Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups

Ethan S. Devinatz and Michael J. Hopkins^{†}

Abstract

Let G be a closed subgroup of the semi-direct product of the n^{th} Morava stabilizer group S_{n} with the Galois group of the field extension 𝔽_{p^{n}}/𝔽_{p}. We construct a “homotopy fixed point spectrum” E_{n}^{hG} whose homotopy fixed point spectral sequence involves the continuous cohomology of G. These spectra have the expected functorial properties and agree with the Hopkins-Miller fixed-point spectra when G is finite.

0 Introduction

Let p be a fixed prime, let n ≥ 1, and let E_{n} denote the spectrum with coefficient ring E_{n_{∗}} = W 𝔽_{p^{n}}[[u_{1}, …, u_{n−1}]][u, u^{−1}] obtained via the Landweber exact functor theorem for BP. W 𝔽_{p^{n}} denotes the ring of Witt vectors with coefficients in the field with p^{n} elements 𝔽_{p^{n}}, and the map BP_{∗} ⟶^{r} E_{n^{∗}}—which also provides E_{n} with the structure of BP -algebra in the stable category—is given by

r(v_{i}) = \{ \begin{matrix}u_{i}u^{1−p^{i}} & i < n \\ u^{1−p^{n}} & i = n \\ 0 & i > n. \end{matrix}

Now let S_{n} denote the n^{th} Morava stabilizer group; i.e., the automorphism group of the height n Honda formal group law Γ_{n} over 𝔽_{p^{n}}. Let Gal ≡ Gal(𝔽_{p^{n}}/𝔽_{p}) denote the Galois group of the field extension 𝔽_{p^{n}}/𝔽_{p}, and let G_{n} = S_{n} ⋊ Gal. Then, according to Morava’s theorem (see [21], [5], and §1 for a resumé), E_{n^{∗}}X is naturally a G_{n}-module for X a finite spectrum. In addition, the pairing E_{n^{∗}}X ⊗_{E_{n^{∗}}} E_{n^{∗}}Y → E_{n^{∗}}(X ∧ Y) is G_{n}-equivariant, where the left side is given the diagonal action. Since E_{n^{∗}} is profinite in each degree, it follows that there exists, in the stable category, a unique action of G_{n} on E_{n} by ring spectrum maps inducing the above action on E_{n^{∗}}X.

If a group acts on a spectrum, one can form the homotopy fixed point spectrum; this of course requires that the group act in an appropriate point-set category and not just up to homotopy. Yet the situation in the case of the homotopy action of G_{n} on E_{n} is not hopeless. Indeed, H. R. Miller and the second author have proved that E_{n} is an A_{∞} ring spectrum and that the space of A_{∞} ring maps from E_{n} to itself has weakly contractible path components. Furthermore, the set of path components of this space is in bijective correspondence with the set of homotopy classes of ring spectrum maps from E_{n} to itself. (See [25] for an account of this theory.) Therefore, G_{n} acts on E_{n} via A_{∞} maps, and although this action is still only an action up to homotopy, it is an honest action up to “all higher A_{∞} homotopies”. Standard techniques then allow one to replace E_{n} by an equivalent spectrum on which G_{n} acts on the nose. Hence, if G is a (finite) subgroup of G_{n}, there is an A_{∞} homotopy fixed point spectrum E_{n}^{hG}. These spectra have already had a number of interesting applications in stable homotopy theory (see e.g. [22]). Subsequently, P. G. Goerss and Hopkins [12], [13],

†Both authors partially supported by the NSF.

[14] extended the machinery of Hopkins-Miller to the E_{∞} setting, and thus E_{n}^{hG} is an E_{∞} ring spectrum.

Unfortunately, this is still not an entirely satisfactory state of affairs. G_{n} is a profinite group, so one might hope to define, for G a closed subgroup of G_{n}, a homotopy fixed point spectrum E_{n}^{hG} whose “homotopy fixed point spectral sequence” starts with the continuous cohomology of G. (This is why we restricted to finite subgroups in the previous paragraph.) Indeed, it is the continuous cohomology of G_{n} that is important in stable homotopy theory— by Morava’s change of rings theorem, the K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence (see Appendix) for π_{∗}L_{K(n)}S^{0} has the form

H_{c}^{∗}(G_{n}, E_{n^{∗}}) = H_{c}^{∗}(S_{n}, E_{n^{∗}})^{Gal} ⇒ π_{∗}L_{K(n)}S^{0}. (0.1)

Here K(n) denotes n^{th} Morava K-theory, L_{K(n)} denotes K(n)_{∗}-localization, and E_{n}^{Gal} denotes the Landweber exact spectrum with coefficient ring ℤ_{p}[[u_{1}, …, u_{n−1}]][u, u^{−1}]. (In general, if H is a subgroup of Gal, E_{n}^{H} denotes the Landweber exact spectrum with coefficient ring W (𝔽_{p^{n}}^{H})[[u_{1}, …, u_{n−1}]][u, u^{−1}], where 𝔽_{p^{n}}^{H} is the subfield of 𝔽_{p^{n}} fixed by the automorphism group H.) The Adams spectral sequence 0.1 thus suggests that L_{K(n)}S^{0} should be the G_{n} homotopy fixed point spectrum of E_{n} in this continuous sense.

In this paper, we construct spectra E_{n}^{hG}, G a closed subgroup of G_{n}, having this desired property. Our construction proceeds in two steps. First we construct E_{n}^{hU} for U an open— and hence closed—subgroup of G_{n}, and then we construct E_{n}^{hG} as an appropriate homotopy colimit of the E_{n}^{hU}’s, for G ⊂ U.

We need to introduce a little more notation in order to state our main results. Let R_{G_{n}}^{+} denote the category whose objects are continuous finite left G_{n}-sets together with the left G_{n}-set G_{n}. The morphisms are continuous G_{n}-equivariant maps, and we denote by r_{g} : G_{n} → G_{n} the map given by right multiplication by g ∈ G_{n}. Let E_{∞} denote the category of commutative S^{0}-algebras in the category of S^{0}-modules of May et al [11]. (A comparison of this category with the category of L-spectra for L the linear isometries operad is carried out in [11; II.4].) Here then is our first main result.

Theorem 1 There is a functor F : (R_{G_{n}}^{+})^{opp} → E_{∞} with the following properties:

i. F(S) is K(n)_{∗}-local for each object S in R_{G_{n}}^{+}. ii. F(G_{n}) = E_{n} and F(r) : E_{n} → E_{n} is the action of g ∈ G_{n} on E_{n}.

iii. There is a natural isomorphism

π_{∗}L_{K(n)}(F(S) ∧ E_{n}^{Gal}) ≈ Map(S, Map_{c}(S_{n}, E_{n^{∗}}))^{G_{n}},

of completed π_{∗}L_{K(n)}(E_{n}^{Gal} ∧ E_{n}^{Gal}) = Map_{c}(S_{n}, E_{n^{∗}})^{Gal}-comodules, where the action of G_{n} on Map_{c}(S_{n}, E_{n^{∗}}), the set of continuous maps from S_{n} to E_{n^{∗}}, is given by

((g, σ)f)(g′) = σ(f((g, σ)^{−1}g′)) = σf(σ^{−1}(g^{−1}g′))

for g, g′ ∈ S_{n}, σ ∈ Gal, and f ∈ Map_{c}(S_{n}, E_{n^{∗}}) (see 1.2). In particular, F(∗) ≃ L_{K(n)}S^{0}.

iv. Define E_{n}^{hU} = F(G_{n}/U), U an open subgroup of G_{n}, and let Z be any CW-spectrum. There is a natural strongly convergent spectral sequence

H_{c}^{∗}(U, E_{n}^{∗}Z) ⇒ (E_{n}^{hU})^{∗}Z

which agrees with the spectral sequence obtained by mapping Z into a K(n)_{∗}-local E_{n}^{Gal}- Adams resolution of E_{n}^{hU}.

Remark 0.2 In what follows—and in the proof of Theorem 1—the necessity of working in a precise point-set category of structured ring spectra will become apparent. However, many of our results, such as iii and iv of Theorem 1, are statements occurring in the stable category. We shall therefore refer to objects as “S^{0}-module spectra” or “commutative S^{0}-algebras” when we wish to emphasize that we need to work at the point-set level and as “CW-spectra” or “ring spectra” when our work takes place in the stable category. Furthermore, [X, Y]^{i} will always denote the group of maps of degree −i between X and Y in the stable category.

Despite our precautions, there are still some ambiguities. For example, suppose X and Y are commutative S^{0}-algebras, but we wish to understand the stable homotopy type of X ∧ Y. Then X ∧ Y might denote the object which is the CW-approximation to the actual point-set level smash product of X and Y. Or, X ∧ Y might denote the derived smash product; that is, the smash product of CW-approximations to X and Y. However, if X and Y are cofibrant objects in the closed model category structure on E_{∞}—called q-cofibrant in [11]—then these two recipes give the same stable object (see [11; VII, 6]). Since there is a functor E_{∞} → E_{∞} sending an object to a weakly equivalent q-cofibrant one—in fact, a cell commutative S^{0}-algebra in the sense of [11]—we may as well assume that F(S), for example, is always q-cofibrant and thus eliminate this ambiguity. We will often use this device of functorially approximating by a cell object, even if we don’t always mention it.

Remark 0.3 Let I_{n} = (p, v_{1}, …, v_{n−1}) ⊂ E_{n^{∗}}, and, if Z is any CW-spectrum, let {Z_{α}} be the directed system of finite CW-subspectra. Then E_{n}^{∗}Z = lim_{⟵_{α,k}} E_{n}^{∗}Z_{α}/I_{n}^{k}E_{n}^{∗}Z_{α} is a profinite

continuous G_{n}-module and we define

H_{c}^{∗}(U, E_{n}^{∗}Z) = lim_{⟵_{α,k}} H_{c}^{∗}(U, E_{n}^{∗}Z_{α}/I_{n}^{k}E_{n}^{∗}Z_{α}).

In general, if G is a p-analytic profinite group and M is a profinite continuous ℤ_{p}[[G]]-module, then lim_{⟵_{α}} H_{c}^{∗}(G, M_{α}) is independent of the presentation M = lim_{⟵_{α}} M_{α}. In fact, it is for example

the cohomology of the usual cochain complex whose j-cochains are Map_{c}(G^{(j)}, M) (cf. proof of Lemma 3.23). We therefore use this as our definition of H_{c}^{∗}(G, M); see also [28] for a more general treatment of this object.

For the next step, let

G_{n} = U_{0} ⊋ U_{1} ⊋ U_{2} ⊋ ⋯ ⊋ U_{i} ⊋ ⋯ (0.4)

be a sequence of normal open subgroups of G_{n} with ⋂_{i} U_{i} = {e}. For example, using the

description of S_{n} in [7; 2.21], we may take U_{i} = V_{i} ⋊ Gal, where V_{i} is the group of power series ∑_{j≥0}^{Γ_{n}}b_{j}x^{p^{j}} with b_{j} = 0 for 0 < j < i and b_{0} = 1 if i > 0.

Definition 0.5 Fix a sequence {U_{i}} as above. For G a closed subgroup of G_{n}, define

E_{n}^{hG} = L_{K(n)} holim_{⟶_{i}} _{E_{∞}}E_{n}^{h(U_{i}G)},

where holim_{⟶}_{E_{∞}} denotes the homotopy colimit taken in the topological model category E_{∞}.

Remark 0.6 More precisely, one should functorially replace holim_{⟶_{i}} _{E_{∞}}E_{n}^{h(U_{i}G)} by a weakly

equivalent cell commutative S^{0}-algebra before applying L_{K(n)}. Then the construction E_{n}^{hG} becomes functorial in E_{∞} (see [11; VIII, 2]).

We will prove the following result in §5.

Theorem 2 The construction E_{n}^{hG} has the following properties:

i. π_{∗}L_{K(n)}(E_{n}^{hG} ∧ E_{n}^{Gal}) is naturally isomorphic to Map_{c}(S_{n}, E_{n^{∗}})^{G} as completed Map_{c}(S_{n}, E_{n^{∗}})^{Gal}-comodules, where G acts on Map_{c}(S_{n}, E_{n^{∗}}) as in Theorem 1iii.

ii. Let Z be any CW-spectrum. The spectral sequence obtained by mapping Z into a K(n)_{∗}-local E_{n}^{Gal}-Adams resolution of E_{n}^{hG} is strongly convergent to (E_{n}^{hG})^{∗}Z and has E_{2}-term naturally isomorphic to H_{c}^{∗}(G, E_{n}^{∗}Z).

Remark 0.7 i. Using the first part of this theorem, it’s easy to see that E_{n}^{hG} is canonically independent (up to weak equivalence in E_{∞}) of the choice of sequence {U_{i}}.

ii. Since G is a closed subgroup of a p-analytic profinite group, it is itself p-analytic (see [9; 10.7]), and therefore H_{c}^{∗}(G, E_{n}^{∗}Z) is defined as in Remark 0.3.

Now by Theorem 1, there are commutative diagrams

\xymatrix{E_{n}^{h(U_{i}G)} = F(G_{n}/U_{i}G) \ar[d] \ar[d] \ar[r] & F(G_{n}) = E_{n} \\ E_{n}^{h(U_{j}G)} = F(G_{n}/U_{j}G) \ar@{-->}[ur]}

and

\xymatrix{E_{n}^{h(U_{i}G)} = F(G_{n}/U_{i}G) \ar[r] \ar@{-->}[dr] & F(G_{n}) = E_{n} \ar[d]^{F(r_{g})} \ar[d]^{g} \\  & F(G_{n}) = E_{n}}

for i ≤ j and g ∈ G, where the unlabeled arrows are induced by the evident projections. There is thus a canonical map E_{n}^{hG} → E_{n}^{G}, the G-fixed points of E_{n}. Let us denote by E_{n}^{h′G} the ordinary homotopy fixed point spectrum for the action of G on E_{n}. Composing the above map with the canonical map E_{n}^{G} → E_{n}^{h′G} ([4; XI, 3.5]) yields a map E_{n}^{hG} → E_{n}^{h′G}. The next result will be proved in §5.

Theorem 3 Let G be a finite subgroup of G_{n}. Then:

i. The map E_{n}^{hG} → E_{n}^{h′G} described above is a weak equivalence.

ii. Let Z be any CW-spectrum. The homotopy fixed point spectral sequence

H^{∗}(G, E_{n}^{∗}Z) ⇒ (E_{n}^{h′G})^{∗}Z = (E_{n}^{hG})^{∗}Z

is naturally isomorphic to the spectral sequence of Theorem 2ii.

We also have a result on iterated homotopy fixed point spectra. Indeed, if K is a closed and U is a normal open subgroup of G_{n}, then the (opposite) Weyl group W (K) = N(K)/K acts on G_{n}/UK via xUK ↦ xhUK for x ∈ G_{n}, h ∈ N(K). This yields an action of W (K) on E_{n}^{h(UK)}, and hence, upon passing to the homotopy colimit, on E_{n}^{hK}. In particular, if F is a finite subgroup of W (K), we may form (E_{n}^{hK})^{hF} in the usual way. We can now state our next result.

Theorem 4 Suppose G is a closed subgroup of G_{n}, K is a closed normal subgroup of G, and F ≡ G/K is finite. Then E_{n}^{hG} is naturally equivalent to (E_{n}^{hK})^{hF}.

Our final result, which will be proven as an application of the machinery developed here, was originally due to the second author and H. R. Miller, who suggested that this is the place where it should logically appear.

Theorem 5 (Hopkins-Miller) Suppose c : S_{n} → ℤ_{p} is a continuous Galois equivariant homomorphism, and let c also denote the composition S_{n} ⟶^{c} ℤ_{p}⟶ E_{n^{∗}}. Then c ∈ H_{c}^{1}(S_{n}, E_{n^{∗}})^{Gal} survives to π_{∗}L_{K(n)}S^{0}.

Let us indicate our strategy for constructing E_{n}^{hU}. We will construct a cosimplicial E_{∞} spectrum corresponding to the K(n)_{∗}-local E_{n}^{Gal}-Adams resolution of E_{n}^{hU}; then we will define E_{n}^{hU} to be Tot of this cosimplicial spectrum. To do this, we need only determine the (expected) homotopy type of L_{K(n)}(E_{n}^{hU} ∧ E_{n}^{Gal}) together with the comodule structure map

L_{K(n)}(E_{n}^{hU} ∧ E_{n}^{Gal}) = L_{K(n)}(E_{n}^{hU} ∧ S^{0} ∧ E_{n}^{Gal}) → L_{K(n)}(E_{n}^{hU} ∧ E_{n}^{Gal} ∧ E_{n}^{Gal}).

Now we might expect a spectral sequence

H_{c}^{∗}(U, π_{∗}L_{K(n)}(E_{n} ∧ E_{n}^{Gal})) ⇒ π_{∗}L_{K(n)}(E_{n}^{hU} ∧ E_{n}^{Gal}),

since there is such a spectral sequence if U is replaced by a finite subgroup of G_{n} (see Theorem 4.3). But π_{∗}L_{K(n)}(E_{n} ∧ E_{n}^{Gal}) = Map_{c}(S_{n}, E_{n^{∗}}), and the action of U is given as in Theorem 1iii (see §1). Since Map_{c}(S_{n}, E_{n^{∗}}) is U-acyclic (cf. proof of Lemma 3.22), we should have

π_{∗}L_{K(n)}(E_{n}^{hU} ∧ E_{n}^{Gal}) = Map_{c}(S_{n}, E_{n^{∗}})^{U}.

The comodule structure map is also determined. But we will see in the proof of Proposition 3.10 that, as E_{n^{∗}}^{Gal}-modules,

Map_{c}(S_{n}, E_{n^{∗}})^{U} = ∏_{i=1}^{m} E_{n^{∗}}^{H_{i}}

for some m and subgroups H_{i} of Gal. We are thus led to take X_{G_{n}/U} = ∏_{i=1}^{m} E_{n}^{H_{i}} as a

model for L_{K(n)}(E_{n}^{hU} ∧ E_{n}^{Gal}). We can also define the corresponding comodule structure map X_{G_{n}/U} → L_{K(n)}(X_{G_{n}/U} ∧ E_{n}^{Gal}). (Note that this map is not—except in a few very special cases—the product of the maps

E_{n}^{H} = L_{K(n)}(SW (𝔽_{p^{n}}^{H}) ∧ E_{n}^{Gal}) = L_{K(n)}(SW (𝔽_{p^{n}}^{H}) ∧ S^{0} ∧ E_{n}^{Gal})
→ L_{K(n)}(SW (𝔽_{p^{n}}^{H}) ∧ E_{n}^{Gal} ∧ E_{n}^{Gal})
= L_{K(n)}(E_{n}^{H} ∧ E_{n}^{Gal}),

where SW (𝔽_{p^{n}}^{H}) denotes the Moore spectrum for the abelian group W (𝔽_{p^{n}}^{H}).) With this construction, we obtain a cosimplicial object, but only in the stable category. However, the technology of Hopkins-Miller as expanded by Goerss-Hopkins is again available to allow us to conclude that the requisite diagrams in fact commute up to all higher E_{∞} homotopies. The original cosimplicial object can now be replaced by an equivalent cosimplicial object in E_{∞}, and Tot can be formed.

The contents of this paper are as follows. In §1, we recall the relevant parts of Morava’s theory, and in §2, we discuss the formation of homotopy inverse limits for certain diagrams commutative up to all higher homotopies. We construct the cosimplicial Adams resolution in §3 and identify the resultant spectral sequence as having the form of a (continuous) homotopy fixed point spectral sequence. In §4 we compute π_{∗}L_{K(n)}(F(S)∧E_{n}^{Gal}); this allows us to identify the preceding spectral sequence as a K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence and completes the proof of Theorem 1. We prove Theorems 2 and 3 in §5, Theorem 4 in §6, and, finally, Theorem 5 in §7. An appendix summarizes the properties of K(n)_{∗}-localizations and K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequences that we need. In particular, we prove the strong convergence of these spectral sequences.

Acknowledgment. The first author would like to thank P. G. Goerss for a helpful discussion about Theorem 2.2.

1 Resumé of Morava’s theory

Start by observing that Map_{c}(S_{n}, W 𝔽_{p^{n}})^{Gal} is a completed Hopf algebra over ℤ_{p}; the diagonal map is given by

Map_{c}(S_{n}, W 𝔽_{p^{n}})^{Gal} → Map_{c}(S_{n} × S_{n}, W 𝔽_{p^{n}})^{Gal} ⟵^{≈} Map_{c}(S_{n}, W 𝔽_{p^{n}})^{Gal} ˆ⊗Map_{c}(S_{n}, W 𝔽_{p^{n}})^{Gal},

where the first map is induced by the multiplication on S_{n} and the second is an isomorphism by [5; AII.3]. There is also a map

η_{L} : E_{n^{∗}}^{Gal} → Map_{c}(S_{n}, W 𝔽_{p^{n}})^{Gal} ˆ⊗_{ℤ_{p}}E_{n^{∗}}^{Gal} ⟶^{≈} Map_{c}(S_{n}, E_{n^{∗}})^{Gal}

whose description is given in [5; 5.5]. In terms of the action of S_{n} on E_{n^{∗}} given in [7], we have η_{L}(x)(g) = g^{−1}x, x ∈ E_{n^{∗}}^{Gal}, g ∈ S_{n}. With these structure maps, we obtain a split completed Hopf algebroid (E_{n^{∗}}^{Gal}, Map_{c}(S_{n}, W 𝔽_{p^{n}})^{Gal} ˆ⊗_{ℤ_{p}}E_{n^{∗}}^{Gal}). A main result of Morava’s theory is the following identification.

Theorem 1.1 (E_{n^{∗}}^{Gal}, π_{∗}L_{K(n)}(E_{n}^{Gal}∧E_{n}^{Gal})) is isomorphic to (E_{n^{∗}}^{Gal}, Map_{c}(S_{n}, W 𝔽_{p^{n}})^{Gal} ˆ⊗_{ℤ_{p}}E_{n^{∗}}^{Gal}) as completed Hopf algebroids.

Proof. We showed in [5; §4] that (E_{n^{∗}}^{Gal}, Map_{c}(S_{n}, W 𝔽_{p^{n}})^{Gal} ˆ⊗E_{n^{∗}}^{Gal}) is isomorphic to a completed Hopf algebroid denoted (E_{n^{∗}}^{Gal}, E_{n^{∗}}^{Gal} ˆ⊗_{U}US ˆ⊗_{U}E_{n^{∗}}^{Gal}). But we observed in [6; 3.4] that this Hopf algebroid is isomorphic to (E_{n^{∗}}^{Gal}, E_{n^{∗}}^{Gal} ˆ⊗_{BP_{∗}}BP_{∗}BP ˆ⊗_{BP_{∗}}E_{n^{∗}}^{Gal}), where the completed tensor product here denotes I_{n}-adic completion. Since π_{∗}L_{K(n)}(E_{n}^{Gal} ∧ E_{n}^{Gal}) is the I_{n}-adic completion of E_{n^{∗}}^{Gal}E_{n}^{Gal} = E_{n^{∗}}^{Gal} ⊗_{BP_{∗}} BP_{∗}BP ⊗_{BP_{∗}} E_{n^{∗}}^{Gal} (see [16]), the proof is complete.

Now if M is a completed left π_{∗}L_{K(n)}(E_{n}^{Gal} ∧ E_{n}^{Gal})-comodule, then W 𝔽_{p^{n}} ˆ⊗M is a G_{n}- module; the action of Gal is given by its action on W 𝔽_{p^{n}}, and the action of S_{n} is given by

g(c ⊗ m) = c · ψ(m)(g^{−1})

where g ∈ S_{n} and

ψ : M → Map_{c}(S_{n}, W 𝔽_{p^{n}})^{Gal} ˆ⊗E_{n^{∗}}^{Gal} ˆ⊗_{E_{n^{∗}}^{Gal}}M → Map_{c}(S_{n}, W 𝔽_{p^{n}} ˆ⊗M)^{Gal}

is the comodule structure map. In particular, if X is a finite CW-spectrum, E_{n^{∗}}X is naturally a G_{n}-module; this allows us to construct an action of G_{n} on E_{n} in the stable category as described in the Introduction. π_{∗}L_{K(n)}(E_{n}∧E_{n}^{Gal}) then becomes a G_{n}-module with G_{n} acting on the left factor E_{n}, and we have

π_{∗}L_{K(n)}(E_{n} ∧ E_{n}^{Gal}) = W 𝔽_{p^{n}} ⊗_{ℤ_{p}} π_{∗}L_{K(n)}(E_{n}^{Gal} ∧ E_{n}^{Gal})
= W 𝔽_{p^{n}} ⊗_{ℤ_{p}} Map_{c}(S_{n}, E_{n^{∗}})^{Gal}
= Map_{c}(S_{n}, E_{n^{∗}}) (see [5; 5.4])

as G_{n}-modules, where G_{n} acts on Map_{c}(S_{n}, E_{n^{∗}}) as in Theorem 1iii. The following notation will be needed for a generalization of this identification.

Notation 1.2 Let M be a continuous Gal-module which is an inverse limit of discrete Galmodules. Then let Map_{c}^{ℓ}(S_{n}^{j}, M) denote the G_{n}-module with action given by

((g, σ)h)(g_{1}, …, g_{j}) = σ(h((g, σ)^{−1}g_{1}, σ^{−1}g_{2}, …, σ^{−1}g_{j}))
= σ(h(σ^{−1}(g^{−1}g_{1}), σ^{−1}g_{2}, …, σ^{−1}g_{j}))

for (g, σ) ∈ G_{n}, (g_{1}, …, g_{j}) ∈ S_{n}^{j}, and h ∈ Map_{c}(S_{n}^{j}, M). If M is a continuous G_{n}-module, Map_{c}^{ℓ}(S_{n}^{j}, M) denotes the above G_{n}-module, where M is regarded as only a continuous Galmodule. Observe that

Map_{c}^{ℓ}(S_{n}^{j}, M) = Map_{c}^{ℓ}(S_{n}^{j−1}, Map_{c}^{ℓ}(S_{n}, M))

for any “pro-discrete” Gal-module M.

If Z is any finite CW-spectrum, then

[Z, L_{K(n)}(E_{n} ∧ (E_{n}^{Gal})^{(j)})]^{∗} = π_{∗}L_{K(n)}(E_{n} ∧ (E_{n}^{Gal})^{(j)} ∧ DZ)
= E_{n^{∗}}E_{n}^{Gal} ˆ⊗_{E_{n^{∗}}^{Gal}}E_{n^{∗}}^{Gal}E_{n}^{Gal} ˆ⊗_{E_{n^{∗}}^{Gal}} ⋯ ˆ⊗_{E_{n^{∗}}^{Gal}}E_{n^{∗}}^{Gal}DZ,

where in the tensor product, E_{n^{∗}}^{Gal} acts on the right of the factor to the left and on the left of the factor to the right. But

E_{n^{∗}}E_{n}^{Gal} ˆ⊗_{E_{n^{∗}}^{Gal}}E_{n^{∗}}^{Gal}E_{n}^{Gal} ˆ⊗_{E_{n^{∗}}^{Gal}} ⋯ ˆ⊗_{E_{n^{∗}}^{Gal}}E_{n^{∗}}^{Gal}DZ =
W 𝔽_{p^{n}} ˆ⊗E_{n^{∗}}^{Gal}E_{n}^{Gal} ˆ⊗_{E_{n^{∗}}^{Gal}} ⋯ ˆ⊗_{E_{n^{∗}}^{Gal}}E_{n^{∗}}^{Gal}DZ =
W 𝔽_{p^{n}} ˆ⊗Map_{c}(S_{n}, W 𝔽_{p^{n}})^{Gal} ˆ⊗ ⋯ ˆ⊗Map_{c}(S_{n}, W 𝔽_{p^{n}})^{Gal} ˆ⊗E_{n^{∗}}^{Gal}DZ =
Map_{c}(S_{n}^{j}, W 𝔽_{p^{n}})^{Gal} ˆ⊗W 𝔽_{p^{n}} ˆ⊗E_{n^{∗}}^{Gal}DZ =
Map_{c}(S_{n}^{j}, W 𝔽_{p^{n}})ˆ⊗E_{n^{∗}}^{Gal}DZ = Map_{c}(S_{n}^{j}, E_{n}^{∗}Z).

Now G_{n} acts on [Z, L_{K(n)}(E_{n}∧(E_{n}^{Gal})^{(j)})]^{∗} via its action on the left factor E_{n}, and, chasing down the identifications, we obtain

[Z, L_{K(n)}(E_{n} ∧ (E_{n}^{Gal})^{(j)})]^{∗} = Map_{c}^{ℓ}(S_{n}^{j}, E_{n}^{∗}Z) (1.3)

as G_{n}-modules. This isomorphism will be generalized to arbitrary Z in §3.

In §7, we will also need to know the formula for the action of G_{n} on the right factor of π_{∗}L_{K(n)}(E_{n}^{Gal} ∧ E_{n}), where we write

π_{∗}L_{K(n)}(E_{n}^{Gal} ∧ E_{n}) = π_{∗}L_{K(n)}(E_{n}^{Gal} ∧ E_{n}^{Gal}) ⊗_{ℤ_{p}} W 𝔽_{p^{n}} = Map_{c}(S_{n}, E_{n^{∗}}).

Let us write g^{R}x for the action of g ∈ S_{n} on x ∈ π_{∗}L_{K(n)}(E_{n}^{Gal} ∧ E_{n}) and g^{L}y for the action of g on y ∈ π_{∗}L_{K(n)}(E_{n} ∧ E_{n}^{Gal}). Let us also write τ for the transposition map L_{K(n)}(E_{n}^{Gal} ∧ E_{n}) → L_{K(n)}(E_{n} ∧ E_{n}^{Gal}) or L_{K(n)}(E_{n} ∧ E_{n}^{Gal}) → L_{K(n)}(E_{n}^{Gal} ∧ E_{n}).

Now consider

h ⊗ e ∈ Map_{c}(S_{n}, ℤ_{p}) ⊗_{ℤ_{p}} E_{n^{∗}} = Map_{c}(S_{n}, E_{n^{∗}}).

Then

g^{R}(h ⊗ e) = τg^{L}(τ(h ⊗ e)) = τg^{L}(η_{L}(e)χ(h)),

where χ(h)(s) = h(s^{−1}). Continuing the computation, we obtain

τg^{L}(η_{L}(e)χ(h)) = τ(η_{L}(ge)g(χ(h))) = χ(g(χ(h))) ⊗ ge.

But

χ(g(χ(h)))(s) = g(χ(h))(s^{−1}) = χ(h)(g^{−1}s^{−1}) = h(sg).

This implies that, for f ∈ Map_{c}(S_{n}, E_{n^{∗}}) and (g, σ) ∈ G_{n}, we have

\begin{aligned}(g^{R}f)(s) & = & g(f(sg)) \\ (σf)(s) & = & σf(σ^{−1}s) \end{aligned} . (1.4)

2 Homotopy inverse limits

Let hE_{∞} denote the homotopy category of commutative S^{0}-algebras. That is, two maps f, g : X → Y in E_{∞} are homotopic if f and g lie in the same path component of E_{∞}(X, Y), the topological space of S^{0}-algebra maps between X and Y. Alternatively, E_{∞} is a tensored category (over the category of unbased topological spaces), and f and g are homotopic if there exists a map h : X ⊗ I → Y restricting to f and g on the ends of the cylinder (see [11; VII, 2]).

Now let J be a small category and suppose X : J → hE_{∞} is a functor. In some cases, X can be replaced by a homotopy equivalent strict diagram, and then its homotopy inverse limit can be formed.

Definition 2.1 Let X : J → hE_{∞} be as above. X is said to be an h_{∞}-diagram if for each α : j_{1} → j_{2} in J, E_{∞}(Xj_{1}, Xj_{2})_{Xα}, the path component of E_{∞}(Xj_{1}, Xj_{2}) containing Xα, is weakly contractible.

The main result of this section is due to W. G. Dwyer, D. M. Kan, and J. H. Smith [10]. We feel that the reader will appreciate an account of the proof.

Theorem 2.2 Let X : J → hE_{∞} be an h_{∞}-diagram. Then there exists a functor X̄ : J → E_{∞} and a natural transformation X̄ → X in hE_{∞} such that X̄(j) → X(j) is a weak equivalence for each j ∈ ObJ.

The proof makes use of a modification of the cosimplicial replacement of a diagram (cf. [4; XI, 5]). We first recall some notation.

If Z is an unbased topological space and Y is a commutative S^{0}-algebra, let F (Z, Y) denote the cotensor product of Z and Y in E_{∞}; its underlying S^{0}-module is just the function S^{0}-module of maps from Σ^{∞}Z_{+} to Y ([11; VII, 2]). We shall also use the notation F ( , ) to denote function spectra in the stable category.

With Z as above, let ΓZ denote the geometric realization of the singular simplicial set of Z. This is of course a functorial cofibrant replacement of Z; it also has the property that ΓV × ΓW is naturally homeomorphic to Γ(V × W) and that this homeomorphism commutes with the projections onto ΓV and ΓW. Hence a pairing V × W → Z induces a pairing ΓV × ΓW → ΓZ.

Construction 2.3 Let X : J → hE_{∞} be an h_{∞}-diagram. Define a cosimplicial commutative S^{0}-algebra ∏_{h}^{∗} X by

∏_{h}^{0}X = ∏_{j∈J} Xj
∏_{h}^{n}X = ∏_{J_{n}} F (ΓE_{∞}Xα, Xj_{0}),

where J_{n} is the set of diagrams

α : j_{0} ⟵^{α_{1}} j_{1} ⟵^{α_{2}} j_{2} ⟵ ⋯ ⟵j_{n−1} ⟵^{α_{n}} j_{n}

in J, and

ΓE_{∞}Xα = ΓE_{∞}(Xj_{1}, Xj_{0})_{Xα_{1}} × ⋯ × ΓE_{∞}(Xj_{n}, Xj_{n−1})_{Xα_{n}}.

If 0 < i < n + 1, the coface d^{i} is defined via the commutative diagram

\xymatrix{∏_{h}^{n} X \ar[r]^{d^{i}} \ar[d]^{π_{α′}} & ∏_{h}^{n+1} X \ar[d]^{π_{α}} \\ F (ΓE_{∞}Xα′, Xj_{0}) \ar[r]^{d_{α}^{i}} & F (ΓE_{∞}Xα, Xj_{0})} (2.4)

where π_{α} denotes the projection onto the factor indexed by

α : j_{0} ⟵^{α_{1}} j_{1} ⟵^{α_{2}} j_{2} ⟵ ⋯ ⟵j_{n} ⟵^{α_{n+1}} j_{n+1},

α′ denotes the diagram

j_{0} ⟵^{α_{1}} j_{1} ⟵ ⋯ ⟵j_{i−1} ⟵^{α_{i}α_{i+1}} j_{i+1} ⟵^{α_{i+2}} j_{i+2} ⟵ ⋯ ⟵j_{n} ⟵^{α_{n+1}} j_{n+1},

and

(d_{α}^{i}g)(f_{1}, …, f_{n+1}) = g(f_{1}, …, f_{i}f_{i+1}, …, f_{n+1})

for (f_{1}, …, f_{n+1}) ∈ ΓE_{∞}Xα. (Here f_{i}f_{i+1} denotes the image of (f_{i}, f_{i+1}) under the map

ΓE_{∞}(Xj_{i}, Xj_{i−1}) × ΓE_{∞}(Xj_{i+1}, Xj_{i}) → ΓE_{∞}(Xj_{i+1}, Xj_{i−1})

induced by the composition pairing

E_{∞}(Xj_{i}, Xj_{i−1}) × E_{∞}(Xj_{i+1}, Xj_{i}) → E_{∞}(Xj_{i+1}, Xj_{i−1}).)

If i = 0, d^{0} is defined as in 2.4, although now α′ is the diagram

j_{1} ⟵^{α_{2}} j_{2} ⟵ ⋯ ⟵^{α_{n+1}} j_{n+1},

and d_{α}^{0} : F (ΓE_{∞}Xa′, Xj_{1}) → F (ΓE_{∞}Xα, Xj_{0}) is defined by

d_{α}^{0}(g)(f_{1}, …, f_{n+1}) = f_{1}(g(f_{2}, …, f_{n+1})),

where f_{1} also denotes the image of f_{1} ∈ ΓE_{∞}(Xj_{1}, Xj_{0}) in E_{∞}(Xj_{1}, Xj_{0}).

Finally, if i = n + 1, α′ is the diagram

j_{0} ⟵^{α_{1}} j_{1} ⟵ ⋯ ⟵j_{n−1} ⟵^{α_{n}} j_{n}

and

d_{α}^{n+1}(g)(f_{1}, …, f_{n+1}) = g(f_{1}, …, f_{n}).

As for the codegeneracies, s^{i} is defined via the commutative diagram

\xymatrix{∏_{h}^{n+2} X \ar[r]^{s^{i}} \ar[d]^{π_{α′}} & ∏_{h}^{n+1} X \ar[d]^{π_{α}} \\ F (ΓE_{∞}Xα′, Xj_{0}) \ar[r]^{s_{α}^{i}} & F (ΓE_{∞}Xα, Xj_{0})} (2.5)

where α′ is the diagram

j_{0} ⟵^{α_{1}} j_{1} ⟵^{α_{2}} j_{2} ⟵ ⋯ ⟵j_{i−1} ⟵^{α_{1}} j_{i} ⟵^{id} j_{i} ⟵^{α_{i+1}} j_{i+1} ⟵ ⋯ ⟵^{α_{n+1}} j_{n+1}

and

s_{α}^{i}(g)(f_{1}, …, f_{n+1}) = g(f_{1}, …, f_{i}, id, f_{i+1}, …, f_{n+1}).

Here id denotes the image of ∗ = Γ(∗) in ΓE_{∞}(Xj_{i}, Xj_{i}) under the evident map.

Recall that a cosimplicial S^{0}-module Y is fibrant if the map

s : Y^{n+1} → M^{n}Y ≡ {(y^{0}, …, y^{n}) ∈ Y^{n} × ⋯ × Y^{n} : s^{i}y^{j} = s^{j−1}y^{i} ∀ 0 ≤ i < j ≤ n}

given by s(y) = (s^{0}(y), …, s^{n}(y)) is a q-fibration—that is, a fibration in the Quillen closed model sense—for all n ≥ −1. (Properly speaking, M^{n}Y should be defined as an equalizer; however we will continue to use this shorthand when to do otherwise would result in more confusion.)

Lemma 2.6 Let X be an h_{∞}-diagram. Then ∏_{h}^{∗} X is fibrant.

Proof. Consider first the map s : ∏_{h}^{1} X → M^{0} ∏_{h}^{∗} X = ∏_{h}^{0} X. The composition

∏_{h}^{1}X ⟶^{s} ∏_{h}^{0}X = ∏_{j} Xj → Xj

is given by

∏_{h}^{1}X = ∏_{j_{0} ⟵^{α}j_{1}} F (ΓE_{∞}(Xj_{1}, Xj_{0})_{Xα}, Xj_{0}) → F (ΓE_{∞}(Xj, Xj)_{id}, Xj) → Xj,

where the last map is evaluation at id ∈ ΓE_{∞}(Xj, Xj). Since id is a vertex of ΓE_{∞}(Xj, Xj), it follows that this last map is a q-fibration and hence so is s.

Now suppose n > 1. Let J_{n,j}^{k} be the subset of J_{n} consisting of those n-tuples of maps with j_{0} = j and α_{k} = id, and set

D_{n,j}^{k}X = ∐_{α∈J_{n,j}^{k}} ΓE_{∞}Xα.

Then let

D_{n,j}X = ⋃_{1≤k≤n} D_{n,j}^{k}X,

and observe that the map s : ∏_{h}^{n} X → M^{n−1} ∏_{h}^{∗} X restricts to an isomorphism

∏_{j} F (D_{n,j}X, Xj) ⟶^{≈} M^{n−1}∏_{h}^{∗}X.

But the inclusion

D_{n,j}X → ∐_{J_{n,j}} ΓE_{∞}Xα

is the inclusion of a summand, where J_{n,j} is the subset of J_{n} consisting of those n-tuples with j_{0} = j. s is therefore a q-fibration in this case as well.

Recall that, given a fibrant cosimplicial S^{0}-module Y, Tot Y = F (Δ[∗], Y), the S^{0}- module of (unpointed) cosimplicial maps from Δ[∗] to Y. Here Δ[∗] is the cosimplicial space which in dimension n is the standard n-simplex Δ[n] with the usual coface and codegeneracy maps. Let Sk_{s}Δ[∗] be the cosimplicial space which in dimension n is the s-skeleton of Δ[n], and define Tot_{s}Y = F (Sk_{s}Δ[∗], Y). The map Tot_{j+1}Y → Tot_{j}Y is a fibration with fiber F (Δ[j + 1]/Δ̇[j + 1], Y^{j+1} ∩ ker s_{0} ∩ ⋯ ∩ ker s_{j}). By mapping a CW-spectrum Z into the tower {Tot_{j}Y }, we obtain a spectral sequence

E_{2}^{s,t} = π^{s}([Z, Y]^{t}) ⟹ [Z, Tot Y]^{t+s} (cf. [4; X.6]). (2.7)

This spectral sequence is strongly convergent in the sense of [4; IX.5.4] provided lim_{⟵_{J}} ^{1}E_{r}^{s,t} = 0

for all s and t.

In particular, if X is an h_{∞}-diagram, we obtain a spectral sequence

E_{2}^{s,t} = lim_{⟵_{J}} ^{s}([Z, X]^{t}) ⟹ [Z, Tot ∏_{h}^{∗}X]^{t+s}. (2.8)

This is proved as in [4; XI.7.1], using the fact that ΓE_{∞}Xα is contractible for all α.

With these constructions in hand, we can now prove the main result of this section.

Proof of Theorem 2.2. If j is an object of J, let J\j be the category whose objects are morphisms j → j′ in J and whose morphisms are the evident commutative diagrams. The evident functor μ_{j} : J\j → J provides us with an h_{∞}-diagram μ_{j}^{∗}X over J\j and hence a functor X̄ : J → E_{∞} defined by X̄(j) = Tot ∏_{h}^{∗} μ_{j}^{∗}X.

Now define the map X̄(j) → X(j) to be the composition

Tot ∏_{h}^{∗}μ_{j}^{∗}X → F (Δ[0], ∏_{h}^{0}μ_{j}^{∗}X) = ∏_{h}^{0}μ_{j}^{∗}X ⟶^{p} Xj, (2.9)

where p is the projection onto the factor indexed by the object j ⟶^{id} j. To prove that

\xymatrix{X̄(j) \ar[d] \ar[r]^{X̄f} & X̄(j′) \ar[d] \\ X(j) \ar[r]^{Xf} & X(j′)}

commutes whenever f : j → j′, we must prove that

\xymatrix{X̄(j) \ar[r] \ar[dr] & X(j) \ar[d]^{Xf} \\ & X(j′)} (2.10)

commutes, where the diagonal map is the projection onto ∏_{h}^{0} μ_{j}^{∗}X followed by the projection onto the factor indexed by f : j → j′. First observe that the two compositions

X̄(j) → F (Δ[1], ∏_{h}^{1}μ_{j}^{∗}X) \overrightarrow{⟶}_{F (d^{1},id)}^{F (d^{0},id)} F (Δ[0], ∏_{h}^{1}μ_{j}^{∗}X) ⟶^{p_{f}} X(j′)

are homotopic, where the last map is the projection onto the factor indexed by f : (j ⟶^{id} j) → (j ⟶^{f} j′). But these compositions are the same as

X̄(j) → F (Δ[0], ∏_{h}^{0}μ_{j}^{∗}X) \overrightarrow{⟶}_{d^{1}}^{d^{0}} ∏_{h}^{1}μ_{j}^{∗}X ⟶^{p_{f}} X(j′).

Now use the definitions of d^{0} and d^{1} to check that these maps give the commutative diagram 2.10.

By 2.8, there is a spectral sequence

E_{2}^{s,t} = lim_{⟵_{J\j}} ^{s}π_{−t}μ_{j}^{∗}X ⇒ π_{−t−s}\overline{X}(j).

But j → j is an initial object of J\j; therefore

lim_{⟵_{J\j}} ^{s}π_{t}μ_{j}^{∗}X = \{ \begin{matrix}0 & s > 0 \\ π_{t}Xj & s = 0 \end{matrix} .

Thus π_{∗}X̄(j) = π_{∗}X(j), and an unraveling of the identifications shows that the map in 2.9 induces the identity on π_{∗}.

3 Construction of the functor F

We begin by stating the following extensions of the Hopkins-Miller theory due to Goerss and Hopkins. These are the results needed to show that, for S ∈ ObR_{G_{n}}^{+}, the cosimplicial object C_{S} we will construct in the stable category lifts to an h_{∞}-diagram, and hence, by Theorem 2.2, to a cosimplicial object in E_{∞}. Then F(S) is defined to be Tot C_{S}.

Theorem 3.1 ([14]) Let E and F be q-cofibrant commutative S^{0}-algebras with E Landweber exact and with E_{∗}F a flat π_{∗}E-module. Suppose that π_{∗}E is concentrated in even dimensions with an invertible element in degree 2. Suppose in addition that π_{0}E is 𝔪-adically complete for some ideal 𝔪, that E_{0}/𝔪 is an algebra of characteristic p, and that the relative frobenius for the homomorphism E_{0}/𝔪 → E_{0}F/𝔪E_{0}F is an isomorphism. Then each path component of E_{∞}(F, E) is weakly contractible, and the Kronecker pairing

π_{0}E_{∞}(F, E) → Hom_{E_{∗}-alg}(E_{∗}F, E_{∗})

is a bijection.

Remark 3.2 Recall that if A is a commutative 𝔽_{p}-algebra, the frobenius homomorphism ϕ_{A} : A → A is defined by ϕ_{A}(x) = x^{p}. If f : A → B is a homomorphism of commutative 𝔽_{p}-algebras, the relative frobenius is defined to be the map given by the dotted arrow in the following diagram, where the square is a push-out in the category of commutative 𝔽_{p}- algebras:

\xymatrix{ &  & B \\ B \ar[r] \ar[urr]^{ϕ_{B}} & C \ar@{.>}[ur] \\ A \ar[u]^{f} \ar[r]_{ϕ_{A}} & A \ar[u] \ar[uur]_{f}}

Remark 3.3 The condition that E be Landweber exact can be weakened. Indeed, E need only satisfy a condition of the sort required by Adams [1; III, 13.3] in his construction of universal coefficient spectral sequences. For example, Property 1.1 of [6] suffices. In particular, any Landweber exact spectrum satisfies this property [6; 1.3].

Theorem 3.4 ([14]) Let H be a subgroup of Gal. Then E_{n}^{H} has a unique structure of commutative S^{0}-algebra descending to its ordinary ring spectrum structure.

Remark 3.5 E_{n}^{H} is in fact the H homotopy fixed point spectrum of E_{n}, so our notation is only slightly abusive.

In order to construct C_{S}, we need to apply Theorem 3.1 to spectra of the form L_{K(n)}(X ∧ (E_{n}^{Gal})^{(j)}), where X is a finite product of E_{n}^{H_{i}}’s. The next result enables us to do this.

Proposition 3.6 Let E = L_{K(n)}(X ∧ (E_{n}^{Gal})^{(j)}) and let F = L_{K(n)}(Y ∧ (E_{n}^{Gal})^{(k)}), where X and Y are (non-empty) finite products of E_{n}^{H_{i}}’s and j, k ≥ 0. Then the pair F, E satisfies the conditions of Theorem 3.1; moreover, the function

π_{0}E_{∞}(F, E) → Hom_{alg}(F_{∗}, E_{∗})

sending a map F → E to the induced map on homotopy groups is one-to-one.

Proof. First observe that E and F are commutative S^{0}-algebras—the key point here is that localization with respect to a homology theory preserves such objects (see [11; VIII]). An easy reduction also shows that it suffices to consider the case where X = E_{n}^{H} and Y = E_{n}^{H′} for subgroups H, H′ of Gal. By Morava’s theory,

E_{∗} = W 𝔽_{p^{n}}^{H} ⊗ π_{∗}L_{K(n)}((E_{n}^{Gal})^{(j+1)}) = Map_{c}(S_{n}^{j}, E_{n^{∗}})^{Gal} ⊗_{ℤ_{p}} W 𝔽_{p^{n}}^{H};

thus E is Landweber exact for an appropriate map BP → E. By the Landweber exact functor theorem, E_{∗}F is a flat E_{∗}-module. E_{0} is also complete with respect to the ideal 𝔪 = Map_{c}(S_{n}^{j}, I)^{Gal} ⊗_{ℤ_{p}} W 𝔽_{p^{n}}^{H}, where I = (p, u_{1}, …, u_{n−1}) ⊂ (E_{n})_{0}. Furthermore,

π_{∗}L_{K(n)}(E ∧ F) = Map_{c}(S_{n}^{j+k+1}, E_{n^{∗}})^{Gal} ⊗_{ℤ_{p}} W 𝔽_{p^{n}}^{H} ⊗_{ℤ_{p}} W 𝔽_{p^{n}}^{H′};

in particular L_{K(n)}(E ∧ F) is Landweber exact. Now E ∧ F is also Landweber exact; this follows from the flatness of E_{∗}F over E_{∗}. Hence, if M(p^{i_{0}}, v_{1}^{i_{1}}, …, v_{n−1}^{i_{n−1}}) is a finite CWspectrum whose Brown-Peterson homology is BP_{∗}/(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}), we have that

E_{∗}F/(p^{i_{0}}, …, v_{n−1}^{i_{n−1}})E_{∗}F = π_{∗}(E ∧ F ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}))
⟶^{≈} π_{∗}(L_{K(n)}(E ∧ F) ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}))
= π_{∗}L_{K(n)}(E ∧ F)/(p^{i_{0}}, …, v_{n−1}^{i_{n−1}})π_{∗}L_{K(n)}(E ∧ F).

Thus

E_{∗}F/I_{n}E_{∗}F = π_{∗}L_{K(n)}(E ∧ F)/I_{n}π_{∗}(L_{K(n)}(E ∧ F))
= Map_{c}(S_{n}^{j+k+1}, 𝔽_{p^{n}}[u, u^{−1}])^{Gal} ⊗_{𝔽_{p}} 𝔽_{p^{n}}^{H} ⊗_{𝔽_{p}} 𝔽_{p^{n}}^{H′}.

Clearly the frobenius is an isomorphism on E_{0}/𝔪, and, from the above equality, it is also an isomorphism on E_{0}F/𝔪E_{0}F. Hence the relative frobenius for E_{0}/𝔪 → E_{0}F/𝔪E_{0}F is an isomorphism.

Finally, to prove that π_{0}E_{∞}(F, E) → Hom_{alg}(F_{∗}, E_{∗}) is one-to-one, it suffices to prove that the function

Hom_{E_{∗}-alg}(E_{∗}F, E_{∗}) → Hom_{alg}(F_{∗}, E_{∗})

given by precomposition with the map

F_{∗} = π_{∗}(S^{0} ∧ F) → π_{∗}(E ∧ F) = E_{∗}F

is one-to-one. But this follows easily from the commutative diagram

[figure: Hom_{E_{∗}-alg}(E_{∗}F, E_{∗})⟶Hom_{alg}(F_{∗}, E_{∗}); Hom_{(E_{∗}⊗ℚ)-alg}(E_{∗}F ⊗ ℚ, E_{∗} ⊗ ℚ) = Hom_{alg}(F_{∗} ⊗ ℚ, E_{∗} ⊗ ℚ).]

It will also be convenient to know that, for E and F as above, the canonical map from π_{0}E_{∞}(F, E) to the set of ring spectrum maps from F to E in the stable category is a bijection. The next result enables us to conclude this.

Lemma 3.7 Let E be as in Proposition 3.6, and let F be a Landweber exact commutative ring spectrum. Then

i. E_{∗}F^{(j)} = \underbrace{E_{∗}F ⊗_{E_{∗}} E_{∗}F ⊗_{E_{∗}} ⋯ ⊗_{E_{∗}} E_{∗}F}_{j times} for any j ≥ 1.

ii. The Kronecker pairing

[F^{(j)}, E]^{∗}⟶Hom_{E_{∗}}(E_{∗}F^{(j)}, E_{∗})

is an isomorphism for all j ≥ 1.

Proof. Since E and F are Landweber exact, E_{∗}F is a flat E_{∗}-module. This immediately implies i. As for ii, begin by observing that F_{1} ∧ F_{2} is Landweber exact if F_{1} and F_{2} are. Hence F^{(j)} satisfies the hypotheses of the lemma, so we may assume j = 1 without loss of generality.

There is a universal coefficient spectral sequence

Ext_{E_{∗}}^{∗∗}(E_{∗}F, E_{∗}) ⇒ E^{∗}F

(see Remark 3.3); it thus suffices to show that Ext_{E_{∗}}^{i}(E_{∗}F, E_{∗}) = 0 for all i > 0. We do this by an argument similar to that in [25; 15.6]. Indeed, E_{∗}F = E_{∗} ⊗_{E_{n^{∗}}^{Gal}} E_{n^{∗}}^{Gal}F and E_{n^{∗}}^{Gal}F is flat over E_{n^{∗}}^{Gal}; hence

Ext_{E_{∗}}^{∗∗}(E_{∗}F, E_{∗}) = Ext_{E_{n^{∗}}^{Gal}}^{∗∗}(E_{n^{∗}}^{Gal}F, E_{∗}).

We thus need only show that Ext_{(E_{n}^{Gal})_{0}}^{i}((E_{n}^{Gal})_{0}F, E_{0}) = 0 for all i > 0, where now E = L_{K(n)}((E_{n}^{Gal})^{(j+1)}) for some j ≥ 0. Let 𝔪 be as in the proof of Proposition 3.6, and write (E_{n}^{Gal})_{0}F = M. Then again by flat base change

Ext_{(E_{n}^{Gal})_{0}}^{i}(M, E_{0}/𝔪) = Ext_{𝔽_{p}}^{i}(M/IM, E_{0}/𝔪).

This last group is trivial for i > 0 since 𝔽_{p} is a field. Similarly, Ext_{(E_{n}^{Gal})_{0}}^{i}(M, 𝔪^{t}/𝔪^{t+1}) = 0 for i > 0; now use the fact that E_{0} = lim_{⟵_{t}} E_{0}/𝔪^{t} to conclude that Ext_{(E_{n}^{Gal})_{0}}^{i}(M, E_{0}) = 0

whenever i > 0.

Now let Δ be the category whose objects are the nonnegative integers—a typical object will be denoted [n]—and whose morphisms from [n] to [m] are the monotone nondecreasing functions from {0, 1, …, n} to {0, 1, …, m}. We will construct an h_{∞}-diagram

C : (R_{G_{n}}^{+})^{opp} × Δ → hE_{∞}

such that, for each S ∈ ObR_{G_{n}}^{+},

C_{S} = C(S, ) : Δ → hE_{∞}

is a cosimplicial K(n)_{∗}-local E_{n}^{Gal}-Adams resolution for the not yet constructed spectrum F(S). As described in the Introduction (for the case S = G_{n}/U), we seek a K(n)_{∗}-local commutative S^{0}-algebra and right E_{n}^{Gal}-module spectrum X_{S} together with an E_{∞}-map d : X_{S} → L_{K(n)}(X_{S} ∧ E_{n}^{Gal}) such that

π_{∗}X_{S} = Map_{G_{n}}(S, Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})) (3.8)

as right E_{n^{∗}}^{Gal}-modules, such that

(3.9) π_{∗}L_{K(n)}(X_{S} ∧ E_{n}^{Gal}) ⟵^{≈} π_{∗}X_{S} ˆ⊗_{E_{n^{∗}}^{Gal}}π_{∗}L_{K(n)}(E_{n}^{Gal} ∧ E_{n}^{Gal})
= Map_{G_{n}}(S, Map_{c}^{ℓ}(S_{n}, E_{n^{∗}}))ˆ⊗_{E_{n^{∗}}^{Gal}}Map_{c}(S_{n}, E_{n^{∗}})^{Gal}
= Map_{G_{n}}(S, Map_{c}^{ℓ}(S_{n} × S_{n}, E_{n^{∗}})),

and such that π_{∗}d corresponds to the map induced by the group multiplication S_{n}×S_{n} → S_{n}. Such spectra do in fact exist.

Proposition 3.10 There exists an h_{∞}-diagram X : (R_{G_{n}}^{+})^{opp} → hE_{∞} and a natural transformation d : X → L_{K(n)}(X ∧ E_{n}^{Gal}) satisfying the above requirements. Moreover, this construction has the following properties:

i. All maps are maps of right E_{n}^{Gal}-module spectra, where E_{n}^{Gal} acts on the right factor of L_{K(n)}(X ∧ E_{n}^{Gal}).

ii. If S is finite, X_{S} ≡ X(S) is a finite product of E_{n}^{H_{i}}’s for some subgroups H_{i} of Gal.

iii. X_{G_{n}} = L_{K(n)}(E_{n} ∧ E_{n}^{Gal}) and if r_{g} : G_{n} → G_{n} is right multiplication by g ∈ G_{n}, X(r_{g}) = L_{K(n)}(g ∧ E_{n}^{Gal}).

iv. d(G_{n}) is given by the composition

L_{K(n)}(E_{n} ∧ E_{n}^{Gal}) = L_{K(n)}(E_{n} ∧ S^{0} ∧ E_{n}^{Gal}) → L_{K(n)}(E_{n} ∧ E_{n}^{Gal} ∧ E_{n}^{Gal}).

Proof. If S is finite, then S is a finite disjoint union of sets of the form G_{n}/U for various open subgroups U of G_{n}. We thus need only define X_{G_{n}/U}; X_{S} can be defined as an appropriate finite product of the X_{G_{n}/U}’s. To define X_{G_{n}/U}, break S_{n} up into U-orbits, say S_{n} = ∐_{i} Ux_{i};

since U is open, there are only a finite number of them. Then

Map_{G_{n}}(G_{n}/U, Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})) = Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})^{U} = ∏_{i} E_{n^{∗}}^{H_{i}},

where U_{i} = Stabx_{i} ⊂ U and H_{i} is the image of U_{i} under the projection G_{n} = S_{n}⋊Gal → Gal. We thus define X_{G_{n}/U} = ∏_{i} E_{n}^{H_{i}}, and hence equations 3.8 and 3.9 are satisfied. As for the

map d(G_{n}/U), observe that, as a right E_{n}^{Gal}-module spectrum, X_{G_{n}/U} is equivalent to a wedge of E_{n}^{Gal}’s. Stable E_{n}^{Gal}-module maps X_{G_{n}/U} → L_{K(n)}(X_{G_{n}/U} ∧ E_{n}^{Gal}) are therefore in bijective correspondence with E_{n^{∗}}^{Gal}-module maps π_{∗}X_{G_{n}/U} → π_{∗}L_{K(n)}(X_{G_{n}/U} ∧ E_{n}^{Gal}).

Furthermore, by the results of Goerss-Hopkins together with Lemma 3.7, any stable ring map X_{G_{n}/U} → L_{K(n)}(X_{G_{n}/U} ∧ E_{n}^{Gal}) lifts to an E_{∞} map, unique up to E_{∞} homotopy. Hence d(G_{n}/U) can be chosen to induce the requisite map on homotopy groups.

If f : S_{1} → S_{2} is a G_{n}-map with S_{2} finite, then there exists a unique map Xf : X_{S_{2}} → X_{S_{1}} of E_{n}^{Gal}-module spectra inducing the map

f^{∗} : Map_{G_{n}}(S_{2}, Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})) → Map_{G_{n}}(S_{1}, Map_{c}^{ℓ}(S_{n}, E_{n^{∗}}))

on stable homotopy groups. As before, Xf has a representative in E_{∞} and is unique up to E_{∞} homotopy.

Finally, the naturality of π_{∗}X and π_{∗}d implies, by Lemma 3.6, that X is a functor and d is a natural transformation. It also follows from 3.6 that X is an h_{∞}-diagram.

Construction 3.11 Let η : S^{0} → E_{n}^{Gal} be the unit map, let μ : E_{n}^{Gal} ∧ E_{n}^{Gal} → E_{n}^{Gal} be the multiplication map, and, for each S ∈ ObR_{G_{n}}^{+}, let s : X_{S} ∧ E_{n}^{Gal} → X_{S} be the module structure map. Define an h_{∞}-diagram

C : (R_{G_{n}}^{+})^{opp} × Δ → hE_{∞}

by

C(S, [j]) ≡ C_{S}^{j} = L_{K(n)}(X_{S} ∧ (E_{n}^{Gal})^{(j)}).

The coface and codegeneracy maps are given by

d^{0}(S) = L_{K(n)}(d(S) ∧ (E_{n}^{Gal})^{(j)}) : C_{S}^{j} → C_{S}^{j+1}
d^{i}(S) = L_{K(n)}(X_{S} ∧ (E_{n}^{Gal})^{(i−1)} ∧ η ∧ (E_{n}^{Gal})^{(j−i+1)}) : C_{S}^{j} → C_{S}^{j+1},  1 ≤ i ≤ j + 1
s^{0}(S) = L_{K(n)}(s ∧ (E_{n}^{Gal})^{(j)}) : C_{S}^{j+1} → C_{S}^{j}
s^{i}(S) = L_{K(n)}(X_{S} ∧ (E_{n}^{Gal})^{(i−1)} ∧ μ ∧ (E_{n}^{Gal})^{(j−1)} : C_{S}^{j+1} → C_{S}^{j},  1 ≤ i ≤ j.

That C is in fact a functor follows from Proposition 3.6; this proposition along with the Goerss-Hopkins results also implies that C is an h_{∞}-diagram. Hence by Theorem 2.2, we may lift C to a diagram in E_{∞}. From now on, when we refer to C, we actually mean this lift of C.

Definition 3.12 The functor F : (R_{G_{n}}^{+})^{opp} → E_{∞} of Theorem 1 is defined by F(S) = Tot (∏^{∗} C_{S}), where, as before, ∏^{∗} C_{S} is the cosimplicial replacement of the (cosimplicial) diagram C_{S}. Since each C_{S}^{j} is K(n)_{∗}-local, so is F(S).

By 2.8, there is a spectral sequence

lim_{⟵_{Δ}} ^{s}[Z, C_{S}]^{t} ⟹ [Z, F(S)]^{t+s} (3.13)

for any CW-spectrum Z. We next claim that

lim_{⟵_{Δ}} ^{s}[Z, C_{S}]^{t} = π^{s}[Z, C_{S}^{∗}]^{t}, (3.14)

where the last term denotes the s^{th} cohomology group of the cosimplicial abelian group [Z, C_{S}]^{t}. This result is of course well-known; however we provide a quick proof for the convenience of the reader.

Proposition 3.15 Let Ab^{Δ} be the category of cosimplicial abelian groups. Then the δfunctors lim_{⟵_{Δ}} ^{∗} : Ab^{Δ} → gr_{+}Ab and π^{∗} : Ab^{Δ} → gr_{+}Ab are naturally equivalent, where gr_{+}Ab

denotes the category of nonnegatively graded abelian groups.

Proof. Since lim_{⟵_{Δ}} ^{0} and π^{0} are naturally equivalent, it suffices to prove that π^{s} is effaceable.

We begin this proof by making a few recollections. The forgetful functor U : Ab^{Δ} → gr_{+}Ab has a right adjoint V —if D is an object of gr_{+}Ab, set

(V D)[n] = ∏_{[n]→[m]} D^{m},

where the product is taken over all morphisms [n] → [m] in Δ. It thus follows that if D_{m} is injective for each m, then V D is injective in Ab^{Δ}. Now let C be a cosimplicial abelian group, and consider the monomorphism C → V UC. We will show that π^{i}(V UC) = 0 for all i > 0. Indeed, first observe that V D = ∏_{n} V (D(n)), where

D(n)^{m} = \{ \begin{matrix}D^{n} & m = n \\ 0 & otherwise \end{matrix} .

It then suffices to show that π^{i}(V (UC(n))) = 0 for all n and all i > 0. But V (UC(n)) is just the simplicial cochain complex of the standard n-simplex with coefficients in C^{n}. Therefore its cohomology is trivial in positive degrees.

In general, if J is a small category and A is a J-diagram of abelian groups, then lim_{⟵_{J}} ^{∗}A

may be computed as the cohomology of the cochain complex of the cosimplicial group ∏^{∗} A, the cosimplicial replacement of A (see [4; XI, 6.2]). Thus the δ-functors π^{∗}C and π^{∗}(∏^{∗} C) are naturally equivalent on the category of cosimplicial abelian groups. The next result will be needed to prove Proposition A.5; the reader may prefer to skip its proof now.

Proposition 3.16 Let C be a cosimplicial abelian group. There is a natural map T (C) : C → ∏^{∗} C of cochain complexes which induces an isomorphism on π^{∗}. Moreover, T can be chosen so that the composition

C^{0} ⟶^{T (C)} ∏_{j} C^{j} ⟶^{π} C^{j_{0}}

sends x to d^{0} ⋯ d^{0}x for any x ∈ C^{0}, j_{0} ≥ 0.

The proof of this proposition requires some preparation.

Definition 3.17 If k ≥ 0, let D_{k}^{∗} be the cosimplicial abelian group with D_{k}^{n} = ⨁_{[k]→[n]} ℤ, the

sum ranging over all morphisms [k] → [n] in Δ. Let ι_{k} ∈ D_{k}^{k} denote the element 1 in the summand corresponding to the identity [k] → [k].

D_{k}^{∗} has a convenient universal property: If C is a cosimplicial abelian group and x ∈ C^{k}, there exists a unique map τ_{x} : D_{k}^{∗} → C of cosimplicial abelian groups with τ_{x}(ι_{k}) = x. We can also put the D_{k}^{∗}’s together.

Definition 3.18 Let D_{∗}^{∗} be the simplicial cosimplicial abelian group whose cosimplicial group of k-simplices is D_{k}^{∗}. If [m] → [k] is a morphism in Δ, the map D_{k}^{∗} → D_{m}^{∗} is defined by sending the summand indexed by [k] → [n] to the summand indexed by [m] → [k] → [n] via the identity.

Lemma 3.19 There exists a sequence of maps T_{k} : D_{k}^{∗} → ∏^{∗} D_{k}^{∗} of cosimplicial abelian groups such that:

i. The composition

\xymatrix{D_{0}^{0} \ar[r]^{T_{0}} & ∏_{j} D_{0}^{j} \ar[r]^{π} & D_{0}^{j_{0}}}

sends ι_{0} to d^{0} ⋯ d^{0}ι_{0} for all j_{0} ≥ 0.

ii. The diagram

\xymatrix{D_{k+1}^{∗} \ar[r]^{∂} \ar[d]^{T_{k+1}} & D_{k}^{∗} \ar[d]^{T_{k}} \\ ∏^{∗} D_{k+1}^{∗} \ar[r]|{∂} & ∏^{∗} D_{k}^{∗}}

commutes for all k ≥ 0, where D_{∗}^{∗} and ∏^{∗} D_{∗}^{∗} are here regarded as chain complexes of cosimplicial abelian groups and ∂ denotes their respective boundary maps.

Proof. We construct T_{k} by induction on k. There is a unique cosimplicial map T_{0} satisfying i. To construct T_{k+1}, it suffices to prove that T_{k}(∂ι_{k+1}) is a boundary in the chain complex ∏^{k+1} D_{∗}^{∗}; we may then define T_{k+1}(ι_{k+1}) = c, where ∂c = T_{k}(ι_{k+1}).

We claim that H_{i}(∏^{k+1} D_{∗}^{∗}) = 0 for all i > 0. Indeed, for fixed i, H_{i}(D_{∗}^{∗})is a cosimplicial group and H_{i}(∏^{k+1} D_{∗}^{∗}) = ∏^{k+1} H_{i}(D_{∗}^{∗}).. But the chain complex D_{∗}^{j} is just the simplicial chain complex of the standard j-simplex; therefore

H_{i}(D_{∗}^{j}) = \{ \begin{matrix}ℤ & i = 0 \\ 0 & otherwise \end{matrix} .

Since 0 = T_{k−1}(∂∂ι_{k+1}) = ∂T_{k}(∂ι_{k+1}) by the inductive hypothesis, it now follows that T_{k}(∂ι_{k+1}) is a boundary if k > 0. If k = 0, use the fact that the maps

\xymatrix{D_{0}^{j} \ar[r]^{T_{0}^{j}} & ∏^{j} D_{0}^{∗} \ar[r]^{π} & D_{0}^{i}}

are augmentation preserving to conclude that T_{1}(∂ι_{1}) is a boundary as well. This completes the induction and the proof.

Proof of Proposition 3.16. For x ∈ C^{k}, define T (C)x to be the image of ι_{k} under the maps

\xymatrix{D_{k}^{k} \ar[r]^{T_{k}} & ∏^{k} D_{k}^{∗} \ar[r]^{∏^{∗} τ_{x}} & ∏^{k} C.}

Since ∂ : D_{k+1}^{∗} → D_{k}^{∗} maps ι_{k+1} to ∑_{i=0}^{k+1}(−1)^{i}d^{i}ι_{k}, it follows that τ_{δx} = τ_{x} ∘ ∂. By 3.19ii, we

then have that T (C)(δx) is given by the image of ι_{k+1} under the composition

\xymatrix{D_{k+1}^{k+1} \ar[r]^{∂} & D_{k}^{k+1} \ar[r]^{T_{k}} & ∏^{k+1} D_{k}^{∗} \ar[r]^{∏^{∗} τ_{x}} & ∏^{k+1} C.}

But (∏^{∗} τ_{x}) ∘ T_{k} is a cosimplicial map; therefore

((∏^{∗}τ_{x}) ∘ T_{k} ∘ ∂)(ι_{k+1}) = ((∏^{∗}τ_{x}) ∘ T_{k})(∑_{i=0}^{k+1}(−1)^{i}d^{i}ι_{k})

= ∑_{i=0}^{k+1} d^{i}((∏^{∗}τ_{x}) ∘ T_{k})(ι_{k})
= δT (C)(x)

Thus T (C) is a cochain map.

Now it is also clear from the definition of T that T^{0} is as required and hence that T is an isomorphism on π^{0}. This implies that T is an isomorphism on π^{∗}, completing the proof.

We can use spectral sequence 3.13 to verify Theorem 1ii. This will follow easily from the next result.

Proposition 3.20 Consider the maps E_{n} → L_{K(n)}(E_{n} ∧ E_{n}^{Gal}) and F(G_{n}) → L_{K(n)}(E_{n} ∧ E_{n}^{Gal}) given by the maps

E_{n} = E_{n} ∧ S^{0} ⟶^{E_{n}∧η} L_{K(n)}(E_{n} ∧ E_{n}^{Gal})

and

F(G_{n}) = Tot ∏^{∗}C_{G_{n}} → ∏^{0}C_{G_{n}} → C_{G_{n}}^{0} = L_{K(n)}(E_{n} ∧ E_{n}^{Gal})

respectively. Let Z be any CW-spectrum. Then there is a unique bijection E_{n}^{∗}Z → F(G_{n})^{∗}Z such that the diagram

\xymatrix{E_{n}^{∗}Z \ar[rr] \ar[dr] &  & F(G_{n})^{∗}Z \ar[dl] \\  & L_{K(n)}(E_{n} ∧ E_{n}^{Gal})^{∗}Z}

commutes.

Proof. the map E_{n} → L_{K(n)}(E_{n} ∧ E_{n}^{Gal}) is an augmentation of the cosimplicial object C_{G_{n}}^{∗} in the stable category. Furthermore, C_{G_{n}}^{∗} is chain contractible to E_{n} in the stable category (cf. proof of Lemma 4.4); hence π^{∗}[Z, C_{G_{n}}^{∗}]^{t} = E_{n}^{t}Z concentrated in degree 0. The desired result now follows from 3.13, 3.14, and an unraveling of the identifications.

Proof of Theorem 1ii. The preceding proposition provides us with a canonical weak equivalence E_{n} → F(G_{n}). It also implies that the map is G_{n}-equivariant and is a map of ring spectra (in the stable category). By Lemma 3.7 and Theorem 3.1 this weak equivalence lifts to an E_{∞} map.

Finally, we show that, for S = G_{n}/U, the spectral sequence 3.13 has the form of a homotopy fixed point spectral sequence. The proof requires some preparation.

Definition 3.21 Say that a discrete G_{n}-module N is a discrete twisted G_{n} − W 𝔽_{p^{n}} module if it is in addition a torsion W 𝔽_{p^{n}}-module and σ(en) = σ(e)σ(n) for all σ ∈ Gal, e ∈ W 𝔽_{p^{n}}, and n ∈ N. Now let M be an inverse limit of discrete twisted G_{n} − W 𝔽_{p^{n}} modules, and let H be a closed subgroup of G_{n}. Define a cochain complex D_{H}^{∗}(M) by

D_{H}^{j}(M) = (Map_{c}^{ℓ}(S_{n}^{j+1}, M))^{H}

with differential δ : D_{H}^{j}(M) → D_{H}^{j+1}(M) given by

δh(s_{0}, s_{1}, …, s_{j+1}) = (∑_{i=0}^{j}(−1)^{i}h(s_{0}, …, s_{i}s_{i+1}, …, s_{j+1})) + (−1)^{j+1}s_{j+1}^{−1}h(s_{0}, …, s_{j}).

Lemma 3.22 The δ-functor H^{∗}(D_{H}^{∗}(?)) is equivalent to H_{c}^{∗}(H,?) on the abelian category of discrete twisted G_{n} − W 𝔽_{p^{n}} modules.

Proof. First observe that H^{∗}(D_{H}^{∗}(?)) is in fact a δ-functor. Indeed, it follows from the proof of [5; 5.4] that if N is a discrete twisted G_{n} − W 𝔽_{p^{n}} module, then N = N^{Gal} ⊗_{ℤ_{p}} W 𝔽_{p^{n}} as twisted Gal − W 𝔽_{p^{n}} modules; this implies that D_{H}^{∗}(?) is exact. Since H^{0}(D_{H}^{∗}(M)) = M^{H} (by a slightly tricky verification), it suffices to prove that H^{∗}D_{H}^{∗}(?) is effaceable. To this end, let N be a discrete twisted G_{n} − W 𝔽_{p^{n}} module, and consider Map_{c}^{ℓ}(S_{n}, N). Map_{c}^{ℓ}(S_{n}, N) is a discrete twisted G_{n} − W 𝔽_{p^{n}} module, and there is a G_{n}-equivariant W 𝔽_{p^{n}}-module homomorphism N → Map_{c}^{ℓ}(S_{n}, N) defined by n ↦ h_{n}, where h_{n}(s) = s^{−1}n. Now

D_{H}^{j}(Map_{c}^{ℓ}(S_{n}, N)) = Map_{c}^{ℓ}(S_{n}^{j+1} × S_{n}, N)^{H}

(see 1.2), and there is a contracting homotopy

q : D_{H}^{∗+1}(Map_{c}^{ℓ}(S_{n}, N)) → D_{H}^{∗}(Map_{c}^{ℓ}(S_{n}, N))

given by

(qh)(s_{0}, …, s_{j}, t) = (−1)^{j+1}h(s_{0}, …, s_{j}, t, 1).

Thus H^{i}(D_{H}^{∗}(Map_{c}^{ℓ}(S_{n}, N)) = 0 for all i > 0, completing the proof.

We will identify the cochain complex [Z, C_{G_{n}/U}^{∗}]^{t} with D_{U}^{∗}(E_{n}^{t}Z). We must thus understand the cohomology of D_{H}^{∗}(?) for profinite modules.

Lemma 3.23 Let H be a closed subgroup of G_{n} and let M be a profinite twisted G_{n} − W 𝔽_{p^{n}} module, say M = lim_{⟵_{α}} M_{α}, where α ranges over a directed set I. Then

i. lim_{⟵_{α}} ^{s}D_{H}^{j}(M_{α}) = \{ \begin{matrix}D_{H}^{j}(M) & s = 0 \\ 0 & s > 0 \end{matrix}

ii. H^{∗}D_{H}^{∗}(M) = lim_{⟵_{α}} H^{∗}D_{H}^{∗}(M_{α}) = lim_{⟵_{α}} H_{c}^{∗}(H, M_{α}).

Proof. We have

lim_{⟵_{α}} ^{s}D_{H}^{j}(M_{α}) = π^{s}∏^{∗}D_{H}^{j}(M),

where M is the I-diagram α ↦ M_{α} and ∏^{∗} D_{H}^{j}(M) is the cosimplicial replacement of the I-diagram D_{H}^{j}(M). Now

∏^{∗}D_{H}^{j}(M) = D_{H}^{j}(∏^{∗}M),

and since each M_{α} is finite, ∏^{q} M is profinite. We claim that

D_{H}^{j} : profinite twisted G_{n} − W 𝔽_{p^{n}} modules → Ab

is exact. Indeed, if A → B is an epimorphism of such modules, then A^{Gal} → B^{Gal} is an epimorphism of profinite abelian groups, since, as in the proof of Lemma 3.22, A = A^{Gal} ⊗_{ℤ_{p}} W 𝔽_{p^{n}} and B = B^{Gal} ⊗_{ℤ_{p}} W 𝔽_{p^{n}}. Hence by [27; I, Theorem 3], there is a continuous (settheoretic) cross-section B^{Gal} → A^{Gal}. This yields a continuous Galois equivariant function B → A and thus a cross-section D_{H}^{j}(B) → D_{H}^{j}(A). Therefore

lim_{⟵_{α}} ^{s}D_{H}^{j}(M_{α}) = D_{H}^{j}(π^{s}∏^{∗}M)
= D_{H}^{j}(lim_{⟵_{α}} ^{s}M_{α}) = \{ \begin{matrix}D_{H}^{j}(M) & s = 0 \\ 0 & s > 0 \end{matrix}

by the vanishing of lim_{⟵}^{s}, s > 0, for directed systems of profinite groups.

As for the second part, consider the double cochain complex ∏^{∗} D_{H}^{∗}(M). This yields two spectral sequences

lim_{⟵_{α}} ^{s}H_{c}^{t}(H, M_{α}) ⟹ H^{s+t}(∏^{∗}D_{H}^{∗}(M))
H^{s}(lim_{⟵_{α}} ^{t}D_{H}^{∗}(M_{α}) ⟹ H^{s+t}(∏^{∗}D_{H}^{∗}(M))

By i, the second spectral sequence collapses to give

H^{∗}(∏^{∗}D_{H}^{∗}(M)) = H^{∗}(D_{H}^{∗}(M)).

On the other hand, H is a closed subgroup of a p-analytic profinite group and is therefore itself a p-analytic group (see [9, 10.7]). Hence H contains an open normal subgroup U which is a Poincaré pro-p-group ([19]; see also [26] for a summary). This implies that H_{c}^{t}(H, M_{α}) is finite for each α and so lim_{⟵_{α}} ^{s}H_{c}^{t}(H, M_{α}) = 0 for all s > 0. The first spectral sequence then

collapses to give

H^{∗}(∏^{∗}D_{H}^{∗}(M)) = lim_{⟵_{α}} H_{c}^{∗}(H, M_{α}).

This completes the proof.

Lemma 3.24 There is a canonical isomorphism [Z, C_{G_{n}/U}^{∗}]^{t} ≈ D_{U}^{∗}E_{n}^{t}Z of cochain complexes, where E_{n}^{∗}Z is topologized as in Remark 0.3.

Proof. As in the proof of 1.3, we have, for Z_{α} a finite CW-spectrum,

[Z_{α}, C_{G_{n}/U}^{j}]^{−∗} = π_{∗}L_{K(n)}(X_{G_{n}/U} ∧ (E_{n}^{Gal})^{(j)} ∧ DZ_{α})
= Map_{c}(S_{n}, E_{n^{∗}})^{U} ˆ⊗_{E_{n^{∗}}^{Gal}}Map_{c}(S_{n}, E_{n^{∗}})^{Gal} ˆ⊗_{E_{n^{∗}}^{Gal}} ⋯
ˆ⊗_{E_{n^{∗}}^{Gal}}Map_{c}(S_{n}, E_{n^{∗}})^{Gal} ˆ⊗_{E_{n^{∗}}^{Gal}}E_{n^{∗}}DZ_{α}
= Map_{c}^{ℓ}(S_{n}^{j+1}, E_{n^{∗}}DZ_{α})^{U}
= Map_{c}^{ℓ}(S_{n}^{j+1}, E_{n}^{−∗}Z_{α})^{U}.

One can also check that the coboundary maps correspond, so that

[Z_{α}, C_{G_{n}/U}^{∗}]^{t} = D_{U}^{∗}E_{n}^{t}Z_{α}. (3.25)

In general, there is a spectral sequence

lim_{⟵_{α}} ^{s}[Z_{α}, C_{G_{n}/U}^{j}]^{t} ⟹ [Z, C_{G_{n}/U}^{j}]^{t},

where Z_{α} ranges over the finite CW-subspectra of Z. But by 3.25 and Lemma 3.23, lim_{⟵_{α}} ^{s}[Z_{α}, C_{G_{n}/U}^{j}]^{t} = 0 for all s > 0. Therefore,

[Z, C_{G_{n}/U}^{∗}]^{t} = lim_{⟵_{α}}[Z_{α}, C_{G_{n}/U}^{∗}]^{t} = D_{U}^{∗}E_{n}^{t}Z.

Finally, we obtain our desired result.

Proposition 3.26 Let S = G_{n}/U. The spectral sequence 3.13 has E_{2}-term canonically isomorphic to H_{c}^{s}(U, E_{n}^{t}Z).

This is the spectral sequence of Theorem 1iv. It is strongly convergent because H_{c}^{s}(U, E_{n}^{t}Z) is a profinite group for each s, t, and therefore lim_{⟵_{r}} ^{1}E_{r}^{s,t} = 0.

4 The Morava module of E_{n}^{hU}

In this section we complete the proof of Theorem 1. The key step is the identification of L_{K(n)}(F(S) ∧ E_{n}^{Gal}) with X_{S}—this not only immediately implies Theorem 1iii but enables us to identify the spectral sequence 3.13 with the K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence converging to [Z, F(S)]^{∗}. If S is finite, C_{S} = ∏_{i} C_{G_{n}/U_{i}} for a finite number of open subgroups

U_{i}; thus we may assume from the beginning that S = G_{n}/U. (If S = G_{n}, Theorem 1iii is a consequence of 1ii.)

The techniques involved in our computation of L_{K(n)}(F(S) ∧ E_{n}^{Gal}) will be applied in other contexts in §§5, 6; we therefore proceed in a little more generality.

Notation 4.1 Write C : J → E_{∞} for any of the following diagrams:

i. J = Δ and C = C_{G_{n}/U} for U an open subgroup of G_{n}.

ii. J = G for G a finite subgroup of G_{n} and C : G → E_{∞} the diagram given by the action of G on E_{n}.

iii. J = F, where F and K are as in Theorem 4 of the Introduction, and C : F → E_{∞} is the diagram given by the action of F on (the left factor of) L_{K(n)}(E_{n}^{hK} ∧ (E_{n}^{Gal})^{(j+1)}), j ≥ 0.

Given a diagram C : J → E_{∞}, there is a diagram L_{K(n)}(C ∧ E_{n}^{Gal}) : J → E_{∞} defined by

L_{K(n)}(C ∧ E_{n}^{Gal})(j) = L_{K(n)}(C(j) ∧ E_{n}^{Gal})
L_{K(n)}(C ∧ E_{n}^{Gal})(f) = L_{K(n)}(C(f) ∧ E_{n}^{Gal})

for j an object and f a morphism in J. There is also a canonical map

(4.2) L_{K(n)}[(Tot ∏^{∗}C) ∧ E_{n}^{Gal}] → Tot (∏^{∗}L_{K(n)}(C ∧ E_{n}^{Gal})).

We will prove the following result.

Theorem 4.3 If C is one of the diagrams in Notation 4.1, then the map 4.2 is a weak equivalence.

Before proving this theorem, we determine its consequence for C = C_{G_{n}/U}. The left side of 4.2 is L_{K(n)}(E_{n}^{hU} ∧ E_{n}^{Gal}). To identify the right side, we examine the spectral sequence ^{II}E_{r}^{∗,∗}(Z, C) obtained by mapping a CW-spectrum Z into the tower of fibrations {Tot_{k}(∏^{∗} L_{K(n)}(C ∧ E_{n}^{Gal}))}.

Lemma 4.4 Let C = C_{G_{n}/U}. Then

IIE_{2}^{s,t}(Z, C) = π^{s}[Z, L_{K(n)}(C^{∗} ∧ E_{n}^{Gal})]^{t} = \{ \begin{matrix}0 & s > 0 \\ [Z, X_{G_{n}/U}]^{t} & s = 0 \end{matrix} .

In particular, the map

Tot (∏^{∗}L_{K(n)}(C ∧ E_{n}^{Gal})) ⟶ ∏^{0}L_{K(n)}(C ∧ E_{n}^{Gal})
⟶ L_{K(n)}(C^{0} ∧ E_{n}^{Gal})
≃ L_{K(n)}(X_{G_{n}/U} ∧ E_{n}^{Gal})
⟶^{s} X_{G_{n}/U}

is a weak equivalence.

Proof. By Proposition 3.15, ^{II}E_{2}^{s,t}(Z, C) = π^{s}[Z, L_{K(n)}(C^{∗} ∧ E_{n}^{Gal})]^{t}. Now let X_{G_{n}/U}^{∗} be the constant cosimplicial spectrum with X_{G_{n}/U}^{j} = X_{G_{n}/U}. There are cosimplicial—in the stable category—maps X_{G_{n}/U}^{∗} → L_{K(n)}(C∧E_{n}^{Gal}) and L_{K(n)}(C∧E_{n}^{Gal}) → X_{G_{n}/U}^{∗} given on 0-simplices by d : X_{G_{n}/U} → L_{K(n)}(X_{G_{n}/U} ∧ E_{n}^{Gal}) (see Proposition 3.10) and s : L_{K(n)}(X_{G_{n}/U} ∧ E_{n}^{Gal}) → X_{G_{n}/U} respectively. Furthermore, these maps are chain homotopy equivalences; use the chain homotopy

h : L_{K(n)}(C^{j} ∧ E_{n}^{Gal}) → L_{K(n)}(C^{j−1} ∧ E_{n}^{Gal})

defined by h = (−1)^{j}X_{G_{n}/U} ∧(E_{n}^{Gal})^{(j−1)}∧μ. This then implies that π^{∗}[Z, L_{K(n)}(C^{∗}∧E_{n}^{Gal})]^{t} is as claimed. The weak homotopy equivalence Tot (∏^{∗} L_{K(n)}(C ∧ E_{n}^{Gal})) → X_{G_{n}/U} is obtained by tracking down the identifications.

Corollary 4.5 There is a natural weak equivalence L_{K(n)}(E_{n}^{hU} ∧ E_{n}^{Gal}) ⟶^{ν} X_{G_{n}/U} of commutative S^{0}-algebras and right E_{n}^{Gal}-module spectra such that the diagram

[figure: L_{K(n)}(E_{n}^{hU} ∧ E_{n}^{Gal}); ν; X_{G_{n}/U}; d; L_{K(n)}(E_{n}^{hU} ∧ S^{0} ∧ E_{n}^{Gal}); L_{K(n)}(E_{n}^{hU} ∧ E_{n}^{Gal} ∧ E_{n}^{Gal}); L_{K(n)}(ν∧E_{n}^{Gal}); L_{K(n)}(X_{G_{n}/U} ∧ E_{n}^{Gal})]

is homotopy commutative. In particular, Theorem 1iii holds.

We now turn to the proof of Theorem 4.3. Let ^{I}E_{r}^{∗,∗}(Z, C) denote the spectral sequence obtained by mapping a CW-spectrum Z into the tower

⋯ → L_{K(n)}(Tot_{k+1}(∏^{∗}C) ∧ E_{n}^{Gal}) → L_{K(n)}(Tot_{k}(∏^{∗}C) ∧ E_{n}^{Gal}) → ⋯;

we have

IE_{1}^{s,t}(Z, C) = [Z, L_{K(n)}(F_{s}(C) ∧ E_{n}^{Gal})]^{t+s}, (4.6)

where F_{s}(C) is the fiber of Tot_{s}(∏^{∗} C) → Tot_{s−1}(∏^{∗} C). There is a canonical stable map from the (unraveled exact couple of the) tower {L_{K(n)}(Tot_{k}(∏^{∗} C)∧E_{n}^{Gal})} to {Tot_{k}(∏^{∗} L_{K(n)}(C∧ E_{n}^{Gal}))}; on fibers this map is the canonical map

L_{K(n)}(F_{s}(C) ∧ E_{n}^{Gal}) → F_{s}(L_{K(n)}(C ∧ E_{n}^{Gal})). (4.7)

Lemma 4.8 Let ^{I}E_{2}^{∗,∗}(S^{0}, C) → ^{II}E_{2}^{∗,∗}(S^{0}, C) be the map of spectral sequences described above. If C is one of the diagrams in 4.1, then this map is an isomorphism.

Proof. If C is a diagram of the form 4.1ii or 4.1iii, then the map 4.7 is an equivalence and hence the desired result follows immediately.

Now let C = C_{G_{n}/U} and examine ^{I}E_{2}^{∗,∗}(S^{0}, C). By 3.14, H^{∗}([Z, F_{∗}(C)]^{t+∗}) = π^{∗}[Z, C^{∗}]^{t} for any CW-spectrum Z, and therefore Lemma 4.4 implies that

H^{s}π_{t−∗}(F_{∗}(C) ∧ E_{n}^{Gal} ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}})) = π^{s}π_{t}(C^{∗} ∧ E_{n}^{Gal} ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}))
= \{ \begin{matrix}π_{t}(X_{G_{n}/U} ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}})) & s = 0 \\ 0 & s ≠ 0. \end{matrix}

In particular, these cohomology groups are all finite. Here M(p^{i_{0}}, v_{1}^{i_{1}}, …, v_{n−1}^{i_{n−1}}) is as in [6]; the multi-index I = (i_{0}, …, i_{n−1}) varies over a cofinal sequence as in [6; §4], so that

L_{K(n)}Y = holim_{⟵_{I}}(Y ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}))

for any E(n)_{∗}-local spectrum Y.

We claim that

π_{∗} holim_{⟵_{I}}(F_{k}(C) ∧ E_{n}^{Gal} ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}})) = lim_{⟵_{I}} π_{∗}(F_{k}(C) ∧ E_{n}^{Gal} ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}));

that is

lim_{⟵_{I}} ^{1}π_{∗}(F_{k}(C) ∧ E_{n}^{Gal} ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}})) = 0 (4.9)

Assuming this, it follows from the vanishing of lim_{⟵_{I}} ^{1}H^{∗}π_{t−∗}(F_{∗}(C)∧E_{n}^{Gal} ∧M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}))

that

H^{s}π_{t−∗}L_{K(n)}(F_{∗}(C) ∧ E_{n}^{Gal}) = lim_{⟵_{I}} H^{s}π_{t−∗}(F_{∗}(C) ∧ E_{n}^{Gal} ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}))

= \{ \begin{matrix}π_{t}X_{G_{n}/U} & s = 0 \\ 0 & s ≠ 0. \end{matrix}

Furthermore, one checks easily that the map ^{I}E_{2}^{s,t}(S^{0}, C) → ^{II}E_{2}^{s,t}(S^{0}, C) is the identity under this identification and the identification of Lemma 4.4.

We now verify the claim. F_{k}(C) is equivalent to a product of C_{G_{n}/U}^{j}’s and is therefore Landweber exact. Hence π_{∗}(F_{k}(C) ∧ E_{n}^{Gal}) is a flat E_{n^{∗}}^{Gal}-module, so

π_{∗}(F_{k}(C) ∧ E_{n}^{Gal} ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}})) = π_{∗}(F_{k}(C) ∧ E_{n}^{Gal})/(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}).

In particular, the inverse system {π_{∗}(F_{k}(C)∧E_{n}^{Gal} ∧M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}))} is Mittag-Leffler and therefore 4.9 holds. This completes the proof.

Corollary 4.10 If C is one of the diagrams in Notation 4.1, then

holim_{⟵_{k}} L_{K(n)}(Tot_{k}(∏^{∗}C) ∧ E_{n}^{Gal}) ⟶^{≃} Tot (∏^{∗}L_{K(n)}(C ∧ E_{n}^{Gal})).

Proof. If C is a diagram of the form 4.1ii or 4.1iii, the result follows from the fact that the map 4.7 is an equivalence and hence the towers {L_{K(n)}(Tot_{k}(∏^{∗} C) ∧ E_{n}^{Gal})} and {Tot_{k}(∏^{∗} L_{K(n)}(C ∧ E_{n}^{Gal}))} are equivalent.

If C = C_{G_{n}/U}, the result follows from Lemma 4.8 together with the fact that both spectral sequences are strongly convergent in the sense of [4; IX, 5.4].

The proof of Theorem 4.3 will now be completed by showing that

L_{K(n)}[(Tot ∏^{∗}C) ∧ E_{n}^{Gal}] ⟶^{≃} holim_{⟵_{k}} L_{K(n)}(Tot_{k}(∏^{∗}C) ∧ E_{n}^{Gal}).

We separate off the following key ingredient.

Let

⋯ → Y_{k} → Y_{k−1} → ⋯ → Y_{0} → ∗

be a tower of fibrations of S^{0}-modules, so that the canonical map lim_{⟵_{k}} Y_{k} → holim_{⟵_{k}} Y_{k} is a

weak equivalence. Define Y^{k} to be the fiber of lim_{⟵_{i}} Y_{i} → Y_{k}; there is then an inverse system of fibrations

[figure: Y^{k+1}; lim_{⟵_{i}} Y_{i}; Y_{k+1}; Y^{k}; lim_{⟵_{i}} Y_{i}; Y_{k}] .

According to [4; XI, 5.5], the map

holim_{⟵_{k}}(lim_{⟵_{i}} Y_{i}) → holim_{⟵_{k}} Y_{k}

is a fibration with fiber holim_{⟵_{k}} Y^{k}. But the commutative diagram

\xymatrix{holim_{⟵_{k}}(lim_{⟵_{i}} Y_{i}) \ar[r] & holim_{⟵_{k}} Y_{k} \\ lim_{⟵_{k}}(lim_{⟵_{i}} Y_{i}) \ar[r]^{=} \ar[u]_{≃} & lim_{⟵_{k}} Y_{k} \ar[u]_{≃}}

shows that this map is a weak equivalence, and thus holim_{⟵_{k}} Y^{k} is stably trivial. In certain

cases, we can say a good deal more.

Lemma 4.11 Let {Y_{k}} be as above, and let E_{r}^{∗,∗}(Z) denote the spectral sequence obtained by mapping the CW-spectrum Z into this tower. Suppose that there exist natural numbers r_{0} and s_{0} such that E_{r_{0}}^{s,∗}(Z) = 0 for all spectra Z whenever s > s_{0}. Then given k, there exists q such that the map Y^{k+q} → Y^{k} is stably trivial.

Proof. Let F_{s} be the fiber of Y_{s} → Y_{s−1}. There is a diagram

\xymatrix{Y^{k} \ar[dr] &  & Y^{k+1} \ar[dr] \ar[ll] &  & Y^{k+2} \ar[ll] & ⋯ \ar[l] \\  & F_{k+1} \ar[ur] &  & F_{k+2} \ar[ur]}

of exact triangles; let _{k}E_{r}^{∗∗}(Z) denote the spectral sequence obtained by mapping Z into this diagram. This spectral sequence is isomorphic to the spectral sequence obtained by mapping Z into the tower {Y_{i}}_{i≥k}. Hence, _{k}E_{r_{0}}^{s,∗}(Z) = 0 for s > max{r_{0} − 2, s_{0} − k − 1}.

Since holim_{⟵_{i}} Y^{i} ≃ ∗, _{k}E_{r}^{∗∗}(Z) is conditionally convergent (in the sense of [2]) to [Z, Y^{k}]^{∗}, and

thus the horizontal vanishing line implies that

\frac{im([Z, Y^{k+s}]^{∗} → [Z, Y^{k}]^{∗})}{im([Z, Y^{k+s+1}]^{∗} → [Z, Y^{k}]^{∗})} = _{k}E_{∞}^{s,∗}(Z)

and that {im([Z, Y^{k+s}]^{∗} → [Z, Y^{k}]^{∗})}_{s≥0} is a complete Hausdorff filtration of [Z, Y^{k}]^{∗}. It then follows that im([Z, Y^{k+s}]^{∗} → [Z, Y^{k}]^{∗}) = 0 for s > max{r_{0} − 2, s_{0} − k − 1}. But Z is arbitrary; therefore Y^{k+s} → Y^{k} is trivial for these values of s, completing the proof.

Lemma 4.12 Let {Y_{k}} satisfy the hypotheses of Lemma 4.11. Then, if W and F are any spectra, there is an equivalence

L_{F} [(holim_{⟵_{k}} Y_{k}) ∧ W] ⟶^{≃} holim_{⟵_{k}} L_{F} (Y_{k} ∧ W).

Remark 4.13 The above map is of course chosen so that composition with the projection onto L_{F} (Y_{k} ∧ W) yields the canonical map

L_{F} [(holim_{⟵_{k}} Y_{k}) ∧ W] → L_{F} (Y_{k} ∧ W).

We will show that lim_{⟵_{k}} ^{1}[Z, L_{F} (Y_{k} ∧ W)]^{∗} = 0 for any spectrum Z, so the equivalence of the

lemma is uniquely determined.

Proof. We have diagrams (in the stable category)

[figure: L_{F} (Y^{k+1} ∧ W); L_{F} ((holim_{⟵_{i}} Y_{i}) ∧ W); L_{F} (Y_{k+1} ∧ W); L_{F} (Y^{k} ∧ W); L_{F} ((holim_{⟵_{i}} Y_{i}) ∧ W); L_{F} (Y_{k} ∧ W)] (4.14)

of fiber sequences. By the previous lemma,

holim_{⟵_{k}} L_{F} (Y^{k} ∧ W) ≃ ∗;

from this should follow the desired result. However, we prefer to avoid trying to argue that “the homotopy inverse limit of the fibers is the fiber of the homotopy limits”, and instead proceed less generally. Indeed, a diagram chase using 4.14 together with the previous lemma shows that the system {[Z, L_{F} (Y_{k} ∧ W)]^{∗}} is Mittag-Leffler for any Z and therefore lim_{⟵_{k}} ^{1}[Z, L_{F} (Y_{k} ∧ W)]^{∗} = 0. A similar argument also shows that

[Z, L_{F} ((holim_{⟵_{i}} Y_{i}) ∧ W)]^{∗} ⟶^{≈} lim_{⟵_{k}}[Z, L_{F} (Y_{k} ∧ W)]^{∗}.

This completes the proof.

Proof of Theorem 4.3. Start with cases i. and ii. of Notation 4.1. By virtue of the preceding work, we need only show that

(4.15) L_{K(n)}[(Tot ∏^{∗}C) ∧ E_{n}^{Gal}] ∧ X ⟶^{≃} holim_{⟵_{k}} L_{K(n)}(Tot_{k}(∏^{∗}C) ∧ E_{n}^{Gal}) ∧ X

for some p-local finite spectrum X Bousfield equivalent to S_{(p)}^{0}. Nilpotence technology [8; 4.1] tells us that this is the same as requiring X to have torsion free ℤ_{(p)}-homology.

We will prove 4.15 by finding a torsion free X such that E_{2}^{∗,∗}(Z ∧ DX, C) has a horizontal vanishing line independent of Z, where E_{r}^{∗,∗}(Z ∧ DX, C) denotes the spectral sequence obtained by mapping Z ∧ DX into {Tot_{k} ∏^{∗} C}. But

E_{2}^{s,t}(Z ∧ DX, C) = H_{c}^{s}(K, E_{n}^{t}(Z ∧ DX))

for K some closed subgroup of G_{n}. (K = U in case i., and K = G in case ii.) Moreover,

H_{c}^{s}(K, E_{n}^{t}(Z ∧ DX)) = lim_{⟵_{α}} H_{c}^{s,−t}(K, E_{n^{∗}}X ⊗_{E_{n^{∗}}} E_{n^{∗}}DZ_{α}),

where Z = holim_{⟵_{α}} Z_{α} is a presentation of Z as a direct limit of finite CW-spectra.

Now Hopkins and Ravenel have shown that there exists a finite spectrum X with free ℤ_{(p)}- homology—in fact, X can be taken to be S^{0} if p−1 ∤ n and a summand of an iterated smash product of a finite complex projective space if p−1 | n—such that H_{c}^{s,∗}(K, E_{n^{∗}}X/I_{n}E_{n^{∗}}X) = 0 for s bigger than some s_{0}. (This is proved for K = G_{n} in [24; 8.3.5–7]; from this follows the result for K closed in G_{n}. Or, one can observe that the proof for G_{n} applies to closed subgroups as well.) An easy induction then shows that H_{c}^{s,∗}(K, E_{n^{∗}}X) and hence H_{c}^{s,∗}(K, E_{n^{∗}}X/I_{m}E_{n^{∗}}X) vanish for s > s_{0}, m ≤ n, where I_{m} = (p, v_{1}, …, v_{m−1}). Finally, proceed by induction on a Landweber filtration of E_{n^{∗}}DZ_{α} to prove that H_{c}^{s,∗}(K, E_{n^{∗}}X ⊗_{E_{n^{∗}}} E_{n^{∗}}DZ_{α}) = 0 for s > s_{0}. (Note that the cross-section theorem [27; I, Theorem 3] allows us to conclude that H_{c}^{∗}(K,?) takes short exact sequences of profinite K-modules to long exact sequences.)

This vanishing line allows us to apply Lemma 4.12 to the tower {(Tot_{k} ∏^{∗} C) ∧ X} to complete the proof of 4.15.

As for case iii, we prove that E_{2}^{∗,∗}(Z, C) has a horizontal vanishing line independent of Z. Indeed,

E_{2}^{s,t}(Z, C) = H^{s}(F, [Z, L_{K(n)}(E_{n}^{hK} ∧ (E_{n}^{Gal})^{(j+1)}]^{t})
= H^{s}(F, Map_{c}^{ℓ}(S_{n}^{j+1}, E_{n}^{t}Z)^{K})

by Proposition 5.3 and 5.5.

Now if M is any discrete twisted G_{n} − W 𝔽_{p^{n}} module, there is a spectral sequence

H^{∗}(F, H_{c}^{∗}(K, Map_{c}^{ℓ}(S_{n}^{j+1}, M))) ⟹ H^{∗}(G, Map_{c}^{ℓ}(S_{n}^{j+1}, M)).

But Map_{c}^{ℓ}(S_{n}^{j+1}, M) = Map_{c}^{ℓ}(S_{n}, Map_{c}^{ℓ}(S_{n}^{j}, M)) is both K and G-acyclic (see proof of Lemma 3.22); this implies that Map_{c}^{ℓ}(S_{n}^{j+1}, M)^{K} is F -acyclic.

If M is profinite, say M = lim_{⟵_{α}} M_{α}, then there is a spectral sequence

lim_{⟵_{α}} ^{i}H^{s−i}(F, Map_{c}^{ℓ}(S_{n}^{j+1}, M_{α})^{K}) ⟹ H^{s}(F, Map_{c}^{ℓ}(S_{n}^{j+1}, M)^{K}).

But

H^{∗}(F, Map_{c}^{ℓ}(S_{n}^{j+1}, M_{α})^{K}) = (Map_{c}^{ℓ}(S_{n}^{j+1}, M_{α})^{K})^{F}
= Map_{c}^{ℓ}(S_{n}^{j+1}, M_{α})^{G}

concentrated in degree 0, and by Lemma 3.23i, lim_{⟵_{α}} ^{i}Map_{c}^{ℓ}(S_{n}^{j+1}, M_{α})^{G} = 0 for i > 0. Thus

E_{2}^{s,t}(Z, C) = 0 for s > 0, and the proof concludes as before.

We conclude this section by proving Theorem 1iv.

Proposition 4.16 Let U be an open subgroup of G_{n}, and let S = G_{n}/U. The spectral sequence 3.13 is naturally isomorphic to the K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence converging to [Z, E_{n}^{hU}]^{∗}.

Proof. Consider the cosimplicial S^{0}-module C_{G_{n}/U}. By Corollary 4.5,

C_{G_{n}/U}^{j} ≃ L_{K(n)}(E_{n}^{hU} ∧ (E_{n}^{Gal})^{(j+1)})

and thus by Remark A.9,

∗ → E_{n}^{hU} → C_{G_{n}/U}^{0} →^{δ} Σ^{−1}C_{G_{n}/U}^{1} →^{δ} Σ^{−2}C_{G_{n}/U}^{2} → ⋯

is a K(n)_{∗}-local E_{n}^{Gal}-resolution of E_{n}^{hU}, where δ = ∑_{i}(−1)^{i}d^{i}. The desired result now follows

from Proposition A.5.

5 Homotopy fixed point spectra for closed subgroups of G_{n}

We begin by recalling the construction of the homotopy direct limit in E_{∞} for the case of a direct sequence of commutative S^{0}-algebras.

Definition 5.1 Let

A_{0} ⟶^{f_{0}} A_{1} ⟶^{f_{1}} A_{2} ⟶^{f_{2}} ⋯ ⟶^{f_{i−1}} A_{i} ⟶^{f_{i}} ⋯

be a direct sequence of commutative S^{0}-algebras. Then holim_{⟶_{i}}_{E_{∞}}A_{i} = lim_{⟶_{j}}_{E_{∞}} Ā_{j}, where

Ā_{j} = A_{0} ⊗ I∐_{f_{0}}A_{1} ⊗ I∐_{f_{1}} ⋯ ∐_{f_{j−2}}A_{j−1} ⊗ I∐_{f_{j−1}}A_{j}

is a j-fold mapping cylinder in E_{∞}. That is, all limits (including tensor products) are to be taken in E_{∞}.

The next result is crucial in the homotopical analysis of E_{n}^{hG}.

Lemma 5.2 Let A_{0} ⟶^{f_{0}} A_{1} ⟶^{f_{1}} A_{2}⟶ ⋯ be a sequence of cellular algebra maps between cell commutative S^{0}-algebras. Then there is a natural weak equivalence holim_{⟶_{i}}_{E_{∞}}A_{i} ≃ holim_{⟶_{i}} A_{i} of

spectra, where holim_{⟶_{i}} denotes the ordinary homotopy colimit of {A_{i}} regarded as a sequence

of spectra.

Proof. Let Ā_{j} be as in the previous definition. The evident map Ā_{j} → A_{j} is a homotopy equivalence in E_{∞} and hence in the category of spectra. Moreover, Ā_{j} is a relative cell commutative S^{0}-algebra under Ā_{j−1} for each j ≥ 1. This implies that Ā_{j−1} → Ā_{j} is a cofibration of underlying spectra [11; VII, 4.14]. Thus, by [11; VII, 3.10], it follows that

holim_{⟶_{i}} _{E_{∞}}A_{i} = lim_{⟶_{j}} _{E_{∞}} Ā_{j} = lim_{⟶_{j}} Ā_{j} ≃ holim_{⟶_{j}} Ā_{j} ≃ holim_{⟶_{j}} A_{j}.

We can now identify the K(n)_{∗}-local E_{n}^{Gal}-homology of E_{n}^{hG} and thus prove Theorem 2i.

Proposition 5.3 π_{∗}L_{K(n)}(E_{n}^{hG}∧E_{n}^{Gal}) = Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})^{G} as completed right π_{∗}L_{K(n)}(E_{n}^{Gal}∧ E_{n}^{Gal})-comodule algebras.

Proof. Just compute:

π_{∗}(E_{n}^{hG} ∧ E_{n}^{Gal} ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}})) = lim_{⟶_{j}} π_{∗}(E_{n}^{h(U_{j}G)} ∧ E_{n}^{Gal} ∧ M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}))
= lim_{⟶_{j}} Map_{c}^{ℓ}(S_{n}, E_{n^{∗}}/(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}))^{U_{j}G}
= Map_{c}^{ℓ}(S_{n}, E_{n^{∗}}/(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}))^{G},

since E_{n^{∗}}/(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}) is discrete. The desired result now follows easily.

It is now also a simple matter to identify E_{n}^{hG} with the usual homotopy fixed point spectrum when G is finite. As in the Introduction, we denote this spectrum by E_{n}^{h′G}.

Proposition 5.4 Let G be a finite subgroup of G_{n}. The map E_{n}^{hG} → E_{n}^{h′G} described in the Introduction is a weak equivalence.

Proof. Since E_{n}^{hG} and E_{n}^{h′G} are both K(n)_{∗}-local, it suffices to prove that the map

π_{∗}L_{K(n)}(E_{n}^{hG} ∧ E_{n}^{Gal}) → π_{∗}L_{K(n)}(E_{n}^{h′G} ∧ E_{n}^{Gal})

is an isomorphism. But we have a commutative diagram

\xymatrix{E_{n}^{hG} \ar[rr] \ar[dr] & & E_{n}^{h′G} \ar[dl] \\ & E_{n}} ,

and by the preceding proposition, π_{∗}L_{K(n)}(E_{n}^{hG} ∧ E_{n}^{Gal}) injects into π_{∗}L_{K(n)}(E_{n} ∧ E_{n}^{Gal}) with image Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})^{G}. On the other hand, Theorem 4.3 implies that L_{K(n)}(E_{n}^{h′G} ∧ E_{n}^{Gal}) ⟶^{≃}[L_{K(n)}(E_{n} ∧E_{n}^{Gal})]^{h′G}, where G acts on the left factor E_{n}. But π_{∗}L_{K(n)}(E_{n} ∧E_{n}^{Gal}) = Map_{c}^{ℓ}(S_{n}, E_{n^{∗}}) is G-acyclic (see proof of Lemma 3.22); therefore π_{∗}L_{K(n)}(E_{n}^{h′G} ∧ E_{n}^{Gal}) = Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})^{G} as well. This completes the proof.

Now let G be a closed subgroup of G_{n} and form a K(n)_{∗}-local E_{n}^{Gal}-Adams resolution of E_{n}^{hG} as in Remark A.9; that is, set

C_{G_{n}/G}^{j} = L_{K(n)}(E_{n}^{hG} ∧ (E_{n}^{Gal})^{(j+1)}).

It follows from 3.25 and the proof of Proposition 5.3 that

[Z, C_{G_{n}/G}^{∗}]^{t} = D_{G}^{∗}E_{n}^{t}Z (5.5)

as cochain complexes, where Z is a finite CW-spectrum and D_{G}^{∗}(?) is as in Definition 3.21. The proof of Lemma 3.24 now applies to show that this equality holds for Z arbitrary. Theorem 2ii is thus a consequence of Lemmas 3.22 and 3.23.

Proposition 5.6 Let G be a closed subgroup of G_{n}. The K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence converging to [Z, E_{n}^{hG}]^{∗} is strongly convergent and has E_{2}-term naturally isomorphic to H_{c}^{∗}(G, E_{n}^{∗}Z).

Proof. The only part which hasn’t been proved above is the strong convergence. But this follows from Proposition A.3.

We can also prove a “covariant” version of Proposition 5.6. The next result is perhaps not the most general result which can be achieved, but is sufficient for the work of [15] on Brown-Comenetz duality. Recall that E(n) denotes the Landweber exact spectrum with coefficient ring E(n)_{∗} = ℤ_{(p)}[v_{1}, …, v_{n}, v_{n}^{−1}].

Proposition 5.7 Let X be a CW-spectrum such that, for each E(n)-module spectrum M, there exists a k with I_{n}^{k}M_{∗}X = 0. Then the K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence converging to π_{∗}L_{K(n)}(X ∧ E_{n}^{Gal}) has E_{2}-term naturally isomorphic to H_{c}^{∗}(G, E_{n^{∗}}X).

Remark 5.8 The hypotheses imply that E_{n^{∗}}X is a discrete G_{n}-module, so H_{c}^{∗}(G, E_{n^{∗}}X) makes good sense.

The proof of this proposition requires a little preparation. Let L_{n} denote the E(n)_{∗}- localization functor. There is a cofiber sequence

Σ^{−n}M_{n}S^{0} → L_{n}S^{0} → L_{n−1}S^{0} (5.9)

(see [23; §5]) and

M_{n}S^{0} = L_{n}(holim_{⟶_{I}} Σ^{−n_{I}}M(p^{i_{0}}, v_{1}^{i_{1}}, …, v_{n−1}^{i_{n−1}}), (5.10)

n_{I} = ∑_{r=0}^{n−1} 2i_{r}(p−1), where I = (i_{0}, i_{1}, …, i_{n−1}) ranges over a cofinal sequence of multi-indices

[6; §4].

Lemma 5.11 Let X satisfy the hypotheses of Proposition 5.7, and let M be an E(n)-module spectrum. Then

i. M ∧ X is K(n)_{∗}-local

ii. M ∧ X ∧ Σ^{−n}M_{n}S^{0} ⟶^{≃} M ∧ X ∧ L_{n}S^{0} ≃ M ∧ X.

Proof. i. M is an E(n)-module spectrum and is therefore E(n)_{∗}-local. Since E(n) is smashing ([24; 7.5.6]), M ∧ X is E(n)_{∗}-local. Hence

L_{K(n)}(M ∧ X) = F (Σ^{−n}M_{n}S^{0}, M ∧ X),

and we must show that

M ∧ X = F (L_{n}S^{0}, M ∧ X) → F (Σ^{−n}M_{n}S^{0}, M ∧ X)

is an equivalence; i.e.

(M ∧ X)^{∗}L_{n}S^{0} ⟶^{≈}(M ∧ X)^{∗}(Σ^{−n}M_{n}S^{0}).

But Σ^{−n}M_{n}S^{0} → L_{n}S^{0} is the composite of the maps L_{n}∂_{i}, 0 ≤ i ≤ n − 1, in the cofibration sequences

N_{i}S^{0} → L_{i}N_{i}S^{0} ≡ M_{i}S^{0} → N_{i+1}S^{0} →^{∂_{i}} ΣN_{i}S^{0},

where N_{0}S^{0} = S^{0}; thus we need only show that (M ∧ X)^{∗}M_{i}S^{0} = 0 for 0 ≤ i ≤ n − 1. Consider the universal coefficient spectral sequence converging to (M ∧ X)^{∗}M_{i}S^{0} whose E_{2}-term is

Ext_{E(n)_{∗}}(v_{i}^{−1}E(n)_{∗}/(p^{∞}, …, v_{i−1}^{∞}), M_{∗}X) = Ext_{E(n)_{∗}}(E(n)_{∗}M_{i}S^{0}, M_{∗}X).

Since there exists k with v_{i}^{k}M_{∗}X = 0, it follows that

Ext_{E(n)_{∗}}(v_{i}^{−1}E(n)_{∗}/(p^{∞}, …, v_{i−1}^{∞}), M_{∗}X) = 0,

completing the proof.

ii. This follows from 5.9 and the fact that, since E(n − 1) is smashing, we have

(L_{n−1}S^{0}) ∧ M ∧ X ≃ L_{n−1}(M ∧ X) ≃ ∗.

Proof of Proposition 5.7. By Lemma 5.11, we have

L_{K(n)}(X ∧ E_{n}^{hG} ∧ (E_{n}^{Gal})^{(j+1)}) ≃ X ∧ E_{n}^{hG} ∧ (E_{n}^{Gal})^{(j+1)}
≃ X ∧ Σ^{−n}M_{n}S^{0} ∧ E_{n}^{hG} ∧ (E_{n}^{Gal})^{(j+1)}
≃ holim_{⟶} X_{α} ∧ Σ^{−n′_{I}}M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}) ∧ E_{n}^{hG} ∧ (E_{n}^{Gal})^{(j+1)}
≃ holim_{⟶} L_{K(n)}(X_{α} ∧ Σ^{−n′_{I}}M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}) ∧ E_{n}^{hG} ∧ (E_{n}^{Gal})^{(j+1)}),

where n′_{I} = n_{I} + n, and the homotopy colimit varies over the finite CW-subspectra X_{α} of X and the sequence of generalized V (n − 1)’s of 5.10. But, by Proposition 5.6,

H^{∗}π_{t}L_{K(n)}(X ∧ E_{n}^{hG} ∧ (E_{n}^{Gal})^{(∗+1)}) = lim_{⟶} H^{∗}π_{t}L_{K(n)}(X_{α} ∧ Σ^{−n′_{I}}M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}}) ∧ E_{n}^{hG} ∧ (E_{n}^{Gal})^{(∗+1)})
= lim_{⟶} H_{c}^{∗}(G, (E_{n})_{t}(X_{α} ∧ Σ^{−n′_{I}}M(p^{i_{0}}, …, v_{n−1}^{i_{n−1}})))
= H_{c}^{∗}(G, (E_{n})_{t}(X ∧ Σ^{−n}M_{n}S^{0}))
= H_{c}^{∗}(G, (E_{n})_{t}X).

This completes the proof.

Finally, if G is a finite subgroup of G_{n}, we identify the homotopy fixed point spectral sequence converging to [Z, E_{n}^{h′G}]^{∗} with the K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence. Let us first introduce some notation.

If G acts on (the commutative S^{0}-algebra) X, write ∏_{G}^{∗}X for the cosimplicial replacement

of the G-diagram defined by the action of G on X. Also write F_{s}(G, X) for the fiber of Tot_{s}(∏_{G}^{∗}X) → Tot_{s−1}(∏_{G}^{∗}X). The next result proves Theorem 3ii.

Proposition 5.12 The sequence

∗ → E_{n}^{h′G} → F_{0}(G, E_{n}) → F_{1}(G, E_{n}) → ⋯

is a K(n)_{∗}-local E_{n}^{Gal}-Adams resolution of E_{n}^{h′G}.

Proof. First observe that F_{s}(G, E_{n}) is a product of E_{n}’s and is therefore E_{n}^{Gal}-injective. We thus need only show that

0 → [Z, L_{K(n)}(E_{n}^{h′G} ∧ E_{n}^{Gal})] → [Z, L_{K(n)}(F_{0}(G, E_{n}) ∧ E_{n}^{Gal})] → ⋯ (5.13)

is exact for all CW-spectra Z. Since G is finite, the cochain complex L_{K(n)}(F_{∗}(G, E_{n})∧E_{n}^{Gal}) is equivalent to F_{∗}(G, L_{K(n)}(E_{n} ∧ E_{n}^{Gal})). Hence

H^{i}[Z, L_{K(n)}(F_{∗}(G, E_{n}) ∧ E_{n}^{Gal})] = H^{i}(G, [Z, L_{K(n)}(E_{n} ∧ E_{n}^{Gal})])
= H^{i}(G, Map_{c}^{ℓ}(S_{n}, E_{n}^{∗}Z))
= \{ \begin{matrix}Map_{c}^{ℓ}(S_{n}, E_{n}^{∗}Z)^{G} & i = 0 \\ 0 & i > 0 \end{matrix}

by the proof of Lemma 3.22. But by Proposition 5.3,

[Z, L_{K(n)}(E_{n}^{h′G} ∧ E_{n}^{Gal})]^{∗} = Map_{c}^{ℓ}(S_{n}, E_{n^{∗}}Z)^{G}.

Tracking down the identifications completes the proof that sequence 5.13 is exact.

6 Proof of Theorem 4

Let G be a closed subgroup of G_{n}, K a closed normal subgroup of G, and suppose F = G/K is finite. Then the canonical map E_{n}^{hG} → E_{n}^{hK} factors through (E_{n}^{hK})^{F}, the F -fixed points of E_{n}^{hK}. The next result proves Theorem 4.

Proposition 6.1 The composition E_{n}^{hG} → (E_{n}^{hK})^{F} → (E_{n}^{hK})^{hF} is a weak equivalence.

Proof. Let F also denote the category with one object ∗ whose automorphism group is F, and consider the functor Y : Δ × F → E_{∞} with

Y([j], ∗) = L_{K(n)}(E_{n}^{hK} ∧ (E_{n}^{Gal})^{(j+1)}).

F acts on E_{n}^{hK}, and Y maps morphisms in Δ as in Remark A.9. Write Y^{j} ≡ Y([j], ∗). Since

E_{n}^{hK} ⟶^{≃} holim_{⟵_{Δ}} Y^{j}

by Corollary A.8, we have that

(E_{n}^{hK})^{hF} ⟶^{∼} holim_{⟵_{F}} holim_{⟵_{Δ}} Y^{j}
≈ holim_{⟵_{Δ}}(Y^{j})^{hF} .

Of course, there is a canonical augmentation (E_{n}^{hK})^{hF} → (Y^{0})^{hF} and hence an augmentation E_{n}^{hG} → (E_{n}^{hK})^{hF} → (Y^{0})^{hF}. We claim that

∗ → E_{n}^{hG} → (Y^{0})^{hF} → Σ^{−1}(Y^{1})^{hF} → ⋯

is a K(n)_{∗}-local E_{n}^{Gal}-Adams resolution of E_{n}^{hG}. Assuming this, it follows from Corollary A.8 that

E_{n}^{hG} →^{∼} holim_{⟵_{Δ}}(Y^{j})^{hF}

and hence that

E_{n}^{hG} →^{∼}(E_{n}^{hK})^{hF} .

To prove the claim, we must show that

0 → [Z, L_{K(n)}(E_{n}^{hG} ∧ E_{n}^{Gal})] → [Z, L_{K(n)}((Y^{0})^{hF} ∧ E_{n}^{Gal})] → [Z, L_{K(n)}((Y^{1})^{hF} ∧ E_{n}^{Gal})] → ⋯

is exact for any CW-spectrum Z. (Since (Y^{i})^{hF} is an E_{n}^{Gal}-module spectrum it is K(n)_{∗}-local E_{n}^{Gal}-injective.) Begin by recalling Theorem 4.3 which asserts that

L_{K(n)}((Y^{i})^{hF} ∧ E_{n}^{Gal}) →^{∼}[L_{K(n)}(Y^{i} ∧ E_{n}^{Gal})]^{hF}

for all i ≥ 0. But

[Z, L_{K(n)}(Y^{i} ∧ E_{n}^{Gal})]^{∗} = Map_{c}^{ℓ}(S_{n}^{i+2}, E_{n}^{∗}Z)^{K}

by 5.5, and since Map_{c}^{ℓ}(S_{n}^{i+2}, E_{n}^{∗}Z)^{K} is F -acyclic (see proof of Theorem 4.3), it follows that

[Z, L_{K(n)}((Y^{i})^{hF} ∧ E_{n}^{Gal})]^{∗} = Map_{c}^{ℓ}(S_{n}^{i+2}, E_{n}^{∗}Z)^{G}
= Map_{c}^{ℓ}(S_{n}^{i+1}, Map_{c}^{ℓ}(S_{n}, E_{n}^{∗}Z))^{G}
= D_{G}^{i}(Map_{c}^{ℓ}(S_{n}, E_{n}^{∗}Z)).

This is in fact an isomorphism of cochain complexes, so

H^{i}([Z, L_{K(n)}((Y^{∗})^{hF} ∧ E_{n}^{Gal})]^{t}) = \{ \begin{matrix}Map_{c}^{ℓ}(S_{n}, E_{n}^{t}Z)^{G} & i = 0 \\ 0 & i > 0. \end{matrix}

Since

[Z, L_{K(n)}(E_{n}^{hG} ∧ E_{n}^{Gal})]^{t} = Map_{c}^{ℓ}(S_{n}, E_{n}^{t}Z)^{G},

the claim is proved.

7 An Application

The results in this section are due to Hopkins and Miller. Let c : S_{n} → ℤ_{p} be a surjective continuous Galois equivariant homomorphism, and consider the exact sequence of groups

0 → K → S_{n} →^{c} ℤ_{p} → 0.

Then ℤ_{p} acts—at least regarded as a discrete group—by S^{0}-algebra maps on E^{h(K⋊Gal)}. We have the following result.

Proposition 7.1 Let t be the topological generator 1 of ℤ_{p}. Then there is a fiber sequence (in the stable category)

L_{K(n)}S^{0} ⟶^{L_{K(n)}η} E_{n}^{h(K⋊Gal)} ⟶^{id−t} E_{n}^{h(K⋊Gal)} ⟶^{∂} ΣL_{K(n)}S^{0}

where η : S^{0} → E_{n}^{h(K⋊Gal)} denotes the unit map.

Proof. Let X be the fiber of id − t. Since t is an S^{0}-algebra map, the unit η factors to give a commutative diagram

\xymatrix{X \ar[rr] & & E_{n}^{h(K⋊Gal)} \\ & S^{0} \ar@{-->}[ul]^{η′} \ar[ur]_{η}} .

We claim that η′ is a K(n)_{∗}-equivalence, and thus L_{K(n)}S^{0} →^{∼} X. To prove this, it suffices to show that

π_{∗}L_{K(n)}(η′ ∧ E_{n}^{Gal}) : E_{n^{∗}}^{Gal} → π_{∗}L_{K(n)}(X ∧ E_{n}^{Gal})

is an isomorphism. There is a commutative diagram

\xymatrix{π_{∗}L_{K(n)}(E_{n}^{h(K⋊Gal)} ∧ E_{n}^{Gal}) \ar[r]^{(id−t)_{∗}} & π_{∗}L_{K(n)}(E_{n}^{h(K⋊Gal)} ∧ E_{n}^{Gal}) \\ Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})^{K⋊Gal} \ar[r]^{id−τ_{t}} & Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})^{K⋊Gal},}

where τ_{t}(f)(g) = f(t^{−1}g) for f ∈ Map_{c}(S_{n}, E_{n^{∗}})^{K⋊Gal} and g ∈ S_{n}. (Here t also denotes any element of S_{n} whose image under c is t ∈ ℤ_{p}.) Since

Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})^{K⋊Gal} = Map_{c}(S_{n}/K, E_{n^{∗}})^{Gal} = Map_{c}(ℤ_{p}, E_{n^{∗}}^{Gal}),

a standard argument shows that id − τ_{t} is surjective and hence

π_{∗}L_{K(n)}(X ∧ E_{n}^{Gal}) = ker(id − τ_{t}).

But ker(id − τ_{t}) just consists of the constant maps from ℤ_{p} to E_{n^{∗}}^{Gal}; this implies that π_{∗}L_{K(n)}(η′ ∧ E_{n}^{Gal}) is an isomorphism, completing the proof.

The next result implies Theorem 5.

Proposition 7.2 Let ∂ be as in Proposition 7.1. Then the composition ∂ ∘ η : S^{0} → ∑ L_{K(n)}S^{0} is detected by ±c ∈ H_{c}^{1}(S_{n}, E_{n^{∗}})^{Gal} in the K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence.

Remark 7.3 Let G be a profinite group and M a discrete G-module, and consider the short exact sequence

0 → M →^{i} Map_{c}(G, M) → Map_{c}(G, M)/M → 0

where i(m)(g) = g^{−1}m. Then the coboundary map provides an isomorphism

(Map_{c}(G, M)/M)^{G} = H_{c}^{0}(G, Map_{c}(G, M)/M) → H_{c}^{1}(G, M).

If M is a trivial G-module, (Map_{c}(G, M)/M)^{G} is just the group of continuous group homomorphisms from G to M; this gives a canonical identification of H_{c}^{1}(G, M) with this group of homomorphisms. The homomorphism c : S_{n} → ℤ_{p} thus defines an element of H_{c}^{1}(S_{n}, ℤ_{p}) = lim_{⟵_{j}} H_{c}^{1}(S_{n}, ℤ/(p^{j})) and hence an element of H_{c}^{1}(S_{n}, E_{n^{∗}}).

Proof of Proposition 7.2. Write f = L_{K(n)}(∂ ∘ η), and consider the cofiber sequence

⋯ → Σ^{−1}L_{K(n)}S^{0} →^{f} L_{K(n)}S^{0} →^{i} C(f) →^{p} L_{K(n)}S^{0} → · · · . (7.4)

By Proposition A.10, f is detected by δ(1) ∈ H_{c}^{1}(G_{n}, E_{n^{∗}}), where δ is the coboundary map H_{c}^{j}(G_{n}, E_{n^{∗}}) → H_{c}^{j+1}(G_{n}, E_{n^{∗}}) for the short exact sequence

0 → E_{n}^{∗}S^{0} →^{p^{∗}} E_{n}^{∗}C(f) →^{i^{∗}} E_{n}^{∗}S^{0} → 0

of G_{n}-modules, and 1 ∈ H^{0}(G_{n}, E_{n^{∗}}) is just the unit in (E_{n^{∗}})^{G_{n}} = ℤ_{p}. Now the sequence 7.4 is self-dual; that is, applying the function spectrum functor F (?, L_{K(n)}S^{0}) yields the same sequence. Hence f is detected by ±δ′(1) ∈ H_{c}^{1}(G_{n}, E_{n^{∗}}), where δ′ denotes the coboundary map for the short exact sequence

0 → E_{n^{∗}}S^{0} →^{i_{∗}} E_{n^{∗}}C(f) →^{p_{∗}} E_{n^{∗}}S^{0} → 0.

(By E_{n^{∗}}X, we here mean π_{∗}L_{K(n)}(X ∧ E_{n}).)

In addition, we have a diagram

[figure: ⋯; L_{K(n)}S^{0}; i; C(f); p; L_{K(n)}S^{0}; −L_{K(n)}η; ⋯; L_{K(n)}S^{0}; E_{n}^{h(K⋊Gal)}; id−t; E_{n}^{h(K⋊Gal)}]

of cofibration sequences; this yields the commutative diagram

[figure: 0; E_{n^{∗}}^{Gal}S^{0}; E_{n^{∗}}^{Gal}C(f); E_{n^{∗}}^{Gal}S^{0}; −η_{∗}; 0; 0; E_{n^{∗}}^{Gal}S^{0}; Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})^{K⋊Gal}; id−τ_{t}; Map_{c}^{ℓ}(S_{n}, E_{n^{∗}})^{K⋊Gal}; 0; Map_{c}(ℤ_{p}, E_{n^{∗}}^{Gal}); Map_{c}(ℤ_{p}, E_{n^{∗}}^{Gal})] (7.5)

Upon tensoring with W 𝔽_{p^{n}}, this diagram becomes a diagram of G_{n}-modules; the action of G_{n} on Map_{c}(ℤ_{p}, E_{n^{∗}}) is given by

((g, σ)h)(s) = g(σh(s · c(g)))

for (g, σ) ∈ G_{n}, h ∈ Map_{c}(ℤ_{p}, E_{n^{∗}}), and s ∈ ℤ_{p}. This follows by naturality from 1.4. There is also a commutative diagram

\xymatrix{0 \ar[r] & E_{n^{∗}} \ar[r] & Map_{c}(ℤ_{p}, E_{n^{∗}}) \ar[r]^{id−τ_{t}} & Map_{c}(ℤ_{p}, E_{n^{∗}}) \ar[r] & 0 \\ 0 \ar[r] & ℤ_{p} \ar[u] \ar[r] & Map_{c}(ℤ_{p}, ℤ_{p}) \ar[r]^{id−τ_{t}} \ar[u] & Map_{c}(ℤ_{p}, ℤ_{p}) \ar[r] \ar[u] & 0} (7.6)

of G_{n}-modules. Then δ′(1) is the image of −δ′′(1) ∈ H_{c}^{1}(G_{n}, ℤ_{p}) in H_{c}^{1}(G, E_{n^{∗}}), where δ′′ is the coboundary map associated to the bottom exact sequence and 1 ∈ Map_{c}(ℤ_{p}, ℤ_{p}) is the constant map with value 1. Finally, use the diagram

[figure: 0; ℤ_{p}; Map_{c}(ℤ_{p}, ℤ_{p}); Map_{c}(c,id); id−τ_{t}; Map_{c}(ℤ_{p}, ℤ_{p}); 0; 0; ℤ_{p}; Map_{c}(S_{n}, ℤ_{p}); Map_{c}(S_{n}, ℤ_{p})/ℤ_{p}; 0]

and Remark 7.3 to complete the proof.

Appendix. The K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence

We begin with some generalities on the Adams-type spectral sequences that we will be considering.

Let E be a commutative ring spectrum (in the stable category), and let F be any spectrum. Then one can construct the E-Adams spectral sequence in the F_{∗}-local category. In more detail, we follow Miller [20] and define an injective class (see [18]) in this category by declaring an F_{∗}-local spectrum X to be E-injective if it is a retract of L_{F} (Y ∧ E) for some spectrum Y. A sequence X′ → X → X′′ is then E-exact if [X′, I] ← [X, I] ← [X′′, I] is exact for every E-injective I. Given X, one may construct an E-exact sequence

∗ → X → I^{0} → I^{1} → ⋯ (A.1)

such that I^{s} is E-injective for all s. One may then construct a diagram

[figure: X = X^{0}; j; X^{1}; i; j; X^{2}; i; ⋯; I^{0}; k; I^{1}; k] (A.2)

of exact triangles; moreover the map

L_{F} (j ∧ E) : L_{F} (X^{i} ∧ E) → L_{F} (I^{i} ∧ E)

is a split monomorphism. Conversely, a diagram of exact triangles as in A.2 with each L_{F} (j ∧ E) split monic yields an E-exact sequence A.1. Such a diagram (with each I^{j} E-injective) is called an F_{∗}-local E-Adams resolution of X and is functorial up to chain homotopy.

By mapping a spectrum Z into an F_{∗}-local E-Adams resolution of X, we obtain a spectral sequence, called the F_{∗}-local E-Adams spectral sequence. The work of Bousfield [3] comes into play in dealing with the convergence question. We define the F_{∗}-local E-nilpotent spectra to be the smallest class C of (F_{∗}-local) spectra such that

i. L_{F} E ∈ C

ii. L_{F} (N ∧ X) ∈ C whenever N ∈ C

iii. C is closed under retracts and cofibrations.

If X is F_{∗}-local E-nilpotent, then the proof of [3; Theorem 6.10] applies to show that the F_{∗}-local E-Adams spectral sequence converges conditionally and strongly to [Z, X]^{∗} for Z any CW-spectrum.

We now specialize to the case E = E_{n}^{Gal} and F = K(n). Here we have the following result.

Proposition A.3 If X is K(n)_{∗}-local, then X is K(n)_{∗}-local E_{n}^{Gal}-nilpotent.

Proof. Consider the Landweber exact spectrum E′(n) with coefficient ring E′(n)_{∗} = ℤ_{(p)}[u_{1}, …, u_{n−1}][u, u^{−1}]. Since E′(n) is equivalent to a wedge of suspensions of E(n) and S^{0} is E(n)-prenilpotent ([17; Theorem 4.1]), it follows that S^{0} is E′(n)-prenilpotent. But X is E′(n)_{∗}-local; therefore X is E′(n)-nilpotent. Since E_{n}^{Gal} = L_{K(n)}E′(n), X is also K(n)_{∗}-local E_{n}^{Gal}-nilpotent.

We now examine some further properties of the K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence which have been used in the text.

Let X be K(n)_{∗}-local, and let C be a cosimplicial S^{0}-module with an augmentation X → C such that

∗ → X → C^{0} →^{δ} Σ^{−1}C^{1} →^{δ} Σ^{−2}C^{2} → ⋯

is a K(n)_{∗}-local E_{n}^{Gal}-resolution of X. (The suspensions appear so that each map δ = ∑(−1)^{i}d^{i} has degree −1.) Consider also the diagram

(A.4)  holim_{⟵_{Δ}} = \xymatrix{Tot^{−1} \ar[dr] &  & Tot^{0} \ar[ll] \ar[dr] &  & Tot^{1} \ar[ll] & ⋯ \ar[l] \\  & F_{0} \ar[ur] &  & F_{1} \ar[ur]}

of exact triangles, where Tot^{i} is the fiber of Tot ∏^{∗} C → Tot_{i} ∏^{∗} C and F_{i} is the fiber of Tot_{i}(∏^{∗} C) → Tot_{i−1}(∏^{∗} C) as in §4. Since F_{i} is the product of various C^{j}’s, it is K(n)_{∗}-local E_{n}^{Gal}-injective. Therefore, the canonical map h : X → holim_{⟵_{Δ}} C extends to a

diagram

[figure: ∗; X; h; C^{0}; δ; h^{0}; Σ^{−1}C^{1}; δ; h^{1}; Σ^{−2}C^{2}; h^{2}; ∗; holim C; F_{0}; F_{1}; F_{2}; ⋯; ⟵; Δ]

of augmented cochain complexes in the stable category, unique up to chain homotopy. This diagram is induced by a map of exact triangles and hence defines a map of spectral sequences.

Proposition A.5 With the notation as above, {h^{i}} induces an isomorphism π^{∗}[Z, C^{∗}] → H^{∗}[Z, F_{∗}] for any spectrum Z. Hence the spectral sequence obtained by mapping Z into diagram A.4 is isomorphic to a K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence.

Proof. The cochain complex

0 → [Z, F_{0}]^{t} → [Z, F_{1}]^{t+1} → [Z, F_{2}]^{t+2} → ⋯

is the normalized cochain complex of the cosimplicial abelian group ∏^{∗}[Z, C]^{t}. There is then a natural cochain equivalence between these two complexes. Hence by Proposition 3.16, there is a cochain map, natural in Z, from [Z, C^{∗}] to [Z, F_{∗}] inducing an isomorphism on cohomology. This map is then induced by a cochain map {g^{i}} from C^{∗} to F_{∗}. It now suffices to show that

[figure: X; h; C^{0}; g^{0}; holim C; F_{0}; ⟵; Δ] (A.6) commutes, for this implies that {g^{i}} is chain homotopic to {h^{i}} and thus induces the same map on cohomology.

To prove the commutativity of A.6, we need only show that

[figure: holim_{⟵_{Δ}} C; C^{0}; g^{0}; holim_{⟵_{Δ}} C; F_{0}]

(A.7)

commutes, where the top map is the canonical map

holim_{⟵_{Δ}} C = Tot (∏^{∗}C) → Tot_{0}(∏^{∗}C) = ∏_{j}C^{j} → C^{0}.

Now F_{0} = ∏_{j} C^{j}, and by Proposition 3.16, the composition

C^{0} ⟶^{g^{0}} ∏_{j}C^{j}⟶ C^{j_{0}}

is given by (d_{0})^{j_{0}}. To prove the commutativity of A.7, we must therefore prove that

Tot (∏^{∗}C) → ∏^{0}C = ∏_{j}C^{j} → C^{j_{0}}

is homotopic to

Tot (∏^{∗}C) → ∏^{0}C = ∏_{j}C^{j} → C^{0} ⟶^{(d_{0})^{j_{0}}} C^{j_{0}}

for each j_{0}. But this follows by a standard argument (cf. proof of Theorem 2.2).

Corollary A.8 The map h : X → holim_{⟵_{Δ}} C is a weak equivalence.

Proof. By the discussion preceding Lemma 4.11, holim_{⟵_{i}} Tot^{i} ≃ ∗. Moreover, the spectral

sequence E_{r}^{∗∗}(S^{0}, C) obtained by applying π_{∗}(?) to A.4 is isomorphic to the K(n)_{∗}-local E_{n}^{Gal}- Adams spectral sequence converging strongly to π_{∗}X. It therefore follows that E_{r}^{∗∗}(S^{0}, C) is strongly convergent and that π_{∗}h is an isomorphism.

Remark A.9 Given a K(n)_{∗}-local S^{0}-module X there is a canonical choice of cosimplicial resolution C as above. Namely, define C^{j} = L_{K(n)}(X ∧ (E_{n}^{Gal})^{(j+1)}) with the coface and codegeneracy maps defined as in Construction 3.11, where X_{S} is replaced by L_{K(n)}(X ∧E_{n}^{Gal}).

We conclude with a “geometric boundary theorem” which was used in §6.

Proposition A.10 Let G be a closed subgroup of G_{n}, and let

⋯ → Σ^{−1}Z →^{∂} X →^{f} Y →^{g} Z → ⋯

be a cofibration sequence with E_{n}^{∗}∂ = 0. Suppose x ∈ [X, E_{n}^{hG}] is detected by u ∈ H_{c}^{∗}(G, E_{n}^{t}X) in the K(n)_{∗}-local E_{n}^{Gal}-Adams spectral sequence. Then x∘∂ is detected by δ(u) ∈ H_{c}^{s+1}(G, E_{n}^{t}Z) up to higher filtration, where δ denotes the coboundary map in H_{c}^{∗}(G,?) associated to the short exact sequence

0 → E_{n}^{∗}Z → E_{n}^{∗}Y → E_{n}^{∗}X → 0.

Remark A.11 The functor D_{G}^{∗}(?) of Definition 3.21 is exact on the category of profinite twisted G_{n} − W 𝔽_{p^{n}} modules by the proof of Lemma 3.23, so that the coboundary map δ of the proposition can be defined.

Proof of A.10. Let C be the cosimplicial resolution of E_{n}^{hG} of A.9, and write

E_{n}^{hG} = [figure: W^{0}; j; W^{1}; i; j; W^{2}; i; j; i; ⋯; C^{0}; k; Σ^{−1}C^{1}; k; Σ^{−2}C^{2}]

for the associated diagram of exact triangles. Recall also from Lemma 3.24 that [?, C^{∗}]^{t} is naturally isomorphic to D_{G}^{∗}(E_{n}^{t}(?)).

Now x lifts to a map x̄ : X → W^{s} such that the composition

X →^{x̄} W^{s} →^{j} Σ^{−s}C^{s}

is a representative of u in D_{G}^{s}(E_{n}^{∗}X). The composition j ∘ x̄ ∘ ∂ is trivial since E_{n}^{∗}∂ = 0. We may therefore construct a diagram

\xymatrix{Σ^{−1}Z \ar[r]^{∂} \ar@{-->}[d]^{z̄} & X \ar[r]^{f} \ar[d]^{x̄} & Y \ar@{-->}[d] \ar[r]^{g} & Z \ar@{-->}[d]^{z̄} \\ W^{s+1} \ar[r]^{i} & W^{s} \ar[r]^{j} & Σ^{−s}C^{s} \ar[r]^{k} & ΣW^{s+1}}

of cofibration sequences. It now follows easily that j ∘ z̄ ∈ [Z, C^{s+1}]^{∗} = D^{s+1}(E_{n}^{∗}Z) is a representative of δ(u). But j ∘ z̄ represents x ∘ ∂ up to higher filtration as well, completing the proof.

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The University of Washington, Seattle, Washington Massachusetts Institute of Technology, Cambridge, Massachusetts