TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9947(XX)0000-0

SOME QUOTIENT HOPF ALGEBRAS OF THE DUAL STEENROD ALGEBRA

J. H. PALMIERI

Abstract. Fix a prime p, and let A be the polynomial part of the dual Steenrod algebra. The Frobenius map on A induces the Steenrod operation 𝒫̃^{0} on cohomology, and in this paper, we investigate this operation. We point out that if p = 2, then for any element in the cohomology of A, if one applies 𝒫̃^{0} enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that “enough times” should be “once.”

The bulk of the paper is a study of some quotients of A in which the Frobenius is an isomorphism of order n. We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about 𝒫̃^{0} . The dual complete Steenrod algebra makes an appearance.

1. Introduction

Fix a prime number p, and let A be the following familiar Hopf algebra: as an algebra,

A = F_{p}[ξ_{1}, ξ_{2}, ξ_{3}, …],

and the coproduct is determined by

(1.1)  Δ(ξ_{n}) = ∑_{i=0}^{n} ξ_{n−i}^{p^{i}} ⊗ ξ_{i},

where ξ_{0} = 1. There is also an antipode – see Lemma 3.6 – but it does not play a central role here. A is graded by putting ξ_{n} in degree 2^{n} − 1 when p = 2, and in degree 2p^{n} − 2 when p is odd. This makes A into a graded connected commutative Hopf algebra.

When p = 2, the graded dual of A is the mod 2 Steenrod algebra; when p is odd, the graded dual of A is the algebra of reduced power operations, which is a quotient of the mod p Steenrod algebra. We will occasionally abuse the language and refer to A as the dual Steenrod algebra, regardless of the prime. We write A^{∗} for its graded dual.

Homotopy theorists would like to understand the cohomology of A, which can be defined as Ext_{A^{∗}}^{∗,∗}(F_{p}, F_{p}), where Ext is computed in the category of graded A^{∗}- modules, or equivalently as Ext_{A}^{∗,∗}(F_{p}, F_{p}), where Ext is computed in the category of graded A-comodules. (In places where it makes a difference, we will use the comodule definition). Note that in this setting, Ext is bigraded: the first grading

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors 5 January 2004.

2000 Mathematics Subject Classification. Primary 55S10, 18G15, 20E18, 20J06.

is the usual homological one, and the second is induced by the grading on the (co)module category. The Frobenius map a ↦ a^{p} on A is a Hopf algebra map, and so it induces an algebra map on cohomology. We denote this map by 𝒫̃^{0} when p is unspecified or odd, and by S̃q^{0} when p = 2 (see Proposition 5.1 for an explanation of this notation). Since the Frobenius does not preserve the grading, neither does 𝒫̃^{0} – it works like this:

𝒫̃^{0} : Ext_{A}^{s,t}(F_{p}, F_{p}) → Ext_{A}^{s,pt}(F_{p}, F_{p}).

We would like to understand how the map 𝒫̃^{0} interacts with the multiplicative structure of Ext_{A}^{∗}(F_{p}, F_{p}). This paper has two goals: to advertise some questions and conjectures related to this interaction, and to study some Hopf algebras related to A that arise when studying 𝒫̃^{0} and the Frobenius.

We start by pointing out that

𝒫̃^{0} : Ext_{A}^{0,0}(F_{p}, F_{p}) → Ext_{A}^{0,0}(F_{p}, F_{p})

is the identity map. Regarding Ext in positive dimensions, one can deduce the following from the main result in [Pal99].

Theorem 1.2. Let p = 2 and fix z ∈ Ext_{A}^{s,t}(F_{2}, F_{2}). If s is positive, then there is

an n so that (S̃q^{0})^{n}(z) is nilpotent.

This fails if A is replaced by the sub-Hopf algebra F_{2}[ξ_{1}] – the cohomology of this is F_{2}[h_{10}, h_{11}, h_{12}, …], with S̃q^{0}(h_{1,n}) = h_{1,n+1} – but holds if A is replaced by any quotient Hopf algebra. See the appendix for the theorem’s proof. One might ask whether there is a simpler proof of this, one which would hold at any prime.

Question 1.3. (a) Is the analogous result true when p is odd?

(b) Can one prove Theorem 1.2 without appealing to the F -isomorphism of [Pal99, Theorem 1.2]?

It is also natural to ask, given z as in Theorem 1.2, how large must n be to make (S̃q^{0})^{n}(z) nilpotent? S̃q^{0} is an algebra map, so if z is nilpotent, then so is S̃q^{0}(z). For every known non-nilpotent element z (except for z = 1), S̃q^{0}(z) is nilpotent as well. This leads us to make the following conjecture. All of our evidence is valid only for the prime 2, but we brashly state it for all primes.

Conjecture 1.4. Fix a prime p and an element z ∈ Ext_{A}^{s,t}(F_{p}, F_{p}) with s > 0. Then 𝒫̃^{0} (z) is nilpotent.

This is rather specific to A: Theorem 1.2 fails for F_{p}[ξ_{1}], and the conjecture implies the theorem. Thus the conjecture does not hold for sub-Hopf algebras of A. It also fails for quotient Hopf algebras of A: for example, the cohomology of F_{2}[ξ_{2}]/(ξ_{2}^{4}) is isomorphic to F_{2}[h_{20}, h_{21}], with S̃q^{0}(h_{20}) = h_{21}. (Note, though, that a modified conjecture may hold for quotients of A: given a quotient Hopf algebra B of A, there exists an n so that for all z ∈ Ext_{A}^{s,t}(F_{p}, F_{p}) with s > 0, (𝒫̃^{0} )^{n}(z) is nilpotent.)

One can interpret many of these questions in terms of the “double” ΦA of A: let ΦA be the sub-Hopf algebra of A generated by the pth powers of the generators:

ΦA = F_{p}[ξ_{1}^{p}, ξ_{2}^{p}, ξ_{3}^{p}, …].

Then A and ΦA are isomorphic via the Frobenius map x ↦ x^{p}, and so their Ext algebras are isomorphic. We write ϕ for the isomorphism:

ϕ : Ext_{A}^{s,t}(F_{p}, F_{p}) ⟶^{≅} Ext_{ΦA}^{s,pt}(F_{p}, F_{p}).

Since ΦA is a sub-Hopf algebra of A, there is also an induced map which preserves the grading:

ρ : Ext_{ΦA}^{s,t}(F_{p}, F_{p}) → Ext_{A}^{s,t}(F_{p}, F_{p}).

It is not hard to see that for any z ∈ Ext_{A}^{s,t}(F_{p}, F_{p}),

ρ(ϕ(z)) = 𝒫̃^{0} (z).

So, for example, Conjecture 1.4 is equivalent to the statement that every positivedimensional element in the image of ρ is nilpotent.

One can also reformulate these questions, conjectures, and results as follows. Let Â be the dual complete Steenrod algebra: Â is the Z[\frac{1}{p}]-graded Hopf algebra obtained from A by adjoining all p^{k}th roots of each indeterminate ξ_{n}:

Â = \underrightarrow{lim} (F_{p}[ξ_{n} : n ≥ 1] → F_{p}[ξ_{n}^{p^{−1}} : n ≥ 1] → F_{p}[ξ_{n}^{p^{−2}} : n ≥ 1] → ⋯) .

(In the language of [Arn94], [Arn00], and [LH96], Â is the “root algebra” associated to A.) The coproduct on A induces one on Â:

Δ(ξ_{n}^{p^{k}}) = ∑_{i=0}^{n} ξ_{n−i}^{p^{i+k}} ⊗ ξ_{i}^{p^{k}},

for all n ≥ 1 and all k ∈ Z, where ξ_{0} = 1 as usual. The antipode for A also induces one for Â, which we will not need. The dual Â^{∗} of Â is the complete Steenrod algebra, which was introduced by Arnon [Arn94, Arn00] and has also been studied by Llerena and Hu’ng [LH96]. (Note that Â^{∗} is not a graded algebra, since it is larger than the direct sum of its graded pieces – see [LH96, Remark 1.5]. Â is a graded algebra, though; the difference is that Â is defined as a direct limit, while Â^{∗} is an inverse limit.)

In other words, one obtains Â from A by inverting the Frobenius map. As a result, one can show – see Proposition 5.3 – that the cohomology of Â is isomorphic to the cohomology of A with the operation 𝒫̃^{0} inverted.

Thus Theorem 1.2 says that every positive dimensional element in Ext∗ ; Â(F_{2}, F_{2}) is nilpotent, and Question 1.3(a) asks whether this is true at odd primes. Â seems as though it should be easier to understand than A. Hence we have the following natural question. Question 1.5. What is the cohomology of Â?

In order to approach all of these questions, we try something simpler: rather than constructing Â by formally inverting the Frobenius and 𝒫̃^{0} , thus making them isomorphisms with infinite order, we will construct Hopf algebras in which they are invertible with finite order.

Thus we study the following quotient Hopf algebras of A: fix n ≥ 1 and define H^{n} by

H^{n} = A/(ξ_{i}^{p^{n}} − ξ_{i} : i ≥ 1).

This is a version of A in which the Frobenius has order n. In this paper, we show that the topological linear dual of H^{n} is a group algebra and we identify the group.

We use this observation to show that Ext_{H^{n}}^{∗,∗}(F_{p}, F_{p}) consists entirely of nilpotent elements. We also try to relate these Ext calculations to the cohomology of the complete Steenrod algebra.

A few words of warning may be warranted for any topologists who may be reading this. First, H^{n} does not inherit the grading from A, since the ideal is not homogeneous. Many topologists are used to dualizing quotient Hopf algebras of A to get sub-Hopf algebras of the Steenrod algebra, but since H^{n} is not graded, its dual sits inside the ungraded dual of A, which is distinctly different from the usual Steenrod algebra. After all, since A is finite-dimensional in each grading, it is isomorphic to its double graded dual; since it is infinite-dimensional as a whole, it is not isomorphic to its double ungraded dual.

For example, let p = 2 and n = 1. If one dualizes with respect to the monomial basis of A, then ξ_{1}^{i} is dual to an element called Sq^{i}. Hence when one dualizes H^{1}, since ξ_{1} is identified with ξ_{1}^{i} for all i, the element dual to ξ_{1} in (H^{1})^{∗} deserves to be called ∑_{j≥1} Sq^{j}. This element is non-nilpotent – compare this to the graded dual of A, in which such infinite sums do not occur and in which every element in the augmentation ideal is nilpotent. The non-nilpotence of x = ∑_{j≥1} Sq^{j} is reflected by the coproduct in H^{1}: Δ(ξ_{2}) contains a term ξ_{1}^{2} ⊗ ξ_{1} = ξ_{1} ⊗ ξ_{1}, and hence x^{2} ≠ 0 – it is detected by ξ_{2}. Δ(ξ_{4}) has a term ξ_{2} ⊗ ξ_{2}, so the iterated coproduct of ξ_{4} has a term ξ_{1} ⊗ ξ_{1} ⊗ ξ_{1} ⊗ ξ_{1}, and hence x^{4} ≠ 0. For each n, Δ(ξ_{2^{n}}) has a term ξ_{2^{n−1}} ⊗ ξ_{2^{n−1}}, and hence inductively x^{2^{n}} ≠ 0 for all n.

In other words, the Hopf algebras H^{n} and their duals may challenge the intuition of any readers (or authors) who are used to working with the Steenrod algebra.

Here is an outline of the paper: in Section 2, we show that H^{n} is dual to a group algebra, and we identify the group – it is the group of strict automorphisms of the additive formal group law over F_{p^{n}}. We accomplish this by imitating work of Ravenel and others on the Morava stabilizer groups. We also note that this automorphism group is a torsion-free pro-p group. In Section 3, we collect some observations about H^{n} and the associated automorphism group; for example, we point out that the group is not p-adic analytic. In Section 4, we use the fact that the group is torsion-free to show that its cohomology consists entirely of nilpotent elements. In Section 5, we fill in some of the details of the ideas in this introduction: the relationship between the Frobenius on A and the Steenrod operation 𝒫̃^{0} , and also the completed Steenrod algebra and its cohomology. There is also an appendix which contains the proof of Theorem 1.2.

Convention. Throughout, an unadorned tensor product ⊗ means tensor product over F_{p}.

Acknowledgments. Much of the work on the Hopf algebras H^{n} was inspired by a brief conversation with Mike Hopkins. I have had many helpful discussions with Ethan Devinatz and Hal Sadofsky. Also, several of the results and proofs owe an obvious debt to Section 6.2 of Ravenel’s green book [Rav86].

2. H^{n} is dual to a group algebra

Let H^{n} be as above. Let B_{j}^{n} be the sub-Hopf algebra of H^{n} generated by ξ_{i} with 1 ≤ i ≤ j. Let H_{j}^{n} be the quotient of H^{n} by B_{j}^{n}. Let F = F_{p^{n}}. That is, for fixed n ≥ 1 and j ≥ 0,

H^{n} = F_{p}[ξ_{1}, ξ_{2}, …]/(ξ_{i}^{p^{n}} − ξ_{i} : i ≥ 1),
H_{j}^{n} = F_{p}[ξ_{j+1}, ξ_{j+2}, …]/(ξ_{i}^{p^{n}} − ξ_{i} : i ≥ j + 1),
B_{j}^{n} = F_{p}[ξ_{1}, ξ_{2}, …, ξ_{j}]/(ξ_{i}^{p^{n}} − ξ_{i} : j ≥ i ≥ 1),
F = F_{p^{n}}.

Note that H_{0}^{n} = H^{n}.

In this section, we show that after tensoring with F, each of these Hopf algebras is dual to a group algebra, we identify the groups, and we observe that the group associated to H_{j}^{n} is torsion-free for each n and j.

First we should verify that H_{j}^{n} is a quotient Hopf algebra of A for each n and j. Given this, it is clear from the formula for the coproduct (1.1) that B_{j}^{n} is a sub-Hopf algebra of H^{n}. Lemma 2.1. Fix n ≥ 1 and j ≥ 0. Then

I_{j}^{n} = (ξ_{1}, …, ξ_{j}; ξ_{i}^{p^{n}} − ξ_{i} : i ≥ j + 1)

is a Hopf ideal of A. Thus H_{j}^{n} is a quotient Hopf algebra of A.

Proof. We have defined I = I_{j}^{n} as an ideal in the algebra A, so we need to show that it interacts well with the coproduct. That is, for each x ∈ I, we need to show that Δ(x) ∈ I ⊗A+A⊗I. It suffices to check this for the generators of the ideal; it is clear when x = ξ_{m} with m ≤ j, so let x = ξ_{m}^{p^{n}} − ξ_{m}, where m > j. We compute:

Δ(ξ_{m}^{p^{n}} − ξ_{m}) = ∑_{i=0}^{m} (ξ_{m−i}^{p^{n+i}} ⊗ ξ_{i}^{p^{n}} − ξ_{m−i}^{p^{i}} ⊗ ξ_{i})

= (ξ_{m}^{p^{n}} − ξ_{m}) ⊗ 1 + 1 ⊗ (ξ_{m}^{p^{n}} − ξ_{m})

+ ∑_{i=1}^{m−1} ((ξ_{m−i}^{p^{n+i}} − ξ_{m−i}^{p^{i}}) ⊗ ξ_{i} + ξ_{m−i}^{p^{n+i}} ⊗ (ξ_{i}^{p^{n}} − ξ_{i})) .

Thus Δ(ξ_{m}^{p^{n}} − ξ_{m}) is in I ⊗ A + A ⊗ I, as desired. □

As in [Rav86, Section 6.2], we can discuss the “topological linear dual” of the Hopf algebras H_{j}^{n}: since H^{n}, for instance, is the direct limit of the finite-dimensional sub-Hopf algebras B_{j}^{n}, then the topological linear dual (H^{n})^{∗} is defined to be (H^{n})^{∗} = \underleftarrow{lim}_{j}(B_{j}^{n})^{∗}, with the topology induced by the inverse limit.

The first part of the proof of [Rav86, Theorem 6.2.3] applies in our setting to give the following. Proposition 2.2. Fix n ≥ 1, j ≥ 0, and ℓ ∈ {1, 2, … } ∪ {∞}, and let H be the following sub-Hopf algebra of H_{j}^{n} ⊗ F:

H = F[ξ_{j+1}, … ξ_{j+ℓ}]/(ξ_{i}^{p^{n}} − ξ_{i} : j + ℓ ≥ i ≥ j + 1).

Then the topological linear dual H^{∗} of H is a group algebra. □

For example, this applies when H = B_{j}^{n} ⊗F or H = H_{j}^{n} ⊗F. By “the topological linear dual” of a finite-dimensional Hopf algebra like B_{j}^{n}, we mean its ordinary vector space dual with the discrete topology.

We would like to identify the groups involved. Let G^{n}, G_{j}^{n}, and P_{j}^{n} be the groups so that

F[G^{n}] ≅ (H^{n} ⊗ F)^{∗},
F[G_{j}^{n}] ≅ (H_{j}^{n} ⊗ F)^{∗},
F[P_{j}^{n}] ≅ (B_{j}^{n} ⊗ F)^{∗}.

Of course, G_{0}^{n} = G^{n} since H_{0}^{n} = H^{n}. Also, note that P_{j}^{n} is a finite p-group and G_{j}^{n} is a pro-p group: H_{j}^{n} is the direct limit of the finite-dimensional sub-Hopf algebras

F_{p}[ξ_{j+1}, …, ξ_{j+ℓ}]/(ξ_{i}^{p^{n}} − ξ_{i}),

and this direct system becomes an inverse system of group algebras, induced by an inverse system of groups, after tensoring up with F and dualizing. Proposition 2.3. Fix n ≥ 1 and j ≥ 0.

(a) G^{n} is the group of strict automorphisms of the additive formal group law over F. That is, G^{n} is the group of power series of the form x + ∑_{i≥1} a_{i}x^{p^{i}} with a_{i} ∈ F, under composition.

(b) G_{j}^{n} is the subgroup of G^{n} consisting of such power series with a_{i} = 0 for 1 ≤ i ≤ j.

(c) P_{j}^{n} is the quotient group G^{n}/G_{j}^{n}. That is, P_{j}^{n} is the group of power series of the form in (a), with x^{p^{j+1}} = 0.

Proof. This follows from the well-known observation that A corepresents the strict automorphisms of the additive formal group law for F_{p}-algebras – see [Rav86, p. 378], for example. That is, A is the mod p group scheme for this strict automorphism group.

In more detail: we prove (c) first. B_{j}^{n} is dual to the group algebra of a finite group, and the group is precisely the group of points in the group scheme defined by B_{j}^{n} ⊗ \overline{F}_{p}, where \overline{F}_{p} is the algebraic closure of F_{p}:

P_{j}^{n} = Hom_{\overline{F}_{p}-alg}(B_{j}^{n} ⊗ \overline{F}_{p}, \overline{F}_{p}).

By imitating the analysis for A, one can see that this is the group of power series of the form

x + a_{1}x^{p} + a_{2}x^{p^{2}} + ⋯ + a_{j}x^{p^{j}}

under composition, subject to x^{p^{j+1}} = 0, where a_{i} ∈ \overline{F}_{p} and a_{i}^{p^{n}} = a_{i}. In particular, a_{i} ∈ \overline{F}_{p} is the image of ξ_{i}. The elements a of \overline{F}_{p} satisfying a^{p^{n}} = a form a subfield isomorphic to F, and this completes the proof of part (c).

To prove (a), take limits: since H^{n} is the direct limit of the finite-dimensional sub-Hopf algebras B_{j}^{n}, H^{n} represents the group scheme

Hom_{\overline{F}_{p}-alg}(\underrightarrow{lim}_{j} B_{j}^{n} ⊗ \overline{F}_{p}, −) = \underleftarrow{lim}_{j} Hom_{\overline{F}_{p}-alg}(B_{j}^{n} ⊗ \overline{F}_{p}, −).

G^{n} is the group of \overline{F}_{p}-points in this group scheme, which is as described in (a). The proof of part (b) is similar. □

We end this section with the following result. Proposition 2.4. Fix n ≥ 1 and j ≥ 0. Then G_{j}^{n} is torsion-free.

Proof. Fix α ∈ G_{j}^{n} different from the identity element, so α = x + ∑_{i>j} a_{i}x^{p^{i}} with at least one a_{i} nonzero. Indeed, suppose that a_{m} is the first nonzero coefficient. Then we claim that in α^{p}, the coefficient of x^{p^{pm}} is the first nonzero term. This implies that the elements {α^{p^{i}}|i ≥ 0} are distinct, and so α is not a torsion element.

To verify the claim, we need to know the coefficient of x^{p^{i}} in α^{p}. Since the product in G_{j}^{n} is determined by the coproduct in H_{j}^{n}, to find any coefficient in α^{p}, we use the pth iterated coproduct in H_{j}^{n}, which is defined in Notation 2.5 and a formula for which is given in Lemma 2.6. The pth iterated coproduct on ξ_{pm} has a term

\underbrace{ξ_{m}^{p^{(p−1)m}} ⊗ ⋯ ⊗ ξ_{m}^{p^{2m}} ⊗ ξ_{m}^{p^{m}} ⊗ ξ_{m}}_{p factors},

which produces the nonzero term

a_{m}^{p^{(p−1)m}+p^{(p−2)m}+···+p^{2m}+p^{m}+1}

in the coefficient of x^{p^{pm}}. Since a_{m} is the first nonzero coefficient in α, then only terms in the iterated coproduct having no ξ_{i} factors with 0 < i < m are relevant. So all of the other relevant terms in the pth iterated coproduct of ξ_{pm}, and all of the relevant terms in the pth iterated coproduct of ξ_{k} for k < pm, will have at least one tensor factor equal to 1, and hence will occur with some multiplicity. It is easy to check that the multiplicity is divisible by p – see Lemma 2.7 – and hence is zero in F. □

Notation 2.5. Given a coassociative coalgebra C with coproduct Δ, we let Δ^{2} = Δ, and Δ^{m} = (Δ ⊗ 1^{⊗m−2}) ∘ Δ^{m−1} for m ≥ 3; we call Δ^{m} the “mth iterated coproduct”. This is indexed so that Δ^{m}(x) lies in C^{⊗m}.

The following is immediate from the formula for the coproduct in H_{j}^{n}. Lemma 2.6. For any m ≥ 1, the mth iterated coproduct in H_{j}^{n} of ξ_{ℓ} is

Δ^{m}(ξ_{ℓ}) = ∑_{\begin{matrix}i_{1}+···+i_{m}=ℓ \\ i_{k}=0 or i_{k}>j \end{matrix}} ξ_{i_{m}}^{p^{i_{1}+···+i_{m−1}}} ⊗ ⋯ ⊗ ξ_{i_{3}}^{p^{i_{1}+i_{2}}} ⊗ ξ_{i_{2}}^{p^{i_{1}}} ⊗ ξ_{i_{1}}.

Lemma 2.7. Given an ordered list (b_{1}, …, b_{m}) and a number p ≥ m, then the number of ways of adding p − m 1’s to the list, while preserving the ordering of the b_{i}’s, is \binom{p}{m}. □

The proof is a simple combinatorial argument.

Thus given a pth iterated coproduct with at p − m 1’s in it, it occurs with multiplicity \binom{p}{m}. If p is prime and 0 < m < p, this multiplicity is congruent to zero mod p.

3. Examples and observations

In this section, we collect some miscellaneous observations about the Hopf algebras H_{j}^{n} and the corresponding groups G_{j}^{n}.

Example 3.1 (H^{1}). Let n = 1. In this case, the Hopf algebra H^{1} is bicommutative. Avinoam Mann pointed out (private communication) that the corresponding abelian group G^{1} is isomorphic to the group of power series of the form 1 + ∑ a_{i}x^{i} with multiplication as the group operation. According to [Cam00, p. 218], this group is a free abelian pro-p group of infinite rank. In particular, it is not p-adic analytic, since such groups must have finite rank. Since G^{1} is a subgroup of G^{n} for every n, no G^{n} is p-adic analytic.

One might also note that for any n, the group G^{n} is a subgroup of the Nottingham group N = N (F), which consists of all power series x + ∑_{j≥2} b_{j}x^{j} with b_{j} ∈ F, under composition. Such an observation is almost a triviality, since every finite p-group, and every countably-based pro p-group, can be embedded as a subgroup of N; the group G^{n} is embedded in a particularly nice way, though – it is an “index subgroup”. See [Cam00, Theorems 10 and 11, and pp. 217-8] for more on the Nottingham group. Example 3.2 (P_{j}^{1} and G^{1}). The group P_{j}^{n} corresponding to the Hopf algebra B_{j}^{n} is a finite p-group of order p^{nj}. P_{1}^{n} is abelian for all n; indeed, P_{1}^{n} is isomorphic to the additive group of F = F_{p^{n}}: P_{1}^{n} ≅ (C_{p})^{n}. (We write C_{d} for the cyclic group of order d.) For each j, P_{j}^{1} is abelian of order p^{j}, and one can show that

P_{j}^{1} ≅ ∏_{i≤j, p∤i} C_{α_{p}( \frac{j+1}{i})},

where α_{p}(r) is the smallest power of p greater than or equal to r:

α_{p}(r) = min_{i}{p^{i} : p^{i} ≥ r} = p^{⌈log_{p}(r)⌉}.

In this decomposition, the cyclic summand of order α_{p}(\frac{j+1}{i}) is generated by the power series 1 + x^{i}. Thus in the inverse system

⋯ → P_{j+2}^{1} → P_{j+1}^{1} → P_{j}^{1} → ⋯

defining G^{1}, the cyclic summands at the jth stage are the images from the later stages of cyclic summands of larger and larger orders. Of course this must happen, since G^{1} is torsion-free; in this particular case, though, we can observe it directly. Example 3.3 (A filtration). Note that B_{j}^{n} is a sub-Hopf algebra of B_{j+1}^{n}, and the (conormal) quotient is F_{p}[ξ_{j}]/(ξ_{j}^{p^{n}} − ξ_{j}) with ξ_{j} primitive. This quotient is isomorphic to B_{1}^{n}. Dually, for each j ≥ 1 there is a group extension

1 → (C_{p})^{n} → P_{j+1}^{n} → P_{j}^{n} → 1.

G^{n} is the inverse limit of the groups P_{j}^{n} with respect to these surjections. So the associated graded consists of a copy of P_{1}^{n} = (C_{p})^{n} = F in each degree.

Equivalently, since G_{j}^{n} is the kernel of the map G^{n} → P_{j}^{n}, G_{j}^{n} is normal in G^{n}. Indeed, each G_{j}^{n} is normal in G_{k}^{n} whenever j ≥ k. This gives a filtration of G^{n}:

G^{n} = G_{0}^{n} ⊇ G_{1}^{n} ⊇ G_{2}^{n} ⊇ · · · .

Again, the associated graded is isomorphic to F in each degree.

We also point out that whenever n divides m, F_{p^{n}} is a subfield of F_{p^{m}}, so G^{n} is a subgroup of G^{m}. For the same reason, the colimit of the groups G^{n} is the group of strict automorphisms of the additive formal group over \overline{F}_{p}, which we denote by G^{∞}. One can define G_{j}^{∞} analogously to G_{j}^{n}: G_{j}^{∞} consists of power series in G^{∞} with the first j coefficients equal to zero. Summarizing, we have the following. Proposition 3.4. (a) Fix j ≥ 0 and k, n ≥ 1. Then G_{j}^{n} is a subgroup of G_{j}^{kn}.

Dually, H_{j}^{n} is a quotient of H_{j}^{kn}: the map sending ξ_{i} ∈ H_{j}^{kn} to ξ_{i} ∈ H_{j}^{n} is a surjective Hopf algebra map.

(b) Fix j ≥ 0. Then with respect to the inclusions in part (a),

\underrightarrow{lim}_{n} G_{j}^{n} ≅ G_{j}^{∞}. □

As far as dualizing part (b), see Proposition 5.4 for a partial result.

Note that it is not necessary to tensor up to F_{p^{kn}} coefficients to get the quotient map H_{j}^{kn} ↠ H_{j}^{n}. We also point out that these subgroups are not normal:

Proposition 3.5. Fix n ≥ 1. G^{n} is not normal in G^{kn} if k ≥ 2. Thus H^{n} is not a conormal quotient of H^{kn} if k ≥ 2; that is, H^{kn} □_{H^{n}}F_{p} and F_{p} □_{H^{n}}H^{kn} are different as sub-vector spaces of H^{kn}.

In order to prove this, we need to know how to invert elements of G^{kn}. We recall the formula for the antipode in A from [Mil58]. Let Part(n) denote the set of ordered partitions of n (so, for example, (3, 1) and (1, 3) are distinct elements of Part(4)). Given an ordered partition α ∈ Part(n), write ℓ(α) for the length of α; for i ≤ ℓ(α), let α(i) be the ith term of α, and let σ(i) be the sum of the first i − 1 terms of α.

Lemma 3.6 (Lemma 10 in [Mil58]). The conjugate of ξ_{n} in A is equal to

χ(ξ_{n}) = ∑_{α∈Part(n)}(−1)^{ℓ(α)} ∏_{i=1}^{ℓ(α)} ξ_{α(i)}^{p^{σ(i)}}.

The same formula applies in H_{j}^{n} and B_{j}^{n} for all n and j.

In particular, the summand corresponding to the partition (1, 1, …, 1) is

(−1)^{n}ξ_{1}^{1+p+p^{2}+···+p^{n−1}}.

Proof of Proposition 3.5. Fix n ≥ 1 and k ≥ 2. To see that G^{n} is not normal in G^{kn}, conjugate α = x + x^{p} ∈ G^{1} ≤ G^{n} by β = x + b_{1}x^{p} for some b_{1} ∈ F_{p^{kn}} \ F_{p^{n}}. Using Lemma 3.6 and the observation following it, one can see that

β^{−1} = x + ∑_{i=1}^{∞}(−1)^{i}b_{1}^{1+p+···+p^{i−1}}x^{p^{i}}.

So

αβ^{−1} = β^{−1} + (β^{−1})^{p}

= x + (−b_{1} + 1)x^{p} + ∑_{i=2}^{∞}(−1)^{i}(b_{1}^{1+p+···+p^{i−1}} − b_{1}^{p+p^{2}+···+p^{i−1}})x^{p^{i}}.

So one can compute βαβ^{−1}:

βαβ^{−1} = (αβ^{−1}) + b_{1}(αβ^{−1})^{p}
= x + x^{p} + (−b_{1}^{p} + b_{1}) (x^{p^{2}} − b_{1}x^{p^{3}} + b_{1}^{1+p}x^{p^{4}} − b_{1}^{2+p}x^{p^{5}} + ⋯) .

We are assuming that b_{1} ∉ F_{p^{n}}, so b_{1}^{p} ≠ b_{1}. If the coefficient −b_{1}^{p} + b_{1} of x^{p^{2}} is in F_{p^{n}}, then the coefficient (−b_{1}^{p} +b_{1})b_{1} of x^{p^{3}} is not. So βαβ^{−1} is not in G^{n}, although α is. □

4. The cohomological variety of H^{n} is trivial

Let H^{n} be as above. In this section, we examine the cohomology of H^{n}. By “cohomology,” we mean Ext_{H^{n}}^{∗}(F_{p}, F_{p}), where Ext is computed in the category of H^{n}-comodules. The main result is that every positive dimensional cohomology class is nilpotent.

Since after tensoring up with F_{p^{n}}, the topological linear dual of H^{n} is the group algebra of the profinite group G^{n}, we have the following result, which is analogous to [Rav86, Corollary 6.2.4]. First note that any H^{n}-comodule M may be viewed as an (H^{n})^{∗}-module via the composite

(H^{n})^{∗} ⊗ M → (H^{n})^{∗} ⊗ H^{n} ⊗ M → M.

This makes M ⊗ F_{p^{n}} into a discrete G^{n}-module. Proposition 4.1. There is an isomorphism

Ext_{H^{n}}^{∗}(F_{p}, F_{p}) ⊗ F_{p^{n}} ≅ H_{c}^{∗}(G^{n}, F_{p^{n}}),

where H_{c}^{∗} denotes continuous group cohomology.

(There should be a similar result for cohomology with coefficients in any H^{n}- comodule M, but we are only interested in the case when M is the trivial comodule.)

Proof. Since G^{n} is the inverse limit of the finite groups P_{j}^{n}, H_{c}^{∗}(G^{n}, F_{p^{n}}) may be defined as follows – see [Ser02, I.2.2, Proposition 8]:

H_{c}^{∗}(G^{n}, F_{p^{n}}) = \underrightarrow{lim}_{j} H^{∗}(P_{j}^{n}, F_{p^{n}}).

Since P_{j}^{n} is finite and its group algebra is dual to B_{j}^{n}, we have an isomorphism H^{∗}(P_{j}^{n}, −) ≅ Ext_{B_{j}^{n}⊗F_{p^{n}}}^{∗} (F_{p^{n}}, −). The result follows from the observation that H^{n} = \underrightarrow{lim} B_{j}^{n}. □

We also need the following. Here, ⊗ means ⊗_{k}. Lemma 4.2. Let C be a coaugmented coalgebra over a field k, and let F be a field extension of k. Then there is an isomorphism of algebras Ext_{C⊗F}^{∗} (F, F) ≅ Ext_{C}^{∗}(k, k) ⊗ F.

Proof. Take an injective resolution of k as a C-comodule, and tensor with F. The result is an injective resolution of k ⊗ F = F as a C ⊗ F -comodule. □

Here is the main result of this section. Theorem 4.3. Every positive-dimensional element in Ext_{H^{n}}^{∗}(F_{p}, F_{p}) is nilpotent.

Proof. As above, let F = F_{p^{n}}. H^{n}⊗F is dual to the group algebra F[G^{n}], where G^{n} is the group defined before Proposition 2.3. By Proposition 4.1, Ext_{H^{n}⊗F}^{∗}(F, F) ≅ H_{c}^{∗}(G^{n}; F). Proposition 2.4 says that G^{n} is torsion-free, and so [Qui71, Proposition 13.4] applies, to say that every positive-dimensional element in Ext_{H^{n}⊗F}^{∗}(F, F) is nilpotent. Hence by Lemma 4.2, the same is true for Ext_{H^{n}}^{∗}(F_{p}, F_{p}). □

Given the description of the group G^{1} in Example 3.1, we expect these Ext algebras to be infinitely generated. This contrasts with the situation for Morava stablizer groups, some of which are Poincaré duality groups, and all of which have reasonably well-behaved cohomology: [Rav86, Theorem 6.2.10] says that the cohomology of S^{n} is always finitely generated, and either satisfies Poincaré duality or is periodic.

Question 4.4. When n = 1, the corresponding group is a free abelian pro-p group, and so every element in its cohomology has nilpotence height p. Is there a uniform nilpotence height for the cohomology of H^{n} when n ≥ 2?

5. On the cohomology of the complete Steenrod algebra

As we discussed in the introduction, the results in this paper arose in part as an attempt to study the cohomology of the complete Steenrod algebra. In this section, we explore this algebra, its cohomology, and the relationship to H^{n}.

We start by recalling from [May70] the relationship between the Frobenius map on a commutative Hopf algebra and the Steenrod operation 𝒫̃^{0} on Ext. As usual, Ext is computed in the category of comodules.

Let B be a graded commutative Hopf algebra over F_{p}. By May’s work [May70], one has Steenrod operations acting on Ext_{B}^{∗}(F_{p}, F_{p}). In this paper, we are only

interested in the operation called 𝒫̃^{0} (also called S̃q^{0} when p = 2), which is an operation

𝒫̃^{0} : Ext_{B}^{s,t}(F_{p}, F_{p}) → Ext_{B}^{s,pt}(F_{p}, F_{p}).

Proposition 5.1 (Proposition 11.10 in [May70]). Let B be a graded connected commutative Hopf algebra over the field F_{p}. The Frobenius map ϕ : b ↦ b^{p} is a Hopf algebra map, and the induced map ϕ_{∗} on Ext is the Steenrod operation 𝒫̃^{0} . □

We defined the dual complete Steenrod algebra Â in the introduction:

Â = \underrightarrow{lim} (F_{p}[ξ_{n} : n ≥ 1] → F_{p}[ξ_{n}^{p^{−1}} : n ≥ 1] → F_{p}[ξ_{n}^{p^{−2}} : n ≥ 1] → ⋯) ,

with the coproduct inherited from A. Arnon [Arn94, Arn00] defined the complete Steenrod algebra, and we note that it is dual to Â. (This is also the assertion in [LH96, Definition 3.6].)

Proposition 5.2. The dual Â^{∗} of Â is the complete Steenrod algebra.

Proof. The dual complete Steenrod algebra Â is defined as a direct limit. The complete Steenrod algebra Â^{∗} is defined in [Arn00, p. 185] as the dual inverse limit. □

Â inherits a grading from A: ξ_{n}^{p^{k}} is in degree 2p^{k}(p^{n}−1) when p is odd, 2^{k}(2^{n}−1) when p = 2. Thus Â is a Z[\frac{1}{p}]-graded Hopf algebra. It is not finite-dimensional in each grading, though, so while the monomial basis is a pleasant one for Â, there is no nice basis for its dual. Furthermore, the dual is not graded anymore, strictly speaking. In the language of [LH96, Definition 1.1], Â^{∗} is a “large” Z[\frac{1}{p}]-graded algebra – additively, it is not isomorphic to the direct sum of its homogeneous pieces, but rather is a (proper) subgroup of the product of those pieces.

As far as we can tell, the cohomology of the complete Steenrod algebra has not been studied in any published paper.

Proposition 5.3. The cohomology of Â is isomorphic to the cohomology of A with the operation 𝒫̃^{0} inverted:

Ext∗ ; Â(F_{p}, F_{p}) ≅ (𝒫̃^{0} )^{−1} Ext_{A}^{∗}(F_{p}, F_{p}).

Proof. We may rewrite the direct limit defining Â as follows:

Â = \underrightarrow{lim}(A ⟶^{ϕ} A ⟶^{ϕ} ⋯).

Coalgebra cohomology commutes with direct limits, so take Ext of this directed system:

Ext∗ ; Â(F_{p}, F_{p}) = \underrightarrow{lim} (Ext_{A}^{∗}(F_{p}, F_{p}) ⟶^{ϕ_{∗}} Ext_{A}^{∗}(F_{p}, F_{p}) ⟶^{ϕ_{∗}} ⋯) .

Proposition 5.1 says that ϕ_{∗} = 𝒫̃^{0} , and this finishes the proof. □

As mentioned in the introduction, much of the work in this paper was motivated, directly or indirectly, by Conjecture 1.4, which is equivalent to the statement that every positive degree element in Ext∗ ; Â(F_{p}, F_{p}) is nilpotent. Theorem 1.2 says that this is true when p = 2.

For example, let p = 2. There is a family of cohomology classes {h_{1,k} : k ∈ Z} in the cohomology of Â, represented in the cobar complex by [ξ_{1}^{2^{k}}]. The classes with k ≥ 0 lift to classes of the same name in Ext_{A}^{∗}(F_{2}, F_{2}), and in the cohomology of A, h_{1,0} is non-nilpotent – its powers produce the spike in the zero stem at the E_{2}-term of the Adams spectral sequence – while h_{1,k} is nilpotent when k ≥ 1. In particular, h_{1,k}^{4} = 0 for all k ≥ 1. These classes are connected by the algebra map

S̃q^{0}: S̃q^{0}(h_{1,k}) = h_{1,k+1}, so if h_{1,k} is nilpotent, so is h_{1,k+1}. In Ext∗ ; Â(F_{2}, F_{2}), S̃q^{0} has been inverted, so the relation h_{1,1}^{4} = 0 implies that h_{1,k}^{4} = 0 for all k ∈ Z.

The multiplicative structure in the cohomology of the polynomial part of the odd primary Steenrod algebra is less well-understood. For example, there are classes h_{1,k} ∈ Ext_{A}^{1}(F_{p}, F_{p}), for k ≥ 0, represented in the cobar complex by [ξ_{1}^{p^{k}}]. These classes are odd-dimensional, so by graded-commutativity of Ext, they square to zero. There are related classes b_{1,k} ∈ Ext^{2}, though, which can be obtained by applying the Bockstein operation to h_{1,k}; equivalently, b_{1,k} is represented in the cobar complex by

∑_{i=1}^{p−1} \frac{1}{p}\binom{p}{i}[ξ_{1}^{ip^{k}}|ξ_{1}^{(n−i)p^{k}}].

One can show that b_{1,0} is non-nilpotent, in analogy with h_{1,0} at the prime 2. The nilpotence (or lack thereof) of b_{1,k} for k > 0 has not been determined in general; a result of Nakamura [Nak75, Proposition 1.1(c)] shows that these classes are all nilpotent when p = 3, and this is the best known result in this direction.

Given Conjecture 1.4, one could conclude that b_{1,k} is nilpotent in Ext∗ ; Â(F_{p}, F_{p}), and hence b_{1,k} is nilpotent in Ext_{A}^{∗}(F_{p}, F_{p}) for sufficiently large k. Then [Pal01, Lemma B.3.3] would imply that b_{1,k} is nilpotent for all k ≥ 1. In general, a positive solution to Conjecture 1.4 would be a helpful step in trying to prove odd primary analogues of the nilpotence and F -isomorphism results in [Pal99]. See [Pal01, Conjecture 5.4.1] and the surrounding discussion.

We end this section by exploring the connection between the dual complete Steenrod algebra Â and the Hopf algebras H^{n}, with an eye toward Conjecture 1.4. The philosophy is that, while Â is defined by formally inverting the Frobenius ϕ, H^{n} is defined by forcing ϕ^{n} to be the identity. At the level of cohomology, 𝒫̃^{0} is invertible of infinite order in Ext∗ ; Â(F_{p}, F_{p}), and it is invertible of order n in Ext_{H^{n}}^{∗}(F_{p}, F_{p}). So one can hope that knowing that every positive-dimensional element in Ext_{H^{n}}^{∗}(F_{p}, F_{p}) is nilpotent would be relevant to studying nilpotence of classes in Ext∗ ; Â(F_{p}, F_{p}).

More formally, we start by noting that one can generalize the definition of Â; in keeping with the definition and notation for H_{j}^{n} from Section 2, for any j ≥ 0 we let

Â_{j} = \underrightarrow{lim} (F_{p}[ξ_{n} : n ≥ j + 1] → F_{p}[ξ_{n}^{p^{−1}} : n ≥ j + 1] → F_{p}[ξ_{n}^{p^{−2}} : n ≥ j + 1] → ⋯) .

Thus Â_{0} = Â, and Â_{j} is a conormal quotient Hopf algebra of Â_{k} whenever j ≥ k. Note that for each n and j, one can view H_{j}^{n} as being a quotient of Â_{j}:

H_{j}^{n} = Â_{j}/(ξ_{i}^{p^{n+k}} − ξ_{i}^{p^{k}} : i ≥ 1, k ∈ Z).

Also, as noted in Proposition 3.4, H_{j}^{n} is a quotient of H_{j}^{m} whenever n divides m. Thus for fixed j, the Hopf algebras H_{j}^{n} form an inverse system.

Proposition 5.4. Fix j ≥ 0. Â_{j} embeds in the inverse limit of the Hopf algebras H_{j}^{n}. This embedding is not an isomorphism.

Proof. The quotient maps Â_{j} ↠ H_{j}^{n} induce a map from Â_{j} to the inverse limit, and one can see that the kernel of the map to the limit is zero.

Let H_{j}^{∞} be the inverse limit. To see that Â_{j} is not isomorphic to H_{j}^{∞}, we note that there are elements in H_{j}^{∞} of unipotence height p: elements x different from 1 satisfying x^{p} = x. To construct these elements, it suffices to construct a family of such elements, one in each H_{j}^{n}, mapping to each other in the inverse system. The relevant elements are ξ_{i}^{p^{n}−1}, for each i > j – computations show that (ξ_{i}^{p^{n}−1})^{p} = ξ_{i}^{p^{n}−1}, and that ξ_{i}^{p^{nd}−1} in H^{nd} maps to ξ_{i}^{p^{n}−1} in H^{n}. □

Question 5.5. (a) What does this imply about the cohomology of Â? (b) What is the cokernel of the map Â_{j} ↪ \underleftarrow{lim} H_{j}^{n}?

(c) How are H_{j}^{∞} and G_{j}^{∞} (from Proposition 3.4) related? How are Â_{j} and G_{j}^{∞} related?

Appendix A. Proof of Theorem 1.2

We restate the theorem. Theorem 1.2. Let p = 2 and fix z ∈ Ext_{A}^{s,t}(F_{2}, F_{2}). If s is positive, then there is

an n so that (S̃q^{0})^{n}(z) is nilpotent.

Proof. This follows from the main result of [Pal99].

In detail, for each pair of integers (S, T) with 0 ≤ S < T, let R_{T,S} be the polynomial algebra

R_{T,S} = F_{2}[h_{ts} : 0 ≤ s ≤ S < T ≤ t].

Let R be the inverse limit of the R_{T,S} under the apparent maps (map each polynomial generator to the generator of the same name if present, and to zero otherwise). Then

R = F_{2}[h_{ts} : s < t]/(h_{ts}h_{vu} : u ≥ t).

Also, R_{n,n−1} is the cohomology of this quotient Hopf algebra of A:

E(n) = A/(ξ_{1}, …, ξ_{n−1}, ξ_{i}^{2^{n}} : i ≥ n),

with the polynomial generator h_{ts} corresponding to the cobar element [ξ_{t}^{2^{s}}]. So there is a restriction map

ρ^{n} : Ext_{A}^{∗}(F_{2}, F_{2}) → R_{n,n−1}.

These can be assembled to give a map

ρ : Ext_{A}^{∗}(F_{2}, F_{2}) → R,

and Theorems 1.3 and 1.4 of [Pal99] say that up to nilpotence, ρ is an isomorphism between the cohomology of A and a certain subring of the codomain. In particular, ρ is a monomorphism mod nilpotence.

Since ρ is the inverse limit of the maps ρ^{n}, and since each ρ^{n} is a restriction map, ρ commutes with the action of S̃q^{0}. Hence there is an induced map

ρ : (S̃q^{0})^{−1} Ext_{A}^{∗}(F_{2}, F_{2}) → (S̃q^{0})^{−1}R,

which is a monomorphism mod nilpotence. But the codomain is zero – the Frobenius map acts nilpotently on each element of the Hopf algebra E(n). Equivalently, at the level of cohomology, there is a multiplicative action of S̃q^{0} on R, defined by S̃q^{0}(h_{ts}) = h_{t,s+1} if s + 1 < t, but S̃q^{0}(h_{t,t−1}) = 0. Thus for each element y ∈ R,

(S̃q^{0})^{n}(y) = 0 for n sufficiently large. □

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Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350

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