MORAVA E-THEORY OF SYMMETRIC GROUPS

N. P. STRICKLAND

1. Introduction

Fix a prime p and an integer n > 0. In this paper we study the { \atop E(n)}-cohomology of the space

DS^{0} = ⋁_{k≥0} BΣ_{k+}.

It will be technically convenient to work with a certain 2-periodic algebraic extension of { \atop E(n)} called Morava E-theory, which we denote by E. See section 2 for details.

It will turn out that E^{0}DS^{0} is related to the theory of subgroups of formal groups. This general theme has its roots in work of Mike Hopkins and Matthew Ando [1]. We will also make heavy use of results of Takuji Kashiwabara [11], and the generalised character theory of Hopkins, Kuhn and Ravenel [8].

Hopkins has proved that E can be made into an E_{∞} ring spectrum, but unfortunately no proof has yet appeared. Our calculation of E^{0}DS^{0} is an important first step towards the exploitation of the E_{∞} structure, as will be explained in a sequel to this paper. We have avoided using the E_{∞} structure here, because of the lack of a published account.

The stable transfer maps BΣ_{k+l} ⟶ BΣ_{k} ×BΣ_{l} and the diagonal maps BΣ_{k} ⟶ BΣ_{k} ×BΣ_{k} give two products on E^{0}DS^{0}, written × and •. The evident maps BΣ_{k} × BΣ_{l} ⟶ BΣ_{k+l} give rise to a coproduct ψ_{∗} on E^{0}DS^{0} (provided that we use a completed tensor product). We will often use the usual permutation representation V_{m} of Σ_{m} on ℂ^{m}, and the Euler class c_{m} = e(V_{m} −1) ∈ E^{0}BΣ_{m}.

The following theorem summarises most of our results.

Theorem 1.1. With the above structure, E^{0}DS^{0} is a Hopf ring. It is a formal power series ring under ×. The ×-indecomposables are

Ind(E^{0}DS^{0}) = ∏_{k≥0} \overline{R}_{k},

where

\overline{R}_{k} = E^{0}BΣ_{p^{k}}/ transfer(E^{0}BΣ_{p^{k−1}}^{p}).

This ring is naturally isomorphic to the ring O_{Sub_{k}(𝔾)} studied in [15], which classifies subgroupschemes of degree p^{k} in the formal group 𝔾 associated to E^{0}ℂP^{∞}. It is a Gorenstein local ring, and a free module over E^{0}, with rank given by the Gaussian binomial coefficient

[ \begin{matrix}n + k − 1 \\ n − 1 \end{matrix} ]_{p} = ∏_{j=1}^{n−1} \frac{p^{k+j} − 1}{p^{j} − 1} = |\{ subgroups A < (ℚ_{p}/ℤ_{p})^{n} with |A| = p^{k}\}|.

In Section 10, we will give an explicit basis consisting of monomials in the Chern classes of V_{p^{k}}. We also have

Prim(E^{0}DS^{0}) = ∏_{k} ker(res: E^{0}BΣ_{p^{k}} ⟶ E^{0}BΣ_{p^{k−1}}^{p}).

This is a free module over Ind(E^{0}DS^{0}) on one generator c, whose component in E^{0}BΣ_{m} is the Euler class c_{m} = e(V_{m} − 1).

We now give a brief outline of the methods used to prove this. Firstly, we appeal to the results of [16]. There, we showed that the maps of spectra that we used to define the products and coproduct on E^{0}DS^{0} interact in the right way to make E^{0}DS^{0} a Hopf ring; this is explained in Section 4. Also, following work of Hopkins, Kuhn and Ravenel [8], we defined an E^{0}-algebra L which is a free module over p^{−1}E^{0}, and we gave a complete description of L ⊗_{E} E^{0}BΣ_{k}. This essentially tells us everything we need to know about p^{−1}E^{0}BΣ_{k}. It is also known that K^{odd}BΣ_{k} = 0 (see [8, Section 6] or [10]). We recall the proof in Section 3, because we need to extract from it the fact that c_{p^{k}}^{(p^{n}−1)/(p−1)} ≠ 0 in K^{0}BΣ_{p^{k}}, which turns out to be crucial later on. It is not hard to deduce from the fact that K^{odd}BΣ_{k} = 0 that E^{∗}BΣ_{k} is a finitely generated free module over E^{∗}, concentrated in even degrees. This of course means that the loss of information in passing from E^{0}BΣ_{k} to p^{−1}E^{0}BΣ_{k} is relatively small.

In Section 5 we recall Kashiwabara’s result [11, Corollary 6.4] that K_{0}DS^{0} is a polynomial ring under the product dual to ψ_{∗}, and we give a simplified proof. It follows that E^{∨}DS^{0} is a completed polynomial ring over E^{0}, and thus that Ind(E^{∨}DS^{0}) is a completed free module and a summand in E^{∨}DS^{0}. Dually, we conclude that Prim(E^{0}DS^{0}) is a product of free modules and a summand in E^{0}DS^{0}.

We next consider E^{0}QS^{2}. There is only one sensible product on E^{0}QS^{2}, which comes from the diagonal map QS^{2} ⟶ QS^{2} × QS^{2}, which is also the map obtained by applying Q to the pinch map S^{2} ⟶ S^{2} ∨ S^{2}. We also prove in Section 5 that E^{0}QS^{2} is a formal power series ring under this product, using an unpublished argument of Kashiwabara.

Our next task is to transfer this result to E^{0}DS^{0}, using the Snaith splitting Σ^{∞}QS^{2} = Σ^{∞}DS^{2} and the Thom isomorphism Ẽ^{0}D_{k}(S^{2}) = Ẽ^{0}BΣ_{k}^{V_{k}} = E^{0}BΣ_{k}. We need a property of the Snaith splitting that does not seem to appear in the literature; in order to prove it, we recall the details of the construction in Section 6. We find that our isomorphism converts the product on E^{0}QS^{2} to the ×-product on E^{0}DS^{0}, so we conclude that E^{0}DS^{0} is a formal power series ring under the ×-product. It follows from this that Ind(E^{0}DS^{0}) is a product of free modules over E^{0}, and a retract of E^{0}DS^{0}.

In Section 8, we study the indecomposables and primitives more closely. We distinguish between two different kinds of primitives, and we define indecomposables and primitives in more elementary terms. We find that the double suspension map E^{∗}QS^{2} ⟶ E^{∗}QS^{0} corresponds to multiplication by c_{k} = e(V_{k} −1) in E^{0}BΣ_{k}. This implies that Prim(E^{0}DS^{0}) is free of rank one over Ind(E^{0}DS^{0}).

In Section 9, we use generalised character theory to show that the representation V_{p^{k}} defines a subgroup scheme of the formal group 𝔾 over the scheme spf(\overline{R}_{k}). This gives us a classifying map O_{Sub_{k}(𝔾)} ⟶ \overline{R}_{k}, and we prove that this is an isomorphism. All this makes heavy use of [15].

In Section 10, we give a basis for O_{Sub_{k}(𝔾)}, consisting of monomials in the Chern classes of V_{p^{k}}. This is a piece of pure algebra that should really have appeared in [15].

2. Morava K-theory and E-theory

In this section, we give more precise definitions of the (co)homology theories which we need. We will consider a completed and extended version of E(n), which we call Morava E-theory.

We start with the graded ring

E_{∗} = W 𝔽_{p^{n}}[[u_{1}, …, u_{n−1}]][u^{±1}]

(where u_{i} ∈ E_{0} and u ∈ E_{2}). We take u_{0} = p, u_{n} = 1 and u_{k} = 0 for k > n. We make E_{∗} into an algebra over BP_{∗} by sending v_{k} to u^{p^{k}−1}u_{k}. This makes E_{∗} into a Landweber exact BP_{∗}-module, so we have a homology theory

E_{∗}X = E_{∗} ⊗_{BP_{∗}} BP_{∗}X = E_{∗} ⊗_{MU_{∗}} MU_{∗}X.

This is represented by a spectrum E.

There is also a ring spectrum K, with

K_{∗} = E_{∗}/(u_{0}, …, u_{n−1}) = 𝔽_{p^{n}}[u^{±1}] = 𝔽_{p^{n}} ⊗ K(n)_{∗}[u]/(u^{p^{n}−1} − v_{n}).

We shall call this the Morava K-theory spectrum. There is a ring map E ⟶ K with the obvious effect on homotopy groups. If p > 2 then K is commutative. If p = 2 then there is a well-known formula

ab − ba = uQ_{n−1}(a)Q_{n−1}(b),

where Q_{n−1} : K ⟶ ΣK is a derivation. It follows that K^{0}X is commutative whenever X is a space with K^{1}X = 0. This will be the case for all spaces that we consider.

We also define L̂X = L_{K(n)}X = L_{K}X, and

E^{∨}(X) = π_{0}L̂(E ∧ X).

This is a more natural thing to consider than E_{0}X when X is K-local. In that case E_{0}X is analogous to something like ℤ_{p} ⊗ ℤ_{p}, which contains a large divisible subgroup, whereas E^{∨}X is analogous to the p-completion of ℤ_{p} ⊗ ℤ_{p}, which is just ℤ_{p}.

Because E^{∗} is periodic, we can choose a degree-zero complex orientation x ∈ Ẽ^{0}ℂP^{∞}. Using this, we define Thom classes, Euler classes and Chern classes for complex bundles, in the usual way (all of them in degree zero).

3. E-Theory of symmetric groups

In this section, we approach the calculation of E^{∗}BΣ_{k} using fairly traditional methods. We begin with some definitions.

Definition 3.1. Let V_{k} denote the space ℂ^{k} with the evident action of Σ_{k}, or the resulting bundle EΣ_{k} ×_{Σ_{k}} V_{k} over BΣ_{k}. Note that V_{k} has a one-dimensional trivial summand, so that V_{k} − 1 is an honest bundle. We let c_{k} ∈ E^{0}(BΣ_{k}) be the Euler class of V_{k} − 1, and we let c ∈ E^{0}DS^{0} = ∏_{k} E^{0}BΣ_{k} be the element with k’th component c_{k}.

Definition 3.2. Define Λ = (ℚ_{p}/ℤ_{p})^{n}, and Λ^{∗} = Hom(Λ, ℚ_{p}/ℤ_{p}) = ℤ_{p}^{n}. Let 𝔹_{k} be the set of isomorphism classes of sets of order k with an action of Λ^{∗}, and write d(k) = |𝔹_{k}|. Let 𝕃_{k} be the set of lattices of index p^{k} in Λ^{∗}, which bijects with the set of transitive Λ^{∗}-sets of order p^{k}, or with the set of subgroups of Λ of order p^{k}. Write \overline{d}(k) = |𝕃_{k}|. It is shown in [15, Section 10.1] that

\overline{d}(k) = [ \begin{matrix}n + k − 1 \\ n − 1 \end{matrix} ]_{p} = ∏_{j=1}^{n−1} \frac{p^{k+j} − 1}{p^{j} − 1}.

Our conclusions are as follows.

Theorem 3.3. E^{0}BΣ_{k} is a Noetherian local ring and a free module over E^{0} of rank d(k). Moreover, we have

E^{∨}BΣ_{k} = Hom_{E^{0}}(E^{0}BΣ_{k}, E^{0})
K^{0}BΣ_{k} = 𝔽_{p^{n}} ⊗_{E} E^{0}BΣ_{k}
K_{0}BΣ_{k} = Hom_{𝔽_{p^{n}}} (K^{0}BΣ_{k}, 𝔽_{p^{n}})
E^{1}BΣ_{k} = E_{1}^{∨}BΣ_{k} = K^{1}BΣ_{k} = K_{1}BΣ_{k} = 0.

Finally, if k = p^{m} then c_{k}^{(p^{n}−1)/(p−1)} is nonzero in K^{0}BΣ_{k} (this will be important later).

Proof. Assemble Proposition 3.5, Proposition 3.7, and Corollary 3.9.

Definition 3.4. Let W_{m} be the m-fold iterated wreath product C_{p} ≀…≀C_{p}. It is well-known that this is the Sylow p-subgroup of Σ_{p^{m}}, and that for any k the Sylow subgroup of Σ_{k} is a product of W_{m}’s.

Proposition 3.5. The group K^{∗}BW_{m} is generated by transfers of Euler classes of complex representations of subgroups, and thus is concentrated in even degrees. Moreover, the element b_{m} = c_{p^{m}}^{(p^{n}−1)/(p−1)} is nonzero.

Proof. The first sentence is proved in [8]; we need to recall the details in order to prove the second sentence.

Suppose that the proposition holds for W_{m}, this being trivial for m = 0. We can then choose a basis {e_{1}, …, e_{d}} for K^{0}BW_{m} over 𝔽_{p^{n}}, where e_{1} = b_{m} and each e_{k} is a transferred Chern class. There is a spectral sequence

H^{∗}(C_{p}; K^{∗}(BW_{m})^{⊗p}) ⇒ K^{∗}(BW_{m+1}).

Let z be a generator of H^{1}(C_{p}; 𝔽_{p}). Write y = βz ∈ H^{2}(C_{p}; 𝔽_{p}), so that y is the Euler class of the representation L of W_{m+1} associated to the character W_{m+1} ⟶ C_{p} ⊂ S^{1}. Let x ∈ K^{0}BW_{m+1} be the Morava K-theory Euler class for the same representation.

Given a sequence of indices I = (i_{0} ≤ … ≤ i_{p−1}) with i_{0} < i_{p−1}, let e_{I} be the sum of the orbit under C_{p} of e_{i_{0}} ⊗ … ⊗ e_{i_{p−1}}. We also write e′_{i} = e_{i} ⊗ … ⊗ e_{i}. The E_{2} term is

E[z] ⊗ P [y] ⊗ 𝔽_{p}{e_{I}, e′_{i}}/(xe_{I}, ye_{I}).

We shall exhibit elements ẽ_{I} and ẽ′_{i} in K^{0}BW_{m+1} that hit e_{I} and e′_{i} under the restriction map K^{∗}BW_{m+1} ⟶ BW_{m}^{p}, which is the edge map of the spectral sequence. First, we take

ẽ_{I} = tr_{W_{m}^{p}}^{W_{m+1}}(e_{i_{0}} ⊗ … ⊗ e_{i_{p−1}}) ∈ K^{0}BW_{m+1}.

It is easy to check that this hits e_{I}, and also that xẽ_{I} = 0 = yẽ_{I}. Next, suppose that e_{i} = tr_{H}^{W_{m}}(e(V)) for some complex representation V of some subgroup H ≤ W_{m}. There is an obvious way to extend the action of H^{p} on p ⊗ V to get an action of C_{p} ≀ H. Write

ẽ′_{i} = tr_{C_{p}≀H}^{W_{m+1}}(e(p ⊗ V)) ∈ K^{0}(BW).

This hits e′_{i} under the edge map.

Our spectral sequence is a module over the Atiyah-Hirzebruch spectral sequence

H^{∗}(BC_{p}; K^{∗}) = E[z] ⊗ P [y] ⇒ K^{∗}BC_{p} = P [x]/x^{p^{n}}.

This has only one differential, viz. d_{p^{n}−1}(z) = y^{p^{n}}. We conclude that d_{p^{n}−1}(ze′_{i}) = y^{p^{n}}e′_{i}, and after this differential the spectral sequence is concentrated in even bidegrees so nothing else can happen. Thus, we have

K^{∗}BW_{m+1} = K^{∗}{ẽ_{I}, x^{j}ẽ′_{i}},

where j runs from 0 to p^{n} − 1.

We still need to show that b_{m+1} is nonzero. In fact, we will show that it is a unit multiple of x^{p^{n}−1}ẽ′_{1}.

Let U be the regular representation of C_{p} regarded as a representation of W_{m+1} in the obvious way. Note that U ≃ ⨁_{r=0}^{p−1} L^{⊗r}. For 0 < r < p we have e(L^{⊗r}) = [r](x), which is a unit multiple of x. It follows that x^{p−1} is a unit multiple of e(U − 1), and thus that x^{p^{n}−1} is a unit multiple of e(U − 1)^{N} = e(N ⊗ (U − 1)), where N = (p^{n} − 1)/(p − 1).

Next, recall that b_{m} = e(V_{p^{m}} − 1)^{N} = e(N ⊗ V_{p^{m}} − N). If we regard p ⊗ (N ⊗ V_{p^{m}} − N) as a representation of W_{m+1}, it becomes isomorphic to N ⊗ V_{p^{m+1}} − N ⊗ U, so we have ẽ′_{1} = e(V_{p^{m+1}} −U)^{N}. It follows that x^{p^{n}−1}ẽ′_{1} is a unit multiple of e(V_{p^{m+1}} −1)^{N} = b_{m+1}, as required.

Proposition 3.6. If Z is a spectrum such that E^{∗}Z is finitely generated over E^{∗} and K^{∗}Z is concentrated in even degrees then E^{∗}Z is free and concentrated in even degrees.

Proof. One can construct ring spectra E/I_{k} (for k > 0) with

π_{∗}(E/I_{k}) = E_{∗}/I_{k} = 𝔽_{p^{n}}[[u_{k}, …, u_{n−1}]][u^{±1}],

and cofibrations E/I_{k} ⟶^{u_{k}} E/I_{k} ⟶ E/I_{k+1}. In particular, we have E/I_{n} = K, and we put E/I_{0} = E. The cofibrations give short exact sequences

(E/I_{k})^{∗}(X)/u_{k} ↣ (E/I_{k+1})^{∗}(X) ↠ ann(u_{k}, (E/I_{k})^{∗}(Σ^{−1}X)).

By induction on k, we see that (E/I_{k})^{∗}X is finitely generated for all k. We claim that it is actually free, and concentrated in even degrees. This is trivially true for (E/I_{n})^{∗}X. If it is true for E/I_{k+1}, we see from the short exact sequence that (E/I_{k})^{odd}(X)/u_{k} = 0, so (E/I_{k})^{odd}(X) = 0 by Nakayama’s lemma. We also see that ann(u_{k}, (E/I_{k})^{even}(X)) = 0, so u_{k} acts regularly on (E/I_{k})^{∗}X. Finally, we see that (E/I_{k+1})^{∗}X = (E/I_{k})^{∗}(X)/u_{k}. It follows by elementary commutative algebra that (E/I_{k})^{∗}(X) is free, as required.

Proposition 3.7. E^{∗}BΣ_{k} is a free module of rank d(k) over E^{∗}, concentrated in even degrees.

Proof. Because the Sylow subgroup of BΣ_{k} is a product of W_{m}’s, we know from Proposition 3.5 and a transfer argument that K^{∗}BΣ_{k} is concentrated in even degrees. We also know from [7, Corollary 5.4] (for example) that E^{∗}BG is finitely generated as an E^{∗}-module, for any finite group G. It follows using Proposition 3.6 that E^{∗}BΣ_{k} is free and concentrated in even degrees. The rank is necessarily the same as the rank of p^{−1}E^{∗}BΣ_{k} over p^{−1}E^{∗}, and the main theorem of [8] shows that this is the same as the number of conjugacy classes of homomorphisms Λ^{∗} = ℤ_{p}^{n} ⟶ BΣ_{k}. (A version of this theorem is proved in [7, Appendix A], which may be easier to get hold of than [8].) It is easily seen that these biject with isomorphism classes of Λ^{∗}-sets of order k, so the rank is d(k). This is explained in more detail in [16].

It turns out that whenever E^{∗}Z is a finitely generated free module over E^{∗}, the module E_{∗}^{∨}Z is simply the dual of E^{∗}Z. However, we shall avoid having to prove this by giving a special result for classifying spaces.

Proposition 3.8. If G is a finite group then there is a weak equivalence

L̂(E ∧ BG_{+}) = F (BG_{+}, E),

and thus an isomorphism E^{∗}BG = E_{∗}^{∨}BG.

Proof. Let X be a spectrum. The Greenlees-May theory of Tate spectra [4] gives a cofibration of G-spectra

EG_{+} ∧ X ⟶ F (EG_{+}, X) ⟶ t_{G}(X).

We now apply the Lewis-May fixed point functor and use Adams’ isomorphism (EG_{+} ∧ X)^{G} = EG_{+} ∧_{G} X (see [12, Theorem 7.1]). This gives a nonequivariant cofibration

BG_{+} ∧ X ⟶ F (BG_{+}, X) ⟶ P_{G}(X) = t_{G}(X)^{G},

and thus a cofibration

L̂(BG_{+} ∧ X) ⟶ L̂F (BG_{+}, X) ⟶ L̂P_{G}(X).

Moreover, all three terms here are exact functors of X. Let Y be a generalised Moore spectrum of type n, so that E ∧ Y lies in the thick subcategory generated by K. It is shown in [6, 5] that P_{G}K = 0, so we must have Y ∧ P_{G}E = 0. As K_{∗}Y ≠ 0 we conclude that K_{∗}P_{G}E = 0 and thus L̂P_{G}E = 0. Note also that E is K-local, and thus the same is true of F (BG_{+}, E). We thus conclude that L̂(BG_{+} ∧ E) = F (BG_{+}, E), as claimed.

Corollary 3.9. If E^{∗}BG is free then E_{∗}^{∨}BG is its dual, and K^{∗}BG = K^{∗} ⊗_{E^{∗}} E^{∗}BG.

Proof. Proposition 3.8 tells us that E_{∗}^{∨}BG is free over E^{∗}. The cofibrations in the proof of Proposition 3.6 show us that K^{∗}BG = K^{∗} ⊗_{E^{∗}} E^{∗}BG and K_{∗}BG = K_{∗} ⊗_{E_{∗}} E_{∗}^{∨}BG. Both of these are finitely generated over the graded field K^{∗}, so they are dual to each other. Thus, the natural map E_{∗}^{∨}BG ⟶ Hom_{E^{∗}}(E^{∗}BG, E^{∗}) becomes an isomorphism after tensoring with K^{∗}. It is therefore an isomorphism.

4. The extended power functor

We next recall some of the ideas developed in [16]. For any based space X, we can define the total extended power

DX = ⋁_{k≥0} EΣ_{k+} ∧_{Σ_{k}} X^{(k)}.

In [12], it is shown how to define DX when X is a spectrum, so that D(Σ^{∞}X) = Σ^{∞}DX. We are mainly interested in DS^{0} = ⋁_{k≥0} BΣ_{k+}. We will sometimes think of this as a space, and sometimes as a spectrum. It can also be described as Σ^{∞}BC_{+}, where C is the category of finite sets. We observed in [16] that it admits two products and two coproducts:

\begin{matrix}σ: & DS^{0} ∧ DS^{0} ⟶ DS^{0} \\ μ: & DS^{0} ∧ DS^{0} ⟶ DS^{0} \\ θ: & DS^{0} ⟶ DS^{0} ∧ DS^{0} \\ δ: & DS^{0} ⟶ DS^{0} ∧ DS^{0}. \end{matrix}

The map σ is obtained by applying D to the fold map S^{0} ∨ S^{0} ⟶ S^{0}, or by applying B to the coproduct functor ⨿: C^{2} ⟶ C. The map μ is obtained by applying B to the product functor ×: C^{2} ⟶ C. The map θ only exists stably. It is obtained by applying D to the stable pinch map S^{0} ⟶ S^{0} ∨ S^{0}, or by taking the transfer associated to the covering map σ. Finally, δ is just the diagonal map of the space BC. The interaction between these maps is studied in detail in [16].

In order to transfer our results about E_{∗}^{∨}BΣ_{k} to DS^{0}, we need the following lemma.

Lemma 4.1. Let M be an E-module spectrum such that the sequence {u_{0}, …, u_{n−1}} is regular on π_{∗}M. Then π_{∗}L̂M = π_{∗}(M)_{I_{n}}^{∧}.

Proof. By [9, Theorem 2.1 and Corollary 2.2], we see that L̂M is the homotopy inverse limit of spectra S/I ∧ L_{n}M, as S/I runs over a suitable tower of generalised Moore spectra of type n. As M is a module over the E(n)-local spectrum E, we see that L_{n}M = M. If I = (v_{0}^{a_{0}}, …, v_{n−1}^{a_{n−1}}), then regularity implies that π_{∗}(S/I ∧ M) = π_{∗}(M)/I. The Milnor sequence now implies that π_{∗}L̂M = π_{∗}(M)_{I_{n}}^{∧}, as claimed.

In particular, the lemma applies when π_{∗}(M) is a free module over E_{∗}. To understand what it says in that case, observe that the completion of ⨁_{k} E^{0} is the set of sequences \underline{a} ∈ ∏_{k} E^{0} such that a_{k} ⟶ 0 in the I-adic topology as k ⟶ ∞.

It is now easy to deduce the following.

Proposition 4.2. E^{0}DS^{0} and E^{0}(DS^{0} ∧ DS^{0}) are products of countably many copies of E^{0}, and E^{0}(DS^{0} ∧ DS^{0}) = E^{0}DS^{0} ˆ⊗_{E}E^{0}DS^{0}. Also, E^{∨}DS^{0} and E^{∨}(DS^{0} ∧ DS^{0}) are completions of countably generated free modules over E^{0}, and E^{∨}(DS^{0} ∧ DS^{0}) = E^{∨}DS^{0} ˆ⊗_{E}E^{∨}DS^{0}. Moreover, we have E^{0}DS^{0} = Hom_{E}(E^{∨}DS^{0}, E^{0}).

We therefore get (co)products as follows:

∗: E^{∨}DS^{0} ˆ⊗_{E}E^{∨}DS^{0} ⟶ E^{∨}DS^{0} induced by σ
ψ_{×}: E^{∨}DS^{0} ⟶ E^{∨}DS^{0} ˆ⊗_{E}E^{∨}DS^{0} induced by θ
ψ_{•}: E^{∨}DS^{0} ⟶ E^{∨}DS^{0} ˆ⊗_{E}E^{∨}DS^{0} induced by δ
ψ_{∗}: E^{0}DS^{0} ⟶ E^{0}DS^{0} ˆ⊗_{E}E^{0}DS^{0} induced by σ
×: E^{0}DS^{0} ˆ⊗_{E}E^{0}DS^{0} ⟶ E^{0}DS^{0} induced by θ
•: E^{0}DS^{0} ˆ⊗_{E}E^{0}DS^{0} ⟶ E^{0}DS^{0} induced by δ

Theorem 4.3. E^{0}DS^{0} is a Hopf ring with products × and •, and coproduct ψ_{∗}.

Proof. We need various diagrams of cohomology groups to commute. Each one is obtained by applying E^{0} to an obvious diagram of spectra. These diagrams are proved to commute in [16, Theorem 3.2].

5. Morava homology of DS^{0}

Kashiwabara proved in [11] that K_{0}DS^{0} is polynomial under ∗. It seems worthwhile to give the following simplification of part of his argument.

Proposition 5.1 (Kashiwabara). K_{0}DS^{0} is a polynomial algebra under ∗.

Proof. We can make K_{0}DS^{0} into a graded vector space by putting K_{0}BΣ_{m} in degree m. Because σ sends BΣ_{k} × BΣ_{l} to BΣ_{k+l} and θ sends BΣ_{m} to ⋁_{m=k+l}(BΣ_{k} × BΣ_{l})_{+}, we see that K_{0}DS^{0} becomes a graded Hopf algebra over 𝔽_{p^{n}} using ∗ and ψ_{×}. It is clearly connected, in the sense that the degree zero part is just 𝔽_{p^{n}}. It follows from Theorem 3.3 that K_{0}DS^{0} has finite type. It follows in turn from a theorem of Borel that K_{0}DS^{0} is isomorphic as a graded ring to a tensor product of polynomial and truncated polynomial algebras. The proposition will therefore follow if we show that K_{0}DS^{0} has no nontrivial nilpotent elements.

To see this, recall that we have a group-completion map DS^{0} ⟶ QS^{0} and a unit map QS^{0} = Ω^{∞}S^{0} ⟶ Ω^{∞}BP. The composite map DS^{0} ⟶ Ω^{∞}BP induces a ring map K_{0}DS^{0} ⟶ K_{0}Ω^{∞}BP. It is shown in [11] that the dual map K^{0}Ω^{∞}BP ⟶ K^{0}DS^{0} is surjective, so that K_{0}DS^{0} ⟶ K_{0}Ω^{∞}BP is injective. It is also known from work of Wilson that K_{0}Ω^{∞}BP is a tensor product of polynomial rings and Laurent series rings (use [17, Theorem 3.3] and a degenerate Atiyah-Hirzebruch spectral sequence, or see [14]). It follows that K_{0}DS^{0} has no nontrivial nilpotent elements, as required.

Corollary 5.2. K_{0}QS^{0} is the tensor product of a polynomial algebra with the group ring

𝔽_{p^{n}}[ℤ] = 𝔽_{p^{n}}[π_{0}QS^{0}] ≃ 𝔽_{p^{n}}[u, u^{−1}].

Proof. It is well-known that QS^{0} is the group-completion of CS^{0}. Let Q_{0}S^{0} be the component of the basepoint in QS^{0}, so that QS^{0} is a disjoint union of copies of Q_{0}S^{0} indexed by π_{0}QS^{0} = ℤ. There is an evident self-map s: QS^{0} ⟶ QS^{0} which sends the k’th copy to the (k + 1)’st. It is not hard to construct a self-map s: DS^{0} ⟶ DS^{0} compatible with this, and to check that the induced map from the telescope s^{−1}DS^{0} to QS^{0} is a weak equivalence. Moreover, for a ∈ K_{0}DS^{0} we have s_{∗}(a) = [1] ∗ a ∈ K_{0}DS^{0}. It follows that K_{0}QS^{0} is obtained from K_{0}DS^{0} by inverting the element [1]. Also, [1] is indecomposable in K_{0}DS^{0}, so we can take it as one of our polynomial generators. The claim follows.

The next corollary is a straightforward consequence of Propositions 4.2 and 5.1.

Corollary 5.3. E^{∨}DS^{0} is the completion of a polynomial algebra over E, and

K_{0}DS^{0} = 𝔽_{p^{n}} ˆ⊗_{E}E^{∨}DS^{0}.

We next prove that K^{0}QS^{2} is a formal power series ring, using an unpublished argument of Kashiwabara.

Remark 5.4. Here we use the usual product on K^{0}QS^{2}, coming from the diagonal map. One might think that there were two products on K^{0}QS^{2}, by analogy with the two products on K^{0}DS^{0}. Indeed, the two projections S^{2} ∨ S^{2} ⟶ S^{2} give two maps Q(S^{2} ∨ S^{2}) ⟶ QS^{2}, and the resulting map Q(S^{2} ∨ S^{2}) ⟶ QS^{2} × QS^{2} is an equivalence. We can thus use the pinch map S^{2} ⟶ S^{2} ∨ S^{2} to get a map θ: QS^{2} ⟶ QS^{2} × QS^{2}, giving a product on K^{0}QS^{2}. However, it is easy to check that θ is homotopic to the diagonal map, so this is the same product as we considered before. This argument does not show that the two products on K^{0}DS^{0} are the same, because the map θ: DS^{0} ⟶ DS^{0} ∧ DS^{0} only exists stably.

Remark 5.5. For our purposes, the ring of formal power series over 𝔽_{p^{n}} on a countable set of indeterminates x_{0}, x_{1}, x_{2}, … consists of all sums ∑_{α} c_{α}x^{α}. Here α runs over multi-indices α = (α_{0}, α_{1}, …) such that α_{i} = 0 for i ≫ 0, and we impose no condition at all on the coefficients c_{α} ∈ 𝔽_{p^{n}}. In particular, ∑_{i} x_{i} is allowed.

Proposition 5.6 (Kashiwabara). K^{0}QS^{2} is a formal power series algebra over 𝔽_{p^{n}}. Moreover, the double suspension map induces isomorphisms

Ind(K_{0}QS^{0}) ≃ Prim(K_{0}QS^{2})

and

Ind(K^{0}QS^{2}) ≃ Prim(K^{0}QS^{0}).

Proof. We use the bar spectral sequence. Let Y be a connected infinite loop space, and X = ΩY, so that Y = BX. We then have a spectral sequence of Hopf algebras

E_{st}^{2} = Tor_{s,t}^{K_{∗}X}(K_{∗}, K_{∗}) ⇒ K_{s+t}(Y)
d^{r}: E_{st}^{r} ⟶ E_{s−r,t+r−1}^{r}.

Note that E_{st}^{r} = 0 where s < 0; it follows that everything on the line s = 1 survives to E^{∞}. The resulting edge map

Ind(K_{t}X) = Tor_{1,t}^{K_{∗}X}(K_{∗}, K_{∗}) ⇒ K_{1+t}(Y)

is just the homology suspension map.

We first apply this with Y = QS^{1}, so that X = QS^{0}. We know that K_{∗}QS^{0} is a polynomial algebra on generators {x_{i}} in degree zero, with one generator inverted. It follows that Tor_{st}^{K_{∗}QS^{0}}(K_{∗}, K_{∗}) is exterior on generators y_{i} ∈ E_{10}^{2}. These are represented by the homology suspensions of the generators of K_{0}QS^{0}. Because the spectral sequence is multiplicative and the generators survive, there can be no differentials at all. Thus K_{∗}QS^{1} is the exterior algebra on primitive classes in odd degree. Moreover, the suspension map induces an isomorphism Ind(K_{∗}QS^{0}) ≃ Prim(K_{∗}QS^{1}), and the evident composite Prim(K_{∗}QS^{1}) ↣ K_{∗}QS^{1} ↠ Ind(K_{∗}QS^{1}) is also an isomorphism.

We now repeat the process, and consider the bar spectral sequence

Tor_{st}^{K_{∗}QS^{1}}(K_{∗}, K_{∗}) ⇒ K_{∗}(QS^{2}) ≃ K_{∗}(DS^{2}).

The E^{2} term is a divided power algebra on the suspensions of the elements y_{i}, which we shall call z_{i}. The coproduct is just the usual one:

ψ(z_{i}^{[m]}) = ∑_{m=j+k} z_{i}^{[j]} ⊗ z_{i}^{[k]}.

Write Z_{∗} for the divided power algebra over 𝔽_{p^{n}} generated by the elements z_{i} ∈ E_{11}^{2}, which is a Hopf algebra over 𝔽_{p^{n}}. Note that E_{∗∗}^{2} = K_{∗} ⊗_{𝔽_{p^{n}}} Z_{∗} as Hopf algebras over K_{∗}, and that Z_{∗} is connected if we use s as the grading.

We claim that there are no differentials in the spectral sequence. Indeed, suppose that d^{2} = … = d^{r−1} = 0, but that d^{r} ≠ 0. Choose an element x ∈ E_{st}^{r} with d^{r}(x) ≠ 0, with s as small as possible. Because the differentials respect the Hopf algebra structure which comes from the connected Hopf algebra Z_{∗}, it is not hard to see that x has to be primitive. On the other hand, it is easy to see that the primitives in E_{∗∗}^{2} = E_{∗∗}^{r} are just the elements z_{i}, and that these are permanent cycles; this gives a contradiction.

It follows that K_{∗}(QS^{2}) admits a filtration whose associated graded is a divided power Hopf algebra. Dually, we see that K^{∗}(QS^{2}) admits a filtration whose associated graded is a polynomial algebra on primitive generators in even degrees (without loss of generality, in degree zero). If we choose elements z′_{i} ∈ K^{0}(QS^{2}) lifting these generators, we discover in the usual way that K^{0}(QS^{2}) ≃ 𝔽_{p^{n}}[[z′_{i}]]. Moreover, the suspension map induces an isomorphism

Ind(K_{∗}QS^{1}) ≃ Prim(K_{∗}QS^{2}).

By combining this with information from our first spectral sequence, we see that the double suspension induces an isomorphism

Ind(K_{∗}QS^{0}) ≃ Prim(K_{∗}QS^{2}).

Dually, it induces an isomorphism

Ind(K^{∗}QS^{2}) ≃ Prim(K^{∗}QS^{0}).

Using Proposition 4.2, we deduce the following corollary.

Corollary 5.7. E^{0}QS^{2} is a formal power series ring over E.

6. The Snaith splitting

In this section, we recall the fundamental theorem of Snaith, which gives a splitting Σ^{∞}QX ≃ Σ^{∞}DX for connected spaces X. We will need to prove an additional property of this map (see Proposition 6.3) and thus we will need to give some details of its construction.

We will need to use the usual combinatorial approximation to QX, which we now describe briefly. First, we define

F_{k} = F_{k}ℝ^{∞} = {injective maps a: {1, …, k} ↣ ℝ^{∞}}.

It is well-known that this is a contractible free Σ_{k}-space, in other words, a model for EΣ_{k}. Next, we define

B_{k}(ℝ^{∞}; X) = F_{k} ×_{Σ_{k}} X^{k}.

The disjoint union ∐_{k} B_{k}(ℝ^{∞}; X) can be thought of as the set of pairs (A, x), where A is a finite subset of ℝ^{∞} and x: A ⟶ X. We can impose an equivalence relation on this set by identifying (A, x) with (B, x|_{B}) if B ⊆ A and x sends A \ B to the basepoint of X. We define

C(X) = C(ℝ^{∞}; X) = (∐_{k} B_{k}(ℝ^{∞}; X)) / ∼ .

Theorem 6.1. There is a natural map t: C(X) ⟶ Q(X), which is a weak equivalence when X is connected. On the other hand, when X = Y_{+} there is a natural homeomorphism

C(Y_{+}) = ∐_{k≥0} B_{k}(ℝ^{∞}; Y) = ∐_{k≥0} EΣ_{k} ×_{Σ_{k}} Y^{k} = D(Y_{+}).

In that case, the resulting map EΣ_{k} ×_{Σ_{k}} Y^{k} ⟶ Q(Y_{+}) is adjoint to the composite

Σ^{∞}EΣ_{k} ×_{Σ_{k}} Y_{+}^{k} ⟶^{transfer} Σ^{∞}EΣ_{k} ×_{Σ_{k−1}} Y_{+}^{k} ⟶^{proj} Σ^{∞}Y_{+}.

Proof. For the first sentence, see [13, 6.3] (for example). The rest follows easily from the constructions given there.

We next define

C_{k}(X) = image(B_{k}(ℝ^{∞}; X) ⟶ C(X)).

This gives a filtration

0 = C_{0}(X) ⊆ X = C_{1}(X) ⊆ C_{2}(X) ⊆ … ⊆ C(X),

and one can easily see that

C_{k}(X)/C_{k−1}(X) = EΣ_{k+} ∧_{Σ_{k}} X^{(k)} = D_{k}(X).

We can now define the Snaith map. For this, we note that the polynomial rings ℝ[u] and ℝ[u, v] can both be used as models of ℝ^{∞}. A point of a ∈ C(ℝ[u]; X) is a finite set A of polynomials in u, with each polynomial f ∈ A given a label x_{f} ∈ X. For any subset B ⊆ A, we have a polynomial

g_{B} = ∏_{f∈B}(v − f) ∈ ℝ[u, v].

Clearly, if B ≠ B′ then g_{B} ≠ g_{B′}, by unique factorisation. Note also that (B, x|_{B}) defines a point of C_{|B|}(ℝ[u]; X); we write y_{B} for its image in D_{|B|}(ℝ[u]; X). By labelling each g_{B} with the point y_{B}, we obtain a point s′(a) = ({g_{B} | B ⊆ A}, y) ∈ C(ℝ[u, v]; DX). It can be show that this gives a well-defined and continuous map

s′: C(ℝ[u]; X) ⟶ C(ℝ[u, v]; DX).

After composing with the map t: C(ℝ[u, v]; DX) ⟶ QDX and taking adjoints, we obtain a map s: Σ^{∞}CX ⟶ Σ^{∞}DX = DΣ^{∞}X. One finds that this sends Σ^{∞}C_{k}X into Σ^{∞} ⋁_{j≤k} D_{j}X and thus that it induces a map Σ^{∞}(C_{k}X/C_{k−1}X) ⟶ Σ^{∞}D_{k}X, which is an equivalence. This is essentially the proof of the following theorem; for more details, see [3].

Theorem 6.2. The Snaith map s: Σ^{∞}CX ⟶ Σ^{∞}DX = DΣ^{∞}X is a weak equivalence. In the case X = Y_{+}, it can also be obtained by applying Σ^{∞} to the homeomorphism C(Y_{+}) = D(Y_{+}) mentioned in Theorem 6.1.

There is an obvious equivalence QS^{0} ≃ Ω^{2}QS^{2}, which gives by adjunction a map e: Σ^{2}QS^{0} ⟶ QS^{2}. We will need to see how this interacts with the Snaith splitting. Firstly, we have a map e′: Σ^{2}CS^{0} ⟶ CS^{2}, defined as follows: if z ∈ S^{2} and A ∈ CS^{0} is a finite subset of ℝ^{∞}, then e′(z, A) is just the subset A with each point labelled by z.

Next, recall that V_{k} is the bundle over BΣ_{k} corresponding to the usual representation of Σ_{k} on ℂ^{k}. We write BΣ_{k}^{V_{k}} for the corresponding Thom space. One can show directly (see [12, Section IX.5]) that

D_{k}(S^{2}) = BΣ_{k}^{V_{k}} = D_{k}(S^{0})^{V_{k}}.

There is an evident vector v = (1, …, 1) ∈ V_{k} that is fixed under the action of Σ_{k}. Write L = ℂv = V_{k}^{Σ_{k}} < V_{k}, so that BΣ_{k}^{L} = Σ^{2}BΣ_{k}. The inclusion L ⟶ V_{k} thus gives a map Δ: Σ^{2}D_{k}(S^{0}) ⟶ D_{k}(S^{2}) and thus a map Δ: Σ^{2}DS^{0} ⟶ D(S^{2}). We also observe that the connectivity of D_{k}(S^{2}) is at least 2k − 1, and thus that the evident map D(S^{2}) ⟶ ∏_{k≥0} D_{k}(S^{2}) is a weak equivalence.

We will also need a map ϵ_{1} : DS^{0} ⟶ S^{0}. This is the counit map for the coproduct map δ: DS^{0} ⟶ DS^{0} ∧ DS^{0}. It sends each space BΣ_{k} in DS^{0} = ⋁_{k≥0} BΣ_{k+} to the non-basepoint in S^{0}.

Proposition 6.3. The following diagram commutes.

Σ^{∞+2}QS^{0}  Σ^{∞}QS^{2} Σ^{∞+2}CS^{0}  Σ^{∞}CS^{2} Σ^{∞+2}DS^{0}  Σ^{∞+2}DS^{0} ∧ DS^{0}  Σ^{∞+2}DS^{0}  Σ^{∞}D(S^{2})^{e}  w u_{Σ^{∞+2}t}_{Σ^{2}s}  _{^{e′}  w} u_{≃ Σ^{∞}t}_{u_{≃ s}} w_{Σ^{2}θ}  w_{1∧ϵ_{1}}  w_{Σ^{∞}Δ}

Proof. It is immediate from the definition of t that the top rectangle commutes. The two ways round the bottom rectangle give two maps Σ^{∞+2}DS^{0} ⟶ Σ^{∞}D(S^{2}). After composing with the inclusion Σ^{2}BΣ_{k+} ⟶ Σ^{2}DS^{0} and the projection D(S^{2}) = ∏_{m} D_{m}(S^{2}) ⟶ D_{m}(S^{2}), we get two maps Σ^{∞+2}BΣ_{k+} ⟶ Σ^{∞}D_{m}(S^{2}), and it is enough to see that these coincide for all k and m. We shall assume that m ≤ k, leaving it to the reader to see that both maps are zero when m > k. We can also take adjoints and identify QD_{m}(S^{2}) with C(ℝ[u, v]; D_{m}(S^{2})) to get two maps of spaces Σ^{2}BΣ_{k+} ⟶ C(ℝ[u, v]; D_{m}(S^{2})). We write F for the one which comes from s ∘ e′: Σ^{∞+2}DS^{0} ⟶ Σ^{∞}D(S^{2}), and G for the one which comes from Δ ∘ (1 ∧ ϵ_{1}) ∘ θ ∘ t.

Consider a point (z, A) ∈ Σ^{2}BΣ_{k+}, so z ∈ S^{2} and A is a finite subset of ℝ[u] of order k. We see easily from our definition of the Snaith map that F (z, A) ∈ C(ℝ[u, v]; D_{m}(S^{2})) is the collection of polynomials g_{B} (where B ⊆ A and |B| = m) labelled with the points Δ(z, B) ∈ D_{m}(S^{2}).

Next, recall that θ is built from the transfer maps Σ^{∞}BΣ_{(i+j)+} ⟶ Σ^{∞}(BΣ_{i} ×BΣ_{j})_{+}. It follows that we have a commutative diagram

Σ^{∞}BΣ_{k+}  Σ^{∞}(BΣ_{m} × BΣ_{k−m})_{+}  Σ^{∞}BΣ_{m+} DS^{0}  DS^{0} ∧ DS^{0}  DS^{0} v u  w^{transfer}  ^{proj} w w_{θ}  w_{1∧ϵ_{1}} u u

To understand the top composite, we need to recall something about transfers. Let q: X ⟶ Y be a finite covering map, and let i: X ⟶ ℝ^{∞} be a map that is injective on each fibre of q. We then get a map r: Y_{+} ⟶ C(X_{+}) by sending a point y ∈ Y to the set iq^{−1}{y}, with each point i(x) ∈ iq^{−1}{y} labelled by x. It is easy to see from the standard constructions that t ∘ r: Y_{+} ⟶ Q(X_{+}) is adjoint to the transfer map q^{!} : Σ^{∞}Y_{+} ⟶ Σ^{∞}X_{+}.

We now apply this with X = F_{k}(ℝ[u])/(Σ_{m} × Σ_{k−m}) (which is a model for BΣ_{m} × BΣ_{k−m}) and Y = F_{k}(ℝ[u])/Σ_{k} (which is a model for BΣ_{k}). We can also think of Y as the set of subsets A ⊂ ℝ[u] of order k, and X as the set of pairs (B, C) of disjoint subsets of ℝ[u] with |B| = m and |C| = k. The map q just sends (B, C) to B ∪ C. For our map i: X ⟶ ℝ^{∞} = ℝ[u, v], we take i(B, C) = g_{B} = ∏_{f∈B}(v − f). The adjoint to the transfer is thus the map Y_{+} ⟶ C(ℝ[u, v]; X_{+}) that sends A to the set of polynomials g_{B} (for all subsets B ⊆ A of order m), with g_{B} labelled by (B, C). When we compose with the projection X = BΣ_{m} × BΣ_{k−m} ⟶ BΣ_{m}, we just replace the label (B, C) ∈ X by B ∈ BΣ_{m}.

It now follows that adjoint of the double suspension of the top composite in the above diagram sends a point (z, A) ∈ Σ^{2}BΣ_{k} to the set of polynomials g_{B} (for B ⊆ A with |B| = k), labelled by (z, B). To get G(z, A), we just have to apply Δ to each label. By comparing this with our analysis of F, we see that G(z, A) = F (z, A) as claimed.

7. The Thom isomorphism

We now discuss the Thom isomorphism. We saw above that D_{m}(S^{2}) = BΣ_{m}^{V_{m}}, so we have a Thom isomorphism Ẽ^{0}D_{m}(S^{2}) = E^{0}BΣ_{m} and thus E^{0}D(S^{2}) = E^{0}DS^{0}. We need to show that this respects ring structures in a suitable sense. Just as in Remark 5.4, we find that there is only one sensible product on E^{0}D(S^{2}), which comes both from the diagonal map D(S^{2}) ⟶ D(S^{2}) × D(S^{2}) and from the pinch map S^{2} ⟶ S^{2} ∨ S^{2}. We give E^{0}D(S^{0}) the ×-product, coming from the stable pinch map S^{0} ⟶ S^{0} ∨ S^{0}.

Lemma 7.1. With the above product structures, the Thom isomorphism E^{0}D(S^{2}) = E^{0}D(S^{0}) is a ring map.

Proof. For any spectra X and Y, we have a natural map δ: D_{k}(X ∧ Y) ⟶ D_{k}(X) ∧ D_{k}(Y), which is induced in an evident way by the diagonal map EΣ_{k} ⟶ EΣ_{k} × EΣ_{k} when X and Y are suspension spectra. See [2, Section I.2] for discussion of this, and [12, Section VII.1] for proofs. In particular, there is a natural map δ: D_{k}(Σ^{2}X) ⟶ D_{k}(S^{2}) ∧ D_{k}(X). Let u ∈ E^{0}D_{k}(S^{2}) be the Thom class, and define ϕ: E^{0}D_{k}(X) ⟶ E^{0}D_{k}(Σ^{2}X) by ϕ(x) = δ^{∗}(u ⊗ x). When X has the form Σ^{∞}Y_{+} we notice that D_{k}(Σ^{2}Y_{+}) is the Thom space for the pullback of V_{k} to D_{k}(Y_{+}), and that ϕ is the Thom isomorphism. The upshot is that this Thom isomorphism is natural for stable maps of Σ^{∞}Y_{+}, not merely for unstable maps of Y. The lemma follows by considering the stable pinch map S^{0} ⟶ S^{0} ∨ S^{0}.

Corollary 7.2. E^{0}DS^{0} is a formal power series ring under ×.

Proof. By the lemma, E^{0}DS^{0} under × is isomorphic to E^{0}DS^{2} under the product coming from the (unstable) pinch map S^{2} ⟶ S^{2} ∨S^{2}. By the Snaith splitting, E^{0}DS^{2} is isomorphic to E^{0}QS^{2}. By naturality, this converts our product to the product on E^{0}QS^{2} that comes from the pinch map. By Proposition 5.6 and Remark 5.4, E^{0}QS^{2} is a formal power series ring under this product.

8. Primitives and indecomposables

In this section we consider the primitives and indecomposables in the Hopf ring E^{0}DS^{0}. We need to interpret the indecomposables in a completed sense.

Definition 8.1. We topologise E^{0}BΣ_{k}, E^{∨}BΣ_{k} and E^{∨}DS^{0} using the 𝔪-adic topology, where 𝔪 = (u_{0}, …, u_{n−1}) is the maximal ideal in E^{0}. We give E^{0}DS^{0} = ∏_{k} E^{0}BΣ_{k} the product topology. Given an augmented topological algebra A over E^{0}, we define Ind(A) to be the quotient of the augmentation ideal by the closure of its square.

Remark 8.2. With this definition, Ind(E^{0}[[x_{0}, x_{1}, …]]) is a countable product of copies of E^{0}, indexed by the variables x_{i}.

We need to distinguish between two kinds of primitives in E^{0}DS^{0}. Recall that we have two unit elements [0] and [1] in E^{0}DS^{0} = ∏_{k} E^{0}BΣ_{k}. The component of [0] in E^{0}BΣ_{k} is 1 if k = 0 and 0 if k > 0, whereas the component of [1] is 1 for all k. We have [0] × a = a = [1] • a for all a ∈ E^{0}DS^{0}. Note also that [1] is invertible under the ×-product. Its inverse is [−1], which is the image of [1] under the map D(−1): DS^{0} ⟶ DS^{0}.

Definition 8.3. We write

Prim(E^{0}DS^{0}) = {x ∈ E^{0}DS^{0} | ψ_{∗}(x) = x ⊗ [0] + [0] ⊗ x}
Prim′(E^{0}DS^{0}) = {x ∈ E^{0}DS^{0} | ψ_{∗}(x) = x ⊗ [1] + [1] ⊗ x}.

The first of these is the more usual definition for primitives in a Hopf ring. It is easy to see from general thoughts about Hopf rings that the •-product makes Ind(E^{0}DS^{0}) into a ring and Prim(E^{0}DS^{0}) into a module over Ind(E^{0}DS^{0}).

On the other hand, one would like to say that the primitives in E^{0}DS^{0} are dual to Ind(E^{∨}DS^{0}). Here E^{∨}DS^{0} is a Hopf algebra with coproduct ψ_{∗}, and it is natural to use the counit for ψ_{∗} as our augmentation on E^{∨}DS^{0}. This counit is [1] ∈ E^{0}DS^{0} rather than [0], and because of this the natural duality is between Prim′(E^{0}DS^{0}) and Ind(E^{∨}DS^{0}). This becomes clearer by examining the explicit formulae in [16, Section 6]. As mentioned there, the map a ↦ a × [1] gives an isomorphism Prim(E^{0}DS^{0}) ⟶ Prim′(E^{0}DS^{0}).

We will need to relate our primitives and indecomposables in terms of transfers and restrictions, which leads us to our next definitions.

Definition 8.4. Write R_{m} = E(BΣ_{m}). We shall call the subgroups BΣ_{i} × BΣ_{j} ≤ BΣ_{i+j} (with i, j > 0) partition subgroups. For each such subgroup H ≤ BΣ_{m}, we have a transfer map E(BH) ⟶ R_{m}; let I = I_{m} ≤ R_{m} be the sum of the images of these maps. We also have restriction maps R_{m} ⟶ E(BH); let J = J_{m} ≤ R_{m} be the intersection of the kernels of these maps.

The following two results summarise our conclusions. They are proved after Lemma 8.11.

Theorem 8.5. The star-indecomposables and primitives in the Hopf ring E^{0}DS^{0} are as follows:

Ind(E^{0}DS^{0}) = ∏_{m} R_{m}/I_{m} = ∏_{k} R_{p^{k}}/I_{p^{k}}
Prim(E^{0}DS^{0}) = ∏_{m} J_{m} = ∏_{k} J_{p^{k}}

Moreover, Prim(E^{0}DS^{0}) is a free module over Ind(E^{0}DS^{0}) on one generator c (given by Definition 3.1).

Theorem 8.6. Suppose that m = p^{k}. Then I_{m} = ann(c_{m}) = ann(J_{m}), and this is a retract of R_{m} as an E-module. Moreover, J_{m} is a free module on one generator c_{m} over R_{m}/I_{m} (and thus is generated as an ideal by c_{m}). The ideal J_{m} is also a retract of R_{m} as an E-module. The rank of R_{m}/I_{m} over E^{0} is \overline{d}(k) (see Definition 3.2).

In the light of the above, it is natural to make the following definition.

Definition 8.7. \overline{R}_{k} = R_{p^{k}}/I_{p^{k}}.

We now prove a series of lemmas leading to the above theorems.

Lemma 8.8. I_{m} and J_{m} are ideals in R_{m}, and I_{m}J_{m} = 0.

Proof. This follows from the fact that res is a ring map, and the formula x tr(y) = tr(res(x)y).

Lemma 8.9. The Euler class c_{m} = e(V_{m} − 1) lies in J_{m}.

Proof. Suppose that m = i+j with i, j > 0. Then the restriction of V_{m} to Σ_{i} ×Σ_{j} is isomorphic to V_{i} ⊕ V_{j}, which has two trivial summands. Thus, the restriction of V_{m} − 1 has a trivial summands and thus e(V_{m} − 1) maps to zero.

Lemma 8.10. If m does not have the form m = p^{k} then I_{m} = R_{m} and J_{m} = 0.

Proof. In this case it is well-known that (a + b)^{m} ≅ a̸^{m} + b^{m} (mod p), and thus that there exist i, j > 0 with i + j = m and d = |Σ_{m}/H| not divisible by p, where H = Σ_{i} × Σ_{j}. The rest of the argument is well-known: we note that the composite

E^{0}BΣ_{m} ⟶^{res} E^{0}BH ⟶^{tr} E^{0}BΣ_{m}

is just multiplication by tr(1). The double coset formula shows that ϵ(tr(1)) = d ∈ E^{0}, and this is a unit. Because E^{0}BΣ_{m} is a local ring, we conclude that tr(1) is a unit. This implies that tr: E^{0}BH ⟶ E^{0}BΣ_{m} is surjective, and res: E^{0}BΣ_{m} ⟶ E^{0}BH is injective.

Lemma 8.11. If m = p^{k} then

I_{m} = image image(tr: E^{0}BΣ_{p^{k−1}}^{p} ⟶ E^{0}BΣ_{p^{k}})
J_{m} = ker(res: E^{0}BΣ_{p^{k}} ⟶ E^{0}BΣ_{p^{k−1}}^{p}).

Proof. Write H = Σ_{p^{k−1}}^{p} and G = Σ_{p^{k}}. Consider a partition subgroup L = Σ_{i} × Σ_{j}, where i + j = p^{k} and i, j > 0. Using the argument of Lemma 8.10, it will be enough to show that |L/L ∩ H| is not divisible by p, or equivalently that H contains a Sylow p-subgroup of L. There are various ways to see this. The most direct is to use the following well-known formulae for the p-adic valuations of factorials and binomials:

v_{p}(p^{k}!) = (p^{k} − 1)/(p − 1)
v_{p} ( \begin{matrix}p^{k} \\ i \end{matrix} ) = k − v_{p}(i)  if 0 < i < p^{k}.

If i = p^{k−1}r + s with 0 ≤ s < p^{k−1} then

H ∩ L = Σ_{p^{k−1}}^{p−1} × Σ_{s} × Σ_{p^{k−1}−s}

and we calculate directly that

v_{p}|H ∩ L| = (p^{k} − 1)/(p − 1) − k + v_{p}(s) = v_{p}|L|

as required.

Proof of Theorem 8.5. The augmentation for the Hopf ring E^{0}DS^{0} is the counit for the coproduct ψ_{∗}, which is just the restriction map

E^{0}DS^{0} ⟶ E(D_{0}S) = E(BΣ_{0}) = E.

Thus, the augmentation ideal is ∏_{m>0} E(BΣ_{m}). The star product is the product ×, which is derived from the map θ = D(Δ): DS^{0} ⟶ DS^{0} ∧ DS^{0}. As mentioned in section 4, this sends Σ^{∞}BΣ_{m+} to ⋁_{m=i+j} Σ^{∞}(BΣ_{i} × BΣ_{j})_{+} by the sum of the transfer maps. It follows easily (using Lemma 8.10) that

Ind(E^{0}DS^{0}) = ∏_{m} R_{m}/I_{m} = ∏_{k} R_{p^{k}}/I_{p^{k}}.

Similarly, the unit for the star product is the element [0], whose component in E(BΣ_{m}) is 1 if m = 0 and 0 otherwise. The coproduct ψ_{∗} is derived from the map σ: DS^{0} ∧ DS^{0} ⟶ DS^{0}, which sends BΣ_{i} ×BΣ_{j} to BΣ_{i+j} by the usual map, which induces the restriction map in E-cohomology. It follows that an element u ∈ E^{0}DS^{0} is primitive (i.e. ψ_{∗}(u) = [0] ⊗ u + u ⊗ [0]) if and only if the component u_{m} lies in J_{m} for all m. In other words (using Lemma 8.10 again),

Prim(E^{0}DS^{0}) = ∏_{m} J_{m} = ∏_{k} J_{p^{k}}.

The circle product in E^{0}DS^{0} is just the ordinary product • induced by the diagonal map of the space DS^{0}. As I_{m} is an ideal in R_{m}, we see that the circle product induces a ring structure on Ind(E^{0}DS^{0}). We observed above that I_{m}J_{m} = 0, which implies that Prim(E^{0}DS^{0}) is a module over Ind(E^{0}DS^{0}). (Both of these facts hold in an arbitrary Hopf ring, and can be proved by elementary manipulations).

This proves all of Theorem 8.5 except for the last sentence. This is a trivial consequence of Theorem 8.6, which we prove next.

Proof of Theorem 8.6. Firstly, we know from Corollary 7.2 that E^{0}DS^{0} is a formal power series ring over E. It follows that the ideal R_{+}^{2} of decomposables is a retract of the augmentation ideal R_{+} as an E-module. We also know that both ideals are homogeneous with respect to the splitting E^{0}DS^{0} = ∏_{m} R_{m}. By choosing homogeneous bases we conclude that I_{m} = R_{+}^{2} ∩ R_{m} is a retract of R_{m}, as claimed.

Similarly, we know that E^{∨}DS^{0} is a completed polynomial ring; it follows that Ind(E^{∨}DS^{0}) is a retract of E^{∨}DS^{0}. Dually, we see that Prim′(E^{0}DS^{0}) is a retract of E^{0}DS^{0}. By applying the map x ↦ [−1] × x (which is an automorphism of E^{0}DS^{0}), we conclude that Prim(E^{0}DS^{0}) is a retract of E^{0}DS^{0}. Moreover, Prim(E^{0}DS^{0}) is homogeneous with respect to the splitting E^{0}DS^{0} = ∏_{m} R_{m} (although Prim′(E^{0}DS^{0}) is not). It follows as in the previous paragraph that J_{m} is a retract of R_{m}.

In [16, Section 6] it is shown that there is a natural isomorphism L ⊗_{E} R_{m}/I_{m} ≃ F (𝕃_{m}, L), where L is a certain extension ring of E, 𝕃_{m} is the set of lattices of index m in ℤ_{p}^{n}, and F (𝕃_{m}, L) is the ring of functions from the finite set 𝕃_{m} to L (with pointwise operations). In particular, we see that the rank of R_{m}/I_{m} over E^{0} is the number of lattices of index m in ℤ_{p}^{n}, which is one definition of \overline{d}(k) (see Definition 3.2). It is also shown in [16] that L ⊗_{E} J_{m} has rank \overline{d}(k), and we conclude in the same way that J_{m} has rank \overline{d}(k) over E^{0}.

Next, we show that Prim(E^{0}DS^{0}) is freely generated by c over Ind(E^{0}DS^{0}). We apply the functor E^{0}(−) to the diagram in proposition 6.3 and use various Thom isomorphisms to obtain the following diagram.

E(QS^{0})  E(QS^{2}) E(DS^{0})  E(DS^{0})  E(DS^{0})^{σ^{∗2}}  w u  u w_{•c}  w_{×[1]}

(The first map in the bottom row is induced by the inclusions Σ^{2}BΣ_{m} = BΣ_{m}^{ℂ} ⟶^{Δ} BΣ_{m}^{V_{m}} and the Thom isomorphisms E(BΣ_{m}^{ℂ}) ≃ E(BΣ_{m}) ≃ E(BΣ_{m}^{V_{m}}). It is standard that the resulting map is just multiplication by the Euler class of V_{m} − 1, which is just c_{m}. This shows that the first map on the bottom row is just x ↦ x • c, as marked in the diagram.)

The right hand vertical map is an isomorphism (by the Snaith splitting theorem). The top arrow is the double suspension map, so the image is contained in Prim′(E^{0}DS^{0}). Consider the analogous map in K-cohomology. Proposition 5.6 shows that the image of this map is precisely 𝔽_{p^{n}} ˆ⊗_{E} Prim′(E^{0}DS^{0}) = Prim′(K^{0}DS^{0}). As all the modules in question are products of free modules, we conclude that the double suspension E(QS^{0}) ⟶ Prim′(E^{0}DS^{0}) is surjective.

Recall that x ↦ x × [1] is an E-module automorphism of E^{0}DS^{0} which carries Prim(E^{0}DS^{0}) isomorphically to Prim′(E^{0}DS^{0}). It follows from this and the discussion above that the map x ↦ x • c is an epimorphism Ind(E^{0}DS^{0}) ⟶ Prim(E^{0}DS^{0}). We therefore have an epimorphism R_{m}/I_{m} ⟶^{c} J_{m}. Both source and target are finitely generated projective (and hence free) modules over E^{0}, and we have seen that they have the same rank. It follows that our map R_{m}/I_{m} ⟶^{c_{m}} J_{m} is an isomorphism. This means in particular that ann(c_{m}) = I_{m} as claimed.

9. Subgroups of formal groups

In this section, we identify the functor represented by the ring \overline{R}_{k}. We shall assume the results and terminology of [15]. We shall write X = spf(E^{0}) and 𝔾 = spf(E^{0}ℂP^{∞}), so that 𝔾 is a formal group over X. The special fibre of X is X_{0} = spf(𝔽_{p^{n}}), and the restriction of 𝔾 over X_{0} is 𝔾_{0} = spf(K^{0}ℂP^{∞}). Moreover, 𝔾 is the universal deformation of 𝔾_{0}.

Now let Z be a space, and W a complex line bundle over Z. We can define the associated projective bundle

P (W) = {(z, L) | z ∈ Z , L ≤ W_{z} , dim(L) = 1}.

There is a tautological line bundle L(W) over P (W), defined by

L(W)_{(z,L)} = L.

This is classified by a map P (W) ⟶ ℂP^{∞}. We thus obtain a map P (W) ⟶ ℂP^{∞} × Z and hence spf(E^{0}P (W)) ⟶ 𝔾 ×_{X} spf(E^{0}Z). If x is the Euler class of L(W) and a_{i} is the i’th Chern class of W, then a well-known lemma says that

E^{∗}P (W) = E^{∗}Z[[x]]/ ∑_{k=0}^{m} a_{m−k}x^{k}.

This shows that D(W) = spf(E^{0}P (W)) is a divisor of degree m on 𝔾 over spf(E^{0}Z).

We can apply this process to get a divisor D(V_{k}) of spf(E^{0}BΣ_{k}). We write Y_{k} = spf(\overline{R}_{k}), which is a closed subscheme of spf(E^{0}BΣ_{p^{k}}), and we let H_{k} denote the restriction of D(V_{p^{k}}) to Y_{k}.

Proposition 9.1. H_{k} is a subgroup divisor of 𝔾.

Proof. It follows from [15, Theorem 42] that there is a largest closed subscheme Z_{k} of spf(E^{0}BΣ_{p^{k}}) over which D(V_{p^{k}}) is a subgroup divisor; we need to show that Y_{k} ⊆ Z_{k}.

Let A be an Abelian p-subgroup of Σ_{p^{k}} that acts transitively on {1, …, p^{k}}. It is not hard to see that the restriction of V_{p^{k}} to A is the regular representation. This can also be written as ⨁_{L∈A^{∗}} L, the direct sum of all one-dimensional representations of A. Recall from [7, Proposition 4.12] that spf(E^{0}BA) can be naturally identified with Hom(A^{∗}, 𝔾). Over this scheme we have a tautological map ϕ: A^{∗} ⟶ Γ(Hom(A^{∗}, 𝔾), 𝔾) and thus a divisor ∑_{L∈A^{∗}}[ϕ(L)]. It is easy to see that this is the same as the restriction of D(V_{p^{k}}) to Hom(A^{∗}, 𝔾). In [15, Proposition 22], we defined a closed subscheme Level(A^{∗}, 𝔾) of Hom(A^{∗}, 𝔾). It follows from [15, Proposition 32] that the restriction of D(V_{p^{k}}) to Level(A^{∗}, 𝔾) is a subgroup divisor. This means that the image of the map

Level(A^{∗}, 𝔾) ↣ Hom(A^{∗}, 𝔾) = spf(E^{0}BA) ⟶ spf(E^{0}BΣ_{p^{k}})

is contained in Z_{k}. It is thus enough to show that the union of these images is precisely Y_{k}. This is essentially clear from the rational description of E^{0}BΣ_{p^{k}} given in [16], but some translation is required, so we adopt a slightly different approach.

Recall from [15, Theorem 23] that Level(A^{∗}, 𝔾) is a smooth scheme, and thus that D(A) = O_{Level(A^{∗},𝔾)} is an integral domain. Using [15, Proposition 26], we see that the Euler classes e(L) are nonzero in D(A) for each nontrivial L ∈ A^{∗}, and thus that c_{p^{k}} = ∏_{L≠1} e(L) is not a zerodivisor in D(A). On the other hand, if A′ is an Abelian p-subgroup of Σ_{p^{k}} which does not act transitively on {1, …, p^{k}}, then the restriction of V_{p^{k}} − 1 to A′ has a trivial summand, and thus c_{p^{k}} maps to zero in D(A′). Next, we recall the version of generalised character theory described in [7, Appendix A]. It is proved there that for any finite group G, there is a natural isomorphism of rings

p^{−1}E^{0}BΣ_{p^{k}} = (∏_{A} p^{−1}D(A))^{Σ_{p^{k}}} ,

where A runs over all Abelian p-subgroups of Σ_{p^{k}}. As \overline{R}_{k} = E^{0}(BΣ_{p^{k}})/ ann(c_{p^{k}}) and everything in sight is torsion-free, we see that p^{−1}\overline{R}_{k} is the quotient of p^{−1}E^{0}BΣ_{p^{k}} by the annihilator of the image of c_{p^{k}}. Using our analysis of the images of c_{p^{k}} in the rings D(A), we conclude that

p^{−1}\overline{R}_{k} = (∏_{A} p^{−1}D(A))^{Σ_{p^{k}}} ,

where the product is now over all transitive Abelian p-subgroups. This implies that for such A, the map E^{0}BΣ_{p^{k}} ⟶ D(A) factors through \overline{R}_{k}, and that the resulting maps \overline{R}_{k} ⟶ D(A) are jointly injective. This means that Y_{k} is the union of the images of the corresponding schemes Level(A^{∗}, 𝔾), as required.

Recall from [15, Theorem 42] that there is a universal example of a subgroup divisor of 𝔾 of degree p^{k}, defined over a certain scheme Sub_{k}(𝔾). Thus, Proposition 9.1 gives us a map of schemes Y_{k} = spf(\overline{R}_{k}) ⟶ Sub_{k}(𝔾), or equivalently a map O_{Sub_{k}(𝔾)} ⟶ \overline{R}_{k} of E^{0}-algebras.

Theorem 9.2. The classifying map O_{Sub_{k}(𝔾)} ⟶ \overline{R}_{k} is an isomorphism.

Proof. By Theorem 8.6 above and [15, Therem 10.1], we know that O_{Sub_{k}(𝔾)} and \overline{R}_{k} are free modules of the same rank over E^{0}. We shall show that the induced map

O_{Sub_{k}(𝔾_{0})} = 𝔽_{p^{n}} ⊗_{E} O_{Sub_{k}(𝔾)} ⟶ 𝔽_{p^{n}} ⊗_{E} \overline{R}_{k}

is injective. We will conclude that it is an isomorphism by dimension count; it follows easily that the original map O_{Sub_{k}(𝔾)} ⟶ \overline{R}_{k} is an isomorphism.

Recall from [15, Proposition 10.16] that the socle of O_{Sub_{k}(𝔾_{0})} is generated by a certain element a′^{N}, where N = p + … + p^{n−1}. It is easy to see from the definition of a′ that it maps to c_{p^{k}} in 𝔽_{p^{n}} ⊗_{E} \overline{R}_{k}. We showed in Theorem 3.3 that c_{p^{k}}^{N+1} ≠ 0 in 𝔽_{p^{n}} ⊗_{E} E^{0}BΣ_{p^{k}}, but c_{p^{k}}I_{p^{k}} = 0 so c_{p^{k}}^{N} ≠ 0 in 𝔽_{p^{n}} ⊗_{E} E^{0}BΣ_{p^{k}}/I_{k} = 𝔽_{p^{n}} ⊗_{E} \overline{R}_{k}. Thus, the socle of O_{Sub_{k}(𝔾_{0})} is mapped injectively. As any nonzero ideal meets the socle, we see that the whole ring is mapped injectively, as required.

10. A basis for O_{Sub_{m}(𝔾)}

We finish by finding a basis for \overline{R}_{m} = O_{Sub_{m}(𝔾)} over E^{0}.

Definition 10.1. Write

σ(u, v) = ∑_{i=u}^{v−1} p^{i} = (p^{v} − p^{u})/(p − 1).

Let a_{j} be the (p^{m} − p^{j})’th Chern class of V_{p^{m}}. Let μ and ν be sequences of the form

1 = μ_{0} < … < μ_{r} ≤ n
0 ≤ ν_{0} < … < ν_{r} = m.

Write

b = ∏_{i=0}^{r−1} a_{ν_{i}}^{σ(μ_{i+1},μ_{i})}.

We also define ρ_{j} for 0 ≤ j < m as follows: find the unique i such that ν_{i−1} ≤ j < ν_{i}, and then set

ρ_{j} = \begin{cases}p^{μ_{i}} & if μ_{i} < n \\ 1 & if μ_{i} = n \end{cases}

Now write

C(μ, ν) = {ba^{α} | ∀j 0 ≤ α_{j} < ρ_{j}}

and

C = ∐_{μ,ν} C_{m}(μ, ν).

Theorem 10.2. C is a basis for \overline{R}_{m} over E^{0}.

In order to prove this, we need yet more definitions. In [15, Proposition 49] we introduced certain quotient rings E′(k, l) of 𝔽_{p^{n}} ⊗_{E} \overline{R}_{m}, and we need to find bases for these in order to make an inductive argument.

Definition 10.3. Suppose that 0 ≤ k ≤ m and 0 < l ≤ n. Let μ and ν be sequences of the form

l = μ_{0} < … < μ_{r} ≤ n
k ≤ ν_{0} < … < ν_{r} = m.

Write

b = ∏_{i=0}^{r−1} a_{ν_{i}}^{σ(μ_{i+1},μ_{i})}.

We also define ρ_{j} for k ≤ j < m as follows: find the unique i such that ν_{i−1} ≤ j < ν_{i}, and then set

ρ_{j} = \begin{cases}p^{μ_{i}} & if μ_{i} < n \\ 1 & if μ_{i} = n \end{cases}

Now write

C_{kl}(μ, ν) = {ba^{α} | ∀j 0 ≤ α_{j} < ρ_{j}}

and

C_{kl} = ∐_{μ,ν} C(μ, ν).

Proof of Theorem 10.2. We use the notation of [15, Proposition 49]. There we introduced certain quotient rings E′(k, l) of 𝔽_{p^{n}} ⊗_{E} \overline{R}_{m} and elements v, a ∈ E′(k, l) such that

E′(0, 1) = 𝔽_{p^{n}} ⊗_{E} \overline{R}_{m}
E′(m, l) = 𝔽_{p^{n}}
E′(k, n) = 𝔽_{p^{n}}
E′(k, l)/a = E′(k + 1, l)
E′(k, l)/v = E′(k, l + 1)
va^{p^{l}} = 0  if l < n.

From the definitions, it is not hard to check that a is (up to sign) the image of our Chern class a_{k} in E′(k, l). Because of the first property, it will suffice to show that C_{kl} is a basis for E′(k, l), for all 0 ≤ k ≤ m and 0 < l ≤ n. This is easy to check when k = m or l = n. Suppose that we know that C_{k+1,l} is a basis for E′(k+1, l) = E′(k, l)/a_{k} and that C_{k,l+1} is a basis for E′(k, l+1) = E′(k, l)/v. Using the relation va_{k}^{p^{l}} = 0, we conclude that

C′_{kl} = {a_{k}^{j}t | t ∈ C_{k+1,l} and 0 ≤ j < p^{l}} ⨿ {a_{k}^{p^{l}}t | t ∈ C_{k,l+1}}

is a spanning set for E′(k, l). This gives an upper bound for dim_{𝔽_{p^{n}}} E′(k, l) and thus by induction a bound for dim_{𝔽_{p^{n}}} E′(0, 1) = dim_{E^{0}} \overline{R_{m}}. This is the conclusion of [15, Proposition 49]. It is proved in [15, Corollary 53] that this bound is sharp, and it follows easily that the bounds used at each stage must be sharp, so that C′_{kl} is a basis for E′(k, l). Thus, we need only check the purely combinatorial fact that C′_{kl} = C_{kl}. In fact, one can check that

{a_{k}^{j}t | t ∈ C_{k+1,l}(μ, ν) and 0 ≤ j < p^{l}} = C_{kl}(μ, ν).

This shows that the first piece of C′_{kl} is the disjoint union of the sets C_{kl}(μ, ν) for which ν_{0} > k. Next, consider a piece C_{k,l+1}(μ, ν) of C_{k,l+1}. If ν_{0} = k, we define

ν′ = (ν_{0} < ν_{1} < … < ν_{r} = m)
μ′ = (l < μ_{1} < … < μ_{r} ≤ n).

If ν_{0} > k, we define instead

ν′ = (k < ν_{0} < ν_{1} < … < ν_{r} = m)
μ′ = (l < l + 1 < μ_{1} < … < μ_{r} ≤ n).

Either way, one can check that

{a_{k}^{p^{l}}t | t ∈ C_{k,l+1}(μ, ν)} = C_{kl}(μ′, ν′).

Using this, one can conclude that C′_{kl} = C_{kl} as required.

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