property P of the Γ-module M:

P: There are numbers r and d so that E_{r}^{s,t,u}(M) = 0 when as + bt + cu > d. Then we have the following results.

(a) M satisfies P if and only if ΣM satisfies P.

(b) Given a short exact sequence of Γ-modules

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0,

if two of M′, M, and M′′ satisfy property P, then so does the third.

(c) If the Γ-module L ⊕ M satisfies property P, then so do L and M.

(d) Every bounded-above injective Γ-module satisfies property P.

Here, Σ denotes the usual “shift” functor on the module category, and we say that a Γ-module is bounded-above if it is zero in all sufficiently large gradings.

Proof. Part (d) is clear: if M is a bounded-above injective Γ-module, then the spectral sequence collapses, and E_{2}^{s,t,u} = E_{∞}^{s,t,u} is zero unless s = t = 0 and u ≥ i, where i is the largest nonzero grading of M. Since c < 0, then E_{2}^{s,t,u} is certainly zero above some plane with normal vector (a, b, c).

Parts (a) and (c) are also clear: for part (a), the spectral sequence for ΣM is the same as the spectral sequence for M, but with a shift in dimension; so M and ΣM would have a vanishing plane with the same normal vector at the same E_{r}, with slightly different intercepts. For part (c), the naturality of the spectral sequence ensures that the E_{r}-term for the module L is a retract of the E_{r}-term for L ⊕ M. So a vanishing plane for the sum induces one on each summand.

We are left with (b). We start by giving a construction of the spectral sequence; this is essentially the same construction as in [Ben91, 3.5], for instance, except we use injective rather than projective resolutions.

Let I_{∗} be a fixed injective resolution of k as a Γ//Λ-module; let J_{∗} be a fixed injective resolution of k as a Γ-module. Then we let

E_{0}^{s,t,u} = Hom_{Γ//Λ}^{u}(k, Hom_{Λ}^{∗}(k, I_{s} ⊗ J_{t} ⊗ M))
≅ Hom_{Γ}^{u}(k, I_{s} ⊗ J_{t} ⊗ M).

(The isomorphism is as in [Ben91, 3.5.1].) The maps in the complexes I_{∗} and J_{∗} induce differentials on this, making it a double complex; hence we get two spectral sequences. One can see (again, as in [Ben91]) that one collapses to E_{2}^{∗} = H^{∗}(Γ; M) and that the other has E_{2}^{s,t,u} as in the statement of the theorem.

We need to examine the exact couple that produces this spectral sequence: we let

D_{0}^{s,t,u} = ⨁ E_{0}^{i,j,u}.

where the direct sum is over all i + j = s + t, i ≥ s. Then we have

[figure: D_{r}^{s,t,u}; E_{r}^{s,t,u}; D_{r}^{s+1,t−1,u}; (0, 0, 0); (r, 1 − r, 0); (−1, 1, 0)]

where the arrows are labeled by their degrees (i.e., how much they change the superscripts). The map i_{r}: D_{r}^{s+1,t−1,u} → D_{r}^{s−1,t+1,u} is induced by the map for r = 0: given D_{r−1} and i_{r−1}, then D_{r} is the image of i_{r−1}, and i_{r} is the restriction of i_{r−1} to this image. The map i_{0} is the obvious inclusion. This leads to an exact sequence:

⋯ → D_{r}^{stu} → E_{r}^{stu} → D_{r}^{s+r,t−r+1,u} → D_{r}^{s+r−1,t−r+2} → E_{r}^{s+r−1,t−r+2} → …

Consider the following property:

Q: There are numbers r and d so that the composite

\underbrace{i_{0} ∘ ⋯ ∘ i_{0}}_{r}: D_{0}^{s+r,t−r,u} ⟶ D_{0}^{s,t,u}

is zero whenever as + bt + cu > d.

We claim that properties P and Q are essentially equivalent: we have the implications

P_{r,d} ⇒ Q_{r,d+(a−b)(r−1)+b},
Q_{r,d} ⇒ P_{r,d}.

These are easy to see: if M satisfies P at the E_{r}-term, with intercept d, then we have E_{r}^{stu} = 0 when as+bt+cu > d. Since a ≥ b ≥ 0, then we also have E_{r}^{s+r−1,t−r+2,u} = 0 under same conditions, so the map i_{r}: D_{r}^{s+r,t−r+1,u} ⟶ D_{r}^{s+r−1,t−r+2} is an isomorphism under these conditions. Indeed, one can see that for any integer k > 0, the following composite is an isomorphism:

(i_{r})^{k}: D_{r}^{s+r−1+k,t−r+2−k,u} ⟶ D_{r}^{s+r−1,t−r+2,u}.

Since the spectral sequence converges, though, the inverse limit of the groups D_{r} must be zero; hence D_{r}^{s+r−1,t−r+2,u} = 0. By construction of the spectral sequence, this group is equal to the image of

(i_{0})^{r}: D_{0}^{s+2r,t−2r+1,u} ⟶ D_{0}^{s+r−1,t−r+2,u} = 0.

Hence M satisfies property Q at the E_{r}-term, with intercept d + (a − b)(r − 1) + b.

It is easier to show the other implication: given Q at E_{r} with intercept d, then when as + bt + cu > d, we have D_{r}^{stu} = 0 = D_{r}^{s+r,t−r+1,u}, so E_{r}^{stu} = 0, by the exact sequence.

By our construction of the spectral sequence, D_{0}(−) is functorial; in fact, D_{0}(M) = M ⊗ D_{0}(k), so it takes a short exact sequence of modules to a short exact sequence of double complexes. One can easily see that if two of the three complexes in a short exact sequence satisfy condition Q, then so does the third; one may take the intercept d for the third to be the larger of the other two intercepts, and one may take the term r for the third to be the sum of the r’s for the other complexes.

We borrow the following terminology from the analogous situation in a triangulated category (see [HSb], for example).

Definition 3.2. Any property P of Γ-modules which satisfies (a)–(d) of Theorem 3.1 is called a generic property.

Example 3.3. Suppose that Γ is a cocommutative Hopf algebra over a field k, and let Γ_{∗} be the dual of Γ, with the standard left Γ-module structure (i.e., if k is the only simple Γ-module, then Γ_{∗} is the injective hull of k). If M is a bounded-above Γ-module which satisfies a generic property P, then Γ_{∗} ⊗ M is a bounded-above injective, so the module

coker(M ↪ Γ_{∗} ⊗ M)

also satisfies P. We let Ω^{−1}M denote this module. It is standard (and easy to show) that

Ext_{Γ}^{s+1}(Ω^{−1}L, M) = Ext_{Γ}^{s}(L, M),
Ext_{Γ}^{s}(L, Ω^{−1}M) = Ext_{Γ}^{s+1}(L, M),

for s > 0. See [Mar83, 14.8], for example. We point out that if

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

is a short exact sequence, then for some injective modules I and J, so is the following:

0 ⟶ M ⟶ I ⊕ M′′ ⟶ J ⊕ Ω^{−1}M′ ⟶ 0.

(See [Mar83, p. 210].) If the modules M, M′, and M′′ are bounded-above, we may choose I and J to be bounded-above, as well. In any case, we say that

0 ⟶ M ⟶ M′′ ⟶ Ω^{−1}M′ ⟶ 0

is exact up to injective summands.

We will use Theorem 3.1 in combination with the next result.

Proposition 3.4. Fix a graded connected cocommutative Hopf algebra Γ over a field k, and fix a property P which is generic on Γ-modules. Let

0 ⟶ Σ^{j}N ⟶ M ⟶ N ⟶ 0

be a short exact sequence of bounded-above Γ-modules, representing the element

h ∈ Ext_{Γ}^{1,j}(N, N).

If h is nilpotent in Ext_{Γ}^{∗∗}(N, N), then M satisfies P if and only if N satisfies P.

Proof. By the definition of genericity, if N satisfies P, then so does M.

For the converse, we assume that h^{m} = 0. For each integer i ≥ 1, we define a Γ-module M_{i} by the following sequence, which represents h^{i} and is exact up to (bounded-above) injective summands:

0 ⟶ M_{i} ⟶ N ⟶^{h^{i}} Ω^{−i}N ⟶ 0.

(We omit suspensions here—we should write h_{i}: N ⟶ Σ^{ij}Ω^{−i}N.) So M_{i} is only well-defined up to injective summands. The following is also exact up to injective summands:

0 ⟶ Ω^{−(i−1)}N ⟶ M_{i} ⟶ N ⟶ 0.

Since h^{m} = 0, then this sequence splits when i = m; hence we see that N is a summand of M_{m} ⊕ I for some bounded-above injective I.

We claim that if M satisfies P, then so does M_{i} for all i; verifying this claim (and letting i = m) will finish the proof. We prove this by induction on i. We start the induction by noting that M = M_{1}.

So suppose that we know that M_{i−1} satisfies P. Consider the following commutative diagram:

0  0  0
↓  ↓  ↓
0 ⟶ K ⟶ 0 ⟶ Ω^{−(i−1)}M_{1} ⟶ 0
↓  ↓  ↓
0 ⟶ M_{i−1} ⟶ N ⟶^{h^{i−1}} Ω^{−(i−1)}N ⟶ 0
↓  ∥  ↓_{Ω^{−(i−1)}h}
0 ⟶ M_{i} ⟶ N ⟶^{h^{i}} Ω^{−i}N ⟶ 0
↓  ↓  ↓
0  0  0

The rows and columns of this diagram are exact up to injective summands, so we see that K is isomorphic to Ω^{−(i−2)}M_{1}, up to injective summands. By applying Example 3.3, we see that Ω^{−(i−2)}M_{1} satisfies P, so from the left-hand column we conclude that M_{i} does as well.

We will also need the following result, about the survival of powers of elements in our spectral sequence. Note that when using trivial coefficients, the Lyndon-Hochschild-Serre spectral sequence is a spectral sequence of commutative algebras: each E_{r}-term is a (graded) commutative algebra, and each differential d_{r} is a derivation. So if we are working with a field k of characteristic p > 0, then for any z ∈ E_{r}^{∗,∗}, z^{p} survives to the E_{r+1}-term. It turns out that we can greatly improve on this result.

Lemma 3.5. Suppose that there is an extension Λ ⟶ Γ ⟶ Γ//Λ of graded cocommutative Hopf algebras over the field F_{2}, and consider the associated spectral sequence E_{r}^{s,t,u}(F_{2}). Fix y ∈ E_{2}^{0,t,u}. Then for each i, the element y^{2^{i}} survives to E_{2^{i}+1}^{0,2^{i}t,2^{i}u}.

Proof. This follows from properties of Steenrod operations on this spectral sequence, as discussed by Singer in [Sin73]. Suppose that we have a spectral sequence E_{r}^{s,t} which is a spectral sequence of algebras over the Steenrod algebra. Fix z ∈ E_{r}^{s,t}, and fix an integer k. Then [Sin73, 1.4] tells us to which term of the spectral sequence Sq^{k} z survives (the result depends on r, s, t, and k). For instance, if z ∈ E_{r}^{0,t}, then Sq^{t}(z) = z^{2} survives to E_{2r−1}^{0,2t}.

(There are similar results at odd primes—see [Saw82].)

By the way, Singer’s results in [Sin73] are stated in the case of an extension of graded cocommutative Hopf algebras Λ → Γ → Γ//Λ, where Λ is also commutative. This last commutativity condition is not, in fact, necessary, as forthcoming work of Singer [Sin] shows.

4. Reduction of Theorem 1.3

In this section we state a variant of Theorem 1.3 (namely, Theorem 4.1), and we show that Theorem 1.3 follows from it.

Let D be the sub-Hopf algebra of A generated by the P_{t}^{s}’s with s < t; in other words, D is dual to the following quotient Hopf algebra of A_{∗}:

D_{∗} = F_{2}[ξ_{1}, ξ_{2}, ξ_{3}, …]/(ξ_{1}^{2}, ξ_{2}^{4}, ξ_{3}^{8}, …, ξ_{n}^{2^{n}}, …).

By Proposition 2.1(c), every elementary sub-Hopf algebra of A sits inside of D. By Proposition 2.4(a), D is a normal sub-Hopf algebra of A, so there is an action of A//D on H^{∗}(D; F_{2}).

We have the following result.

Theorem 4.1. The restriction map H^{∗}(A; F_{2}) ⟶ H^{∗}(D; F_{2}) factors through

φ: H^{∗}(A; F_{2}) ⟶ H^{∗}(D; F_{2})^{A//D},

and φ is an F -isomorphism.

In this section, we reduce Theorem 1.3 to Theorem 4.1; we prove Theorem 4.1 in Section 5. The first step in the reduction is the following.

Proposition 4.2. There is an F -isomorphism

f: H^{∗}(D; F_{2}) ⟶ \underleftarrow{lim}_{Q}H^{∗}(E; F_{2}).

Proof. We need to show two things: every element in the kernel of f is nilpotent, and we can lift some 2^{n}th power of any element in the codomain of f. The first of these statements follows from the main theorem of [HP93]—restriction to the elementary sub-Hopf algebras of A (and hence of D) detects nilpotence.

The second statement follows, almost directly, from a result of Hopkins and Smith [HSb, Theorem 4.12], stated as Theorem 4.3 below, combined with the fact that the Hopf algebra D is an inverse limit of finite-dimensional Hopf algebras. We explain. For each integer r ≥ 1, we let B(r) be the sub-Hopf algebra of D which is dual to the following quotient Hopf algebra of D_{∗}:

B(r)_{∗} = D_{∗}/(ξ_{1}, ξ_{2}, …, ξ_{r}).

Then each B(r) is normal in D, D//B(r) is finite-dimensional, and we have D = \underleftarrow{lim}_{r}D//B(r). Note that this inverse limit stabilizes in any given degree.

We claim that given any object E of Q and given any y ∈ H^{∗}(E; F_{2}), some 2^{n}th power of y is in the image of the restriction map H^{∗}(D; F_{2}) ⟶ H^{∗}(E; F_{2}). We have a normal sub-Hopf algebra E(r) = E ∩ B(r) of E, and E//E(r) is a sub-Hopf algebra of D//B(r). Furthermore, E = \underleftarrow{lim} E//E(r), so any y ∈ H^{∗}(E; F_{2}) is represented by a class \overline{y} ∈ H^{∗}(E//E(r); F_{2}), for some r.

By Theorem 4.3, then, there is an integer N so that the class \overline{y}^{2^{N}} lifts to H^{∗}(D//B(r); F_{2}), and hence y^{2^{N}} lifts to H^{∗}(D; F_{2}).

We have just used the following theorem:

Theorem 4.3 (Theorem 4.12 in [HSb]). Suppose that Λ ⊆ Γ is an inclusion of finite-dimensional graded connected cocommutative Hopf algebras over a field k of characteristic p > 0. For any λ ∈ H^{∗}(Λ; k), there is a number N so that λ^{p^{N}} is in the image of the restriction map H^{∗}(Γ; k) ⟶ H^{∗}(Λ; k).

Proof of Theorem 1.3, assuming Theorem 4.1. By Proposition 4.2, there is an F - isomorphism

f: H^{∗}(D; F_{2}) ⟶ \underleftarrow{lim}_{Q}H^{∗}(E; F_{2}).

We need to show that this induces an F -isomorphism on the invariants; then composing with the F -isomorphism in Theorem 4.1 gives the desired result (it is clear that the composite of two F -isomorphisms is again an F -isomorphism).

Since each restriction map H^{∗}(D; F_{2}) ⟶ H^{∗}(E; F_{2}) is an A-algebra map, so is the assembly of these into the map f. So we have an induced map on the invariants:

f̃: (H^{∗}(D; F_{2}))^{A} ⟶ (\underleftarrow{lim}_{Q}H^{∗}(E; F_{2}))^{A} .

Any x in the kernel of f̃ is also in the kernel of f, and hence is nilpotent. Given any y ∈ (\underleftarrow{lim}_{Q}H^{∗}(E; F_{2}))^{A}, we know that y^{2^{n}} is in im(f) for some n; by Lemma 4.4 below, we may conclude that y^{2^{n+N}} ∈ im(f̃) for some N. (To apply the lemma, we need to know that for every z ∈ H^{∗}(D; F_{2}), Sq^{2^{k}}(z) is zero for all but finitely many values of k. The action of A decreases degrees, by Proposition 2.3, so this is clear.)

This finishes the proof of Theorem 1.3, given Theorem 4.1.

We have used the following lemma. We say that an A-module M is locally finite if for every element m ∈ M, the submodule generated by m is finite-dimensional.

Lemma 4.4. Suppose that R and S are commutative A-algebras, with an A-algebra map f: R ⟶ S that detects nilpotence: every x ∈ ker f is nilpotent. Suppose also that R is locally finite as an A-module. Given z ∈ R so that f(z) ∈ S is invariant under the A-action, then z^{2^{N}} is also invariant, for some N.

Proof. Since R is locally finite, then Sq^{n}(z) = 0 for all but finitely many values of n. Since f is an A-module map and f(z) is invariant, then we also know that f(Sq^{n}(z)) = 0 for all n; therefore, each Sq^{n}(z) is nilpotent. Recall that the total Steenrod square Sq = ∑_{i≥0} Sq^{i} is an algebra map, and to say that an element y is invariant under the A-action is to say that Sq(y) = y. In our case, Sq(z) has only finitely many nonzero terms, and all but the first (Sq^{0} z = z) are nilpotent; hence Sq(z) − z is nilpotent; i.e., z^{2^{N}} is invariant for some N.

5. Proof of Theorem 4.1

In this section we prove Theorem 4.1. First we show that the restriction map H^{∗}(A; F_{2}) ⟶ H^{∗}(D; F_{2}) factors through the invariants H^{∗}(D; F_{2})^{A}, and then we show that the resulting map is an F -isomorphism.

D is a normal sub-Hopf algebra of A, so by Proposition 2.3(b), there is a spectral sequence

E_{2}^{s,t,u} = H^{s,u}(A//D; H^{t,∗}(D; F_{2})) ⇒ H^{s+t,u}(A; F_{2})

It is well-known, and easy to verify from any reasonable construction of the spectral sequence, that the edge homomorphism

H^{∗}(A; F_{2}) ⟶ E_{2}^{0,∗} = H^{∗}(D; F_{2})^{A//D}

factors the restriction map (see [Ben91, p. 114], for instance). This gives us the desired map

φ: H^{∗}(A; F_{2}) ⟶ H^{∗}(D; F_{2})^{A//D} = H^{∗}(D; F_{2})^{A}.

We have to show two things: that every element of ker φ is nilpotent, and that for every y ∈ H^{∗}(D; F_{2})^{A}, there is an integer n so that y^{2^{n}} ∈ im φ. (The first of these follows from [Pal96, Theorem 3.1]; see also [HP93, Theorem 1.1]. We provide a proof here, anyway.) These two verifications have many common features, so first we establish some tools that we will use in both parts.

For n ≥ 1 we define sub-Hopf algebras D(n) of A by

D(n) = (F_{2}[ξ_{1}, ξ_{2}, ξ_{3}, …]/(ξ_{1}^{2}, ξ_{2}^{4}, …, ξ_{n}^{2^{n}}))^{∗} .

We let D(0) = A. This gives us a diagram of Hopf algebra inclusions:

⋯ ↪ D(2) ↪ D(1) ↪ D(0) = A.

D is the inverse limit of this diagram. Taking cohomology gives

H^{∗}(D(0); F_{2}) ⟶ H^{∗}(D(1); F_{2}) ⟶ H^{∗}(D(2); F_{2}) ⟶ ….

By Proposition 2.4, each D(n) is a normal sub-Hopf algebra of A; hence there is an action of A on H^{∗}(D(n); F_{2}).

Lemma 5.1. (a) There is an isomorphism H^{∗}(D; F_{2}) ≅ \underrightarrow{lim} H^{∗}(D(n); F_{2}).

(b) This induces an isomorphism H^{∗}(D; F_{2})^{A} ≅ \underrightarrow{lim}(H^{∗}(D(n); F_{2}))^{A}.

Proof. Since D is the inverse limit of the D(n)’s, and the diagrams

⋯ ↪ D(3) ↪ D(2) ↪ D(1) ↪ D(0),
H^{∗}(D(0); F_{2}) ⟶ H^{∗}(D(1); F_{2}) ⟶ H^{∗}(D(2); F_{2}) ⟶ …,

stabilize (and are finite-dimensional) in each degree, then part (a) is clear.

Part (b): Since the action of Sq^{k} ∈ A takes H^{s,t}(B; F_{2}) to H^{s,t−k}(B; F_{2}) (for B = D or B = D(n)), then the action of A on any element of H^{∗}(D; F_{2}) ≅ \underrightarrow{lim} H^{∗}(D(n); F_{2}) is completely determined at some finite stage of the direct system. Hence the ring of invariants of the direct limit is equal to the direct limit of the invariants.

This lemma lets us translate information about H^{∗}(D; F_{2}) to H^{∗}(D(n); F_{2}) for some n ≫ 0. We also want to know how to go from D(n) to D(n − 1) (and then apply downward induction on n, ending at D(0) = A). There is a Hopf algebra extension

D(n) ⟶ D(n − 1) ⟶ D(n − 1)//D(n), (**)

and the quotient Hopf algebra is easy to describe: it is dual to F_{2}[ξ_{n}^{2^{n}}] with ξ_{n}^{2^{n}} primitive, so we have an isomorphism of algebras

D(n − 1)//D(n) ≅ E(P_{n}^{n}, P_{n}^{n+1}, P_{n}^{n+2}, …).

(Recall from Section 2 that P_{n}^{n+s} is the element of A dual to ξ_{n}^{2^{n+s}}.) Anyway, by Proposition 2.3, for any D(n − 1)-module M, we have a spectral sequence

E_{2}^{s,t,u}(M) = H^{s,u}(D(n − 1)//D(n); H^{t,∗}(D(n); M)) ⇒ H^{s+t,u}(D(n − 1); M).

Our main tool for the completion of the proof is the following.

Lemma 5.2. Fix an integer N. For some r and some c, there is a vanishing plane “of slope N” at the E_{r}-term of the spectral sequence: E_{r}^{s,t,u}(F_{2}) = 0 when Ns + t − u > c.

To prove this lemma, we show that for a certain finite module M, we have the appropriate vanishing plane at E_{2}^{∗∗∗}(M); then we use Theorem 3.1 to induce a vanishing plane (with the same normal vector, but possibly different intercept) on E_{r}^{∗∗∗}(F_{2}) for some r.

Lemma 5.3. Fix an integer j > n, and let M_{j} be the D(n − 1)//D(n)-module dual to E[P_{n}^{n}, P_{n}^{n+1}, …, P_{n}^{j−1}]. Then E_{2}^{s,t,u}(M_{j}) = 0 when

2^{j}(2^{n} − 1)s + t − u > 0.

(So given an integer N, if we choose j so that 2^{j}(2^{n} − 1) > N, then we will have E_{2}^{s,t,u}(M_{j}) = 0 for Ns + t − u > 0.)

Proof. M_{j} is a trivial D(n)-module (by definition), and as algebras, we have

D(n − 1)//D(n) ≅ M_{j}^{∗} ⊗ E[P_{n}^{j}, P_{n}^{j+1}, P_{n}^{j+2}, …].

Hence the E_{2}-term of the spectral sequence for M_{j} looks like

E_{2}^{s,t,u}(M_{j}) ≅ H^{s,u}(D(n − 1)//D(n); H^{t,∗}(D(n); M_{j}))
≅ H^{s,u}(D(n − 1)//D(n); M_{j} ⊗ H^{t,∗}(D(n); F_{2}))
≅ H^{s,u}(E[P_{n}^{j}, P_{n}^{j+1}, …]; H^{t,∗}(D(n); F_{2})).

The Hopf algebra E[P_{n}^{j}, P_{n}^{j+1}, …] is 2^{j}(2^{n}−1)-connected, so if L is a module which is zero below degree t, then

H^{s,u}(E[P_{n}^{j}, P_{n}^{j+1}, …]; L) = 0

when u < 2^{j}(2^{n} − 1)s + t. Now note that H^{t,∗}(D(n); F_{2}) is zero below degree t.

We want to apply Theorem 3.1 to get a vanishing plane at some term of the spectral sequence E_{r}^{∗∗∗}(F_{2}); first we describe how to build M_{j} out of F_{2}, inductively.

Lemma 5.4. For j > n, let M_{j} be as in Lemma 5.3. Let M_{n} = F_{2}. For each j ≥ n, there is a short exact sequence of D(n − 1)//D(n)-modules

0 ⟶ Σ^{2^{j}(2^{n}−1)}M_{j} ⟶ M_{j+1} ⟶ M_{j} ⟶ 0.

The connecting homomorphism in H^{∗}(D(n − 1); −) is multiplication by h_{nj} ⊗ 1_{M_{j}}, where

h_{nj} = [ξ_{n}^{2^{j}}] ∈ H^{∗}(D(n − 1); F_{2}).

Proof. This is clear.

Proof of Lemma 5.2. Lin has shown [Lin, 3.2] that when j ≥ n, then the element h_{nj} ∈ H^{∗}(D(n − 1); F_{2}) is nilpotent. Therefore Lemma 5.2 follows from the above lemmas, together with Theorem 3.1 and Proposition 3.4.

By the way, Lemma 5.2 has the following obvious corollary; we do not use this in this paper, but it might be useful elsewhere.

Corollary 5.5. Fix an integer N and a finite-dimensional A-module M, and consider the Lyndon-Hochschild-Serre spectral sequence E_{∗}^{s,t,u}(M) associated to the extension (**). For some r and some c, there is a vanishing plane at the E_{r}-term of this spectral sequence: E_{r}^{s,t,u}(M) = 0 when Ns + t − u > c.

Proof. This follows immediately from Theorem 3.1 and induction on dim M.

Now we finish the proof of Theorem 4.1. We have two conditions to verify, and we devote a subsection to each one.

5.1. Every x ∈ ker(φ) is nilpotent. Fix x ∈ H^{∗}(A; F_{2}) so that φ(x) = 0; we want to show that x is nilpotent. As mentioned above, this has been proved elsewhere— see [Pal96, Theorem 3.1] and [HP93, Theorem 1.1]. We present a proof here anyway, because we have done most of the work already in setting up the other half of the proof of Theorem 4.1.

Given Hopf algebras C ⊆ B over a field k, we write res_{B,C}: H^{∗}(B; k) ⟶ H^{∗}(C; k) for the restriction map.

We have an element x ∈ H^{∗}(A; F_{2}) so that φ(x) = 0. Then res_{A,D}(x) = 0, so by Lemma 5.1(a), res_{A,D(n)}(x) = 0 for some n. Since D(0) = A and res_{A,D(0)} is the identity map on H^{∗}(A; F_{2}), it suffices to show that if for some n, res_{A,D(n)}(x) = 0, then res_{A,D(n−1)}(x^{j}) = 0 for some j.

Consider the spectral sequence

E_{2}^{s,t,u}(F_{2}) = H^{s,u}(D(n − 1)//D(n); H^{t,∗}(D(n); F_{2})) ⇒ H^{s+t,u}(D(n − 1); F_{2})

associated to the extension

D(n) ⟶ D(n − 1) ⟶ D(n − 1)//D(n).

Write z for res_{A,D(n−1)}(x), so that res_{D(n−1),D(n)}(z) = res_{A,D(n)}(x) = 0. Because of this, z must be represented by a class z̃ ∈ E_{2}^{p,q,v} with p > 0.

Choose an integer N so that Np + q − v > 0. By Lemma 5.2, for some r and c we have E_{r}^{s,t,u}(F_{2}) = 0 when Ns + t − u > c. Since this is a spectral sequence of commutative algebras, we can find a j so that z̃^{j} survives to E_{r}, and also so that

Npj + qj − vj > c.

Hence at this E_{r}-term, z̃^{j} must be zero. This means that z̃^{j} = 0 at E_{∞}, so that z̃^{j} = 0 in the abutment, mod higher filtrations. But the vanishing plane at E_{r} also exists at E_{∞}, and one can see that the higher filtrations are all above the vanishing plane, and hence zero. So z̃^{j} = 0 in the abutment; hence z^{j} = 0 in H^{∗}(D(n−1); F_{2}), which is what we wanted to show.

5.2. Every y ∈ coker(φ) is nilpotent. Fix y ∈ H^{∗}(D; F_{2})^{A//D}. We want to show that there is an integer m so that y^{2^{m}} ∈ im φ. By Lemma 5.1(b), there is an n so that y lifts to H^{∗}(D(n); F_{2})^{A//D(n)}. Now we show that some power of y lifts to H^{∗}(D(n − 1); F_{2})^{A//D(n−1)}; since D(0) = A, then downward induction on n will finish the proof.

Since y is invariant under the A//D(n)-action, then it is also invariant under the action of D(n − 1)//D(n) (since the latter quotient is a sub-Hopf algebra of the former). So y represents a class at the E_{2}-term of the spectral sequence

E_{2}^{s,t,u}(F_{2}) = H^{s,u}(D(n − 1)//D(n); H^{t,∗}(D(n); F_{2})) ⇒ H^{s+t,u}(D(n − 1); F_{2}).

Also, y lies in the (t, u)-plane, say y ∈ E_{2}^{0,q,v}. Choose N large enough so that N + q − v − 1 and N − 1 are both positive. By Lemma 5.2, we know that for some c and r, we have E_{r}^{s,t,u} = 0 for Ns + t − u > c.

Choose i so that

2^{i} > max(r − 2, \frac{c − N}{N + q − v − 1}).

We claim that y^{2^{i}} is a permanent cycle. By Lemma 3.5, the possible differentials on y^{2^{i}} are

d_{j+1}(y^{2^{i}}) ∈ E_{j+1}^{j+1,2^{i}q−j,2^{i}v},

for j ≥ 2^{i}, so we show that the targets of these differentials lie above the vanishing plane. (Since 2^{i} was chosen to be at least r − 1, then the vanishing plane is present at the E_{j+1}-term for all j ≥ 2^{i}.) We just have to verify the inequality specified by the vanishing plane:

N(j + 1) + 2^{i}q − j − 2^{i}v = (N − 1)j + N + 2^{i}(q − v)

≥ (N − 1)2^{i} + N + 2^{i}(q − v) = 2^{i}(N + q − v − 1) + N

> \frac{c − N}{N + q − v − 1}(N + q − v − 1) + N

= c.

Hence all of the differentials on y^{2^{i}} vanish, so it is a permanent cycle. For degree reasons, it cannot be a boundary; hence it gives a nonzero element of E_{∞}, and hence a nonzero element of H^{∗}(D(n − 1); F_{2}). Lastly, Lemma 4.4 tells us that some 2^{k}th power of y^{2^{i}} is invariant in H^{∗}(D(n − 1); F_{2}). This completes the inductive step, and hence the proof of Theorem 4.1.

References

[Ben91] D. J. Benson, Representations and cohomology, Volume II, Cambridge studies in advanced mathematics, vol. 31, Cambridge University Press, 1991.

[HP93] M. J. Hopkins and J. H. Palmieri, A nilpotence theorem for modules over the mod 2 Steenrod algebra, Topology 32 (1993), 751–756.

[HSb] M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory II, preprint.

[HSa] M. J. Hopkins and J. H. Smith, private communication.

[HPS] M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc., to appear.

[Lin] 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.

[MT68] M. E. Mahowald and M. C. Tangora, An infinite subalgebra of Ext_{A}^{∗∗}(F_{2}, F_{2}), Trans. Amer. Math. Soc. 132 (1968), 263–274.

[Mar83] H. R. Margolis, Spectra and the Steenrod algebra, North-Holland, 1983.

[MPT71] H. R. Margolis, S. B. Priddy, and M. C. Tangora, Another systematic phenomenon in the cohomology of the Steenrod algebra, Topology 10 (1971), 43–46.

[May70] J. P. May, A general algebraic approach to Steenrod operations, The Steenrod Algebra and its Applications (F. P. Peterson, ed.), 1970, Lecture Notes in Mathematics, vol. 168, Springer-Verlag, pp. 153–231.

[Mil58] J. W. Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171.

[Pal96] J. H. Palmieri, Nilpotence for modulex over the mod 2 Steenrod algebra I, Duke Math. J. 82 (1996), 195–208.

[Pal97] J. H. Palmieri, A note on the cohomology of finite dimensional cocommutative Hopf algebras, J. Algebra 188 (1997), 203–215.

[Pal] J. H. Palmieri, Stable homotopy over the Steenrod algebra, in preparation.

[Qui71] D. G. Quillen, The spectrum of an equivariant cohomology ring I, II, Ann. of Math. (2) 94 (1971), 549–572, 573–602.

[Saw82] J. Sawka, Odd primary Steenrod operations in first quadrant spectral sequences, Trans. Amer. Math. Soc. 273 (1982), 737–752.

[Sin73] W. M. Singer, Steenrod squares in spectral sequences I and II, Trans. Amer. Math. Soc. 175 (1973), 327–336, 337–353.

[Sin] W. M. Singer, private communication.

[Wil81] C. Wilkerson, The cohomology algebras of finite dimensional Hopf algebras, Trans. Amer. Math. Soc. 264 (1981), 137–150.

[Zac67] A. Zachariou, A subalgebra of Ext_{A}^{∗∗}(F_{2}, F_{2}), Bull. Amer. Math. Soc. 73 (1967), 647–648.

Room 2-388, Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge (our fair city), MA 02139

   E-mail address: palmieri@math.mit.edu