The Topological Singer Construction

Sverre Lunøe–Nielsen and John Rognes

Received: January 21, 2011

Communicated by Lars Hesselholt

Abstract. We study the continuous (co-)homology of towers of spectra, with emphasis on a tower with homotopy inverse limit the Tate construction X^{tG} on a G-spectrum X. When G = C_{p} is cyclic of prime order and X = B^{∧p} is the p-th smash power of a bounded below spectrum B with H_{∗}(B; 𝔽_{p}) of finite type, we prove that (B^{∧p})^{tC_{p}} is a topological model for the Singer construction R_{+}(H^{∗}(B; 𝔽_{p})) on H^{∗}(B; 𝔽_{p}). There is a stable map ϵ_{B}: B → (B^{∧p})^{tC_{p}} inducing the Ext_{𝒜} -equivalence ϵ: R_{+}(H^{∗}(B; 𝔽_{p})) → H^{∗}(B; 𝔽_{p}). Hence ϵ_{B} and the canonical map Γ: (B^{∧p})^{C_{p}} → (B^{∧p})^{hC_{p}} are p-adic equivalences.

2010 Mathematics Subject Classification: 55P42, 55P91, 55S10, 55T15 Keywords and Phrases: Singer construction, Tate construction, limit of Adams spectral sequences, Ext-equivalence.

1. Introduction

Let p be a fixed prime. We study homology and cohomology groups with 𝔽_{p}-coefficients associated to towers of spectra

(1.1)  Y = holim_{n→−∞} Y_{n} ⟶ ⋯ ⟶ Y_{n−1} ⟶ Y_{n} ⟶ ⋯,

where each Y_{n} is bounded below and of finite type over 𝔽_{p}, and Y is equal to the homotopy inverse limit of the tower. By a result of Caruso, May and Priddy [11], there exists an inverse limit of Adams spectral sequences that calculates the homotopy groups of the p-completion Y_{p} = F (S^{−1}/p^{∞}, Y) of Y, where F (−, −) denotes the function spectrum and S^{−1}/p^{∞} is the Moore spectrum with homology ℤ/p^{∞} in degree −1. The E_{2}-term for this spectral sequence is given by the Ext-groups of the direct limit of cohomology groups

(1.2)  H_{c}^{∗}(Y; 𝔽_{p}) = colim_{n→−∞} H^{∗}(Y_{n}; 𝔽_{p}) ,

arising from the tower (1.1), considered as a module over the mod p Steenrod algebra 𝒜. We shall refer to this colimit as the continuous cohomology groups of Y.

1.1. The algebraic Singer construction. A natural question is: how well do we understand the structure of (1.2) as an 𝒜 -module? There is an interesting example of a tower of spectra where this question has the answer: very well. In fact, this question appeared in the study of Segal’s Burnside ring conjecture for cyclic groups of prime order. At the heart of W. H. Lin’s proof of the case p = 2, published in [20], lies a careful study of the 𝒜 -module

P = H_{c}^{∗}(ℝP_{−∞}^{∞}; 𝔽_{2}) = colim_{m→∞} H^{∗}(ℝP_{−m}^{∞}; 𝔽_{2}) ,

and its associated Ext-groups Ext_{𝒜}^{∗,∗}(P, 𝔽_{2}). It turns out that P = P (x, x^{−1}) = 𝔽_{2}[x, x^{−1}] is isomorphic to the so-called Singer construction R_{+}(M) on the trivial 𝒜 -module M = 𝔽_{2}, up to a degree shift. The Singer construction has an explicit description as a module over 𝒜. More importantly, it has the property that there is a natural 𝒜 -module homomorphism ϵ: R_{+}(M) → M that induces an isomorphism

ϵ^{∗}: Ext_{𝒜}^{∗,∗}(M, 𝔽_{2}) ⟶^{≅} Ext_{𝒜}^{∗,∗}(R_{+}(M), 𝔽_{2}) .

1.2. The topological Singer construction. Our objective is to present a topological realization and a useful generalization of these results. For any bounded below spectrum B of finite type over 𝔽_{p}, we construct a tower of spectra

(B^{∧p})^{tC_{p}} = holim_{n→−∞}(B^{∧p})^{tC_{p}}[n] → ⋯ → (B^{∧p})^{tC_{p}}[n−1] → (B^{∧p})^{tC_{p}}[n] → ⋯

as in (1.1). Here C_{p} is the cyclic group of order p, and B^{∧p} is a C_{p}-equivariant model of the p-th smash power of B, specified in Definition 5.3. The Tate construction (B^{∧p})^{tC_{p}} is recalled in Definition 4.1, and the spectra (B^{∧p})^{tC_{p}}[n] in the Tate tower are introduced in Definition 4.3.

Remark 1.1. Note that the naively equivariant homotopy type of the p-fold smash power B^{∧p}, as a spectrum with C_{p}-action, is the same in all models for the stable homotopy category, but that its genuinely equivariant homotopy type, as a C_{p}-spectrum, depends on the choice of model for the smash product. In particular, the meaning of the fixed point spectrum (B^{∧p})^{C_{p}} and the geometric fixed point spectrum Φ^{C_{p}}(B^{∧p}) is model-dependent. On the other hand, the homotopy orbit spectrum (B^{∧p})_{hC_{p}}, the homotopy fixed point spectrum (B^{∧p})^{hC_{p}} and the Tate construction (B^{∧p})^{tC_{p}} only depend on the naively equivariant homotopy type of B^{∧p}. For definiteness, we work with Lewis–May equivariant spectra, as in [18].

In Definition 5.8 we introduce the notation

R_{+}(B) = (B^{∧p})^{tC_{p}}

for the topological Singer construction on B, and in Theorem 5.9 we prove that there is a natural isomorphism

R_{+}(H^{∗}(B; 𝔽_{p})) ≅ H_{c}^{∗}(R_{+}(B); 𝔽_{p}) = colim_{n→−∞} H^{∗}((B^{∧p})^{tC_{p}}[n]; 𝔽_{p})

of 𝒜 -modules, relating the algebraic and topological Singer constructions.

Theorem 5.9. Let B be a symmetric spectrum that is bounded below and of finite type over 𝔽_{p}. There are natural isomorphisms

ω: H_{c}^{∗}(R_{+}(B)) ⟶^{≅} R_{+}(H^{∗}(B))

and

ω_{∗}: R_{+}(H_{∗}(B)) ⟶^{≅} H_{∗}^{c}(R_{+}(B))

of 𝒜 -modules and complete 𝒜_{∗}-comodules, respectively.

We define a natural stable map ϵ_{B}: B → (B^{∧p})^{tC_{p}} in Definition 5.10, and prove in Proposition 5.12 that it realizes the algebraically defined 𝒜 -module homomorphism ϵ: R_{+}(H^{∗}(B; 𝔽_{p})) → H^{∗}(B; 𝔽_{p}) in continuous cohomology. It is in the definition of ϵ_{B} that the choice of genuinely equivariant model for B^{∧p} begins to play a role.

Proposition 5.12. Let B be a bounded below spectrum of finite type over 𝔽_{p}. Then the homomorphism

(ϵ_{B})^{∗}: H_{c}^{∗}(R_{+}(B)) ⟶ H^{∗}(B)

induced on continuous cohomology by the spectrum map ϵ_{B}: B → R_{+}(B) is equal to Singer’s homomorphism

ϵ_{H^{∗}(B)} : R_{+}(H^{∗}(B)) ⟶ H^{∗}(B)

associated to the 𝒜 -module H^{∗}(B), via the identification ω : H_{c}^{∗}(R_{+}(B)) ≅ R_{+}(H^{∗}(B)) of Theorem 5.9.

The Segal conjecture for C_{p} follows as a special case of this, when B = S is the sphere spectrum, since S^{∧p} is a model for the genuinely C_{p}-equivariant sphere spectrum. More generally, we prove in Theorem 5.13 that for any bounded below spectrum B with H_{∗}(B; 𝔽_{p}) of finite type the canonical map

Γ: (B^{∧p})^{C_{p}} ⟶ (B^{∧p})^{hC_{p}} ,

relating the C_{p}-fixed points and C_{p}-homotopy fixed points for B^{∧p}, becomes a homotopy equivalence after p-completion.

Theorem 5.13. Let B be a bounded below spectrum of finite type over 𝔽_{p}. Then the natural maps

ϵ_{B}: B ⟶ R_{+}(B) = (B^{∧p})^{tC_{p}}

and

Γ: (B^{∧p})^{C_{p}} ⟶ (B^{∧p})^{hC_{p}}

are p-adic equivalences of spectra.

In [7, 1.7], with M. Bökstedt and R. R. Bruner, we deduce from this result that there are p-adic equivalences Γ_{n}: (B^{∧p^{n}})^{C_{p^{n}}} ⟶ (B^{∧p^{n}})^{hC_{p^{n}}} for all n ≥ 1.

1.3. Outline of the paper. In §2, we discuss towers of spectra as above, and the associated limit systems obtained by applying homology or cohomology with 𝔽_{p}-coefficients. When dealing with towers of ring spectra it is convenient to work in homology, while for the formation of the Ext-groups mentioned above it is convenient to work in cohomology. In order to be able to switch back and forth between cohomology and homology we discuss linear topologies arising from filtrations, and continuous dualization. Then we see how an 𝒜 -module structure in cohomology dualizes to an 𝒜_{∗}-comodule structure in a suitably completed sense. In §3 we recall the algebraic Singer construction on an 𝒜 -module M, in its cohomological form, and study a dual homological version, defined for 𝒜_{∗}- comodules M_{∗} that are bounded below and of finite type. We give the details for the form of the Singer construction that is related to the group C_{p}, since most references only consider the smaller version related to the symmetric group Σ_{p}. Then, in §4, we define a specific tower of spectra {X^{tG}[n]}_{n} with homotopy inverse limit equivalent to the Tate construction X^{tG} = [ẼG∧F (EG_{+}, X)]^{G} on a G-spectrum X. We consider the associated (co-)homological Tate spectral sequences, and compare our approach of working with homology groups to earlier papers that focused directly on homotopy groups. In §5, we specialize to the case when X = B^{∧p}. This is also where we discuss the genuinely C_{p}-equivariant model of the spectrum B^{∧p}, given by the p-fold smash product of (symmetric) spectra introduced by Bökstedt [6] in his definition of topological Hochschild homology. It is for this particular C_{p}-equivariant model that we can define the natural stable map ϵ_{B}: B → R_{+}(B) realizing Singer’s homomorphism ϵ.

1.4. Notation. Spectra will usually be named B, X or Y. Here B will be a bounded below spectrum or S-algebra of finite type over 𝔽_{p}. Spectra denoted by X will be equipped with an equivariant structure. The main examples we have in mind are the p-fold smash product X = B^{∧p} treated here, and the topological Hochschild homology spectrum X = THH(B) treated in the sequel [22]. When dealing with generic towers of spectra, we will use Y. The example of main interest is the Tate construction Y = X^{tG} on some G-equivariant spectrum X. We write 𝒜 for the mod p Steenrod algebra and 𝒜_{∗} for its 𝔽_{p}-linear dual. We will work with left modules over 𝒜 and left comodules under 𝒜_{∗}. In the body of the paper we write H_{∗}(B) = H_{∗}(B; 𝔽_{p}) and H^{∗}(B) = H^{∗}(B; 𝔽_{p}), for brevity. Unlabeled Hom means Hom_{𝔽_{p}}, and ⊗ means ⊗_{𝔽_{p}}.

1.5. History and notation of the Singer construction. The Singer construction appeared originally for p = 2 in [25] and [26], and for p odd in [19]. The work presented here concentrates on its relation to the calculations by Lin and Gunawardena and their work on the Segal conjecture for groups of prime order. A published account for the case of the group of order 2 is found in [20]. A further study appears in [1], where a more conceptual definition of the Singer construction is given.

In W. Singer’s paper [25], the following problem is posed: Let M be an unstable 𝒜 -module and let Δ = 𝔽_{2}{Sq^{r} | r ∈ ℤ} be the graded 𝔽_{2}-vector space with a formal generator Sq^{r} in each degree. There is a map of graded 𝔽_{2}-vector spaces

d: Δ ⊗ M ⟶ M

taking Sq^{r} ⊗m to Sq^{r}(m) for r ≥ 0, and to 0 for r < 0. Does there exist a natural 𝒜 -module structure on the source of d rendering this map an 𝒜 - linear homomorphism? Singer answers this question affirmatively, by using an idea of Wilkerson [27] to construct the 𝒜 -module that he denotes R_{+}(M), an 𝒜 -module map d: R_{+}(M) → M of degree 1, and an isomorphism R_{+}(M) ≅ Δ ⊗ M, also of degree 1, that makes the two maps called d correspond. In the end, the construction does not depend on M being unstable. In Li and Singer’s paper [19], the odd-primary version of this problem is solved, with Δ = 𝔽_{p}{β^{i} P^{r} | i ∈ {0, 1}, r ∈ ℤ}. Starting with that paper there is a degree shift in the notation: R_{+}(M) now denotes the suspension of R_{+}(M) from Singer’s original paper, so that the 𝒜 -module map d: R_{+}(M) → M is of degree 0. In connection with the Segal conjecture, Adams, Gunawardena and Miller [1] published an algebraic account of the Singer construction, for all primes p. They write T′(M) for the 𝒜 -module denoted R_{+}(M) in [25] and [26], and let T′′(M) = ΣT′(M) be its suspension, denoted R_{+}(M) in [19]. For the trivial 𝒜 -module 𝔽_{p}, T′′(𝔽_{p}) is isomorphic to the Tate homology Ĥ_{−∗}(Σ_{p}; 𝔽_{p}), which can be obtained by localization from ΣH^{∗}(Σ_{p}; 𝔽_{p}). Hence T′(𝔽_{p}) = Ĥ^{∗}(Σ_{p}; 𝔽_{p}) is a localized form of H^{∗}(Σ_{p}; 𝔽_{p}). Adams, Gunawardena and Miller are not only concerned with the Segal conjecture for the groups of prime order, but also for the elementary abelian groups (C_{p})^{n}, so the cross product in cohomology is important for them. They therefore prefer T′ over T′′. In fact, they really work with an extended functor T (M) = T (𝔽_{p}) ⊗_{T′(𝔽_{p})} T′(M), where T (𝔽_{p}) = Ĥ^{∗}(C_{p}; 𝔽_{p}). In our context, for the cohomological study of towers of ring spectra it will be the coproduct in Tate homology that is most important, which is why we prefer T′′ over T′. Again, our emphasis is on C_{p} instead of Σ_{p}, so that the Singer-type functor we shall work with is the extension

Ĥ_{−∗}(C_{p}; 𝔽_{p}) ⊗_{T′′(𝔽_{p})} T′′(M)

of T′′(M), which is a direct sum of (p − 1) shifted copies of the T′′(M) = R_{+}(M) of Li and Singer. This is the functor we shall denote R_{+}(M), so that R_{+}(𝔽_{p}) = Ĥ_{−∗}(C_{p}; 𝔽_{p}), and R_{+}(M) = ΣT (M) in the notation of [1]. The connection between the Singer construction and the continuous cohomology of a tower of spectra, displayed below as (5.1) for G = Σ_{p}, was found by Haynes Miller, and explained in [9, II.5.1]. There the functor denoted R_{+} is the same as in [19] for odd p, shifted up one degree from [26] for p = 2. In the following we will make use of the fact that the Singer construction on an 𝒜 -module M comes equipped with the homomorphism of 𝒜 -modules ϵ: R_{+}(M) → M. In Singer’s work [26], this map has degree +1 (and was named d), whereas in [19] and [9] it has degree 0. We choose to follow the latter conventions, because this is the functor that with no shift of degrees describes our continuous (co-)homology groups. Furthermore, the homomorphism ϵ will be realized by a map of spectra, and will therefore be of degree zero. We choose to write R_{+}(M) instead of T (M) or any of its variants, because the letter T is heavily overloaded by the presence of THH, the Tate construction and the circle group 𝕋. To add to the confusion, the letter T is also used in Singer’s [26] work, but with a different meaning than the T appearing in [1].

1.6. Acknowledgments. This work started out as a part of the first author’s PhD thesis [21] at the University of Oslo, supervised by the second author. The first author wishes to express the deepest gratitude to Prof. Rognes for offering ideas, help and support during the work of his thesis. Thanks are also due to R. Bruner, G. Carlsson, I. C. Borge, S. Galatius, T. Kro, M. Bökstedt, M. Brun and B. I. Dundas for interest, comments and discussions, and to the referee for his careful reading of the manuscript.

2. Limits of spectra

We introduce our conventions regarding towers of spectra and their associated (co-)homology groups. Our motivation is the result of Caruso, May and Priddy, saying that there is an inverse limit of Adams spectral sequences arising from such towers. The input for this inverse limit of Adams spectral sequences will give us the definition of continuous (co-)homology groups.

2.1. Inverse limits of Adams spectral sequences.

Definition 2.1. Let R be a (Noetherian) ring. A graded R-module M_{∗} is bounded below if there is an integer ℓ such that M_{∗} = 0 for all ∗ < ℓ. It is of finite type if it is finitely generated over R in each degree. A spectrum B is bounded below if its homotopy π_{∗}(B) is bounded below as a graded abelian group. It is of finite type over 𝔽_{p} if its mod p homology H_{∗}(B) = H_{∗}(B; 𝔽_{p}) is of finite type as a graded 𝔽_{p}-vector space. The spectrum B is of finite type over ℤ̂_{p} if its homotopy π_{∗}(B) is of finite type as a graded ℤ̂_{p}-module.

Let {Y_{n}}_{n∈ℤ} be a sequence of spectra, with maps f_{n}: Y_{n−1} → Y_{n} for all integers n. Assume that each Y_{n} is bounded below and of finite type over 𝔽_{p}, and let Y be the homotopy inverse limit of this system:

(2.1)  Y ⟶ ⋯ ⟶ Y_{n−1} ⟶^{f_{n}} Y_{n} ⟶ ⋯

In general, Y will neither be bounded below nor of finite type over 𝔽_{p}. For each n there is an Adams spectral sequence {E_{r}^{∗,∗}(Y_{n})}_{r} with E_{2}-term

E_{2}^{s,t}(Y_{n}) = Ext_{𝒜}^{s,t}(H^{∗}(Y_{n}), 𝔽_{p}) ⟹ π_{t−s}((Y_{n})_{p}) ,

converging strongly to the homotopy groups of the p-completion of Y_{n}. Each map in the tower (2.1) induces a map of Adams spectral sequences f_{n}: {E_{r}^{∗,∗}(Y_{n−1})}_{r} → {E_{r}^{∗,∗}(Y_{n})}_{r}. For every r let

E_{r}^{∗,∗}(\underline{Y}) = lim_{n→−∞} E_{r}^{∗,∗}(Y_{n}) ,

and similarly for the d_{r}-differentials. We now state and prove a slightly sharper version of [11, 7.1].

Proposition 2.2. Let {Y_{n}}_{n} be a tower of spectra such that each Y_{n} is bounded below and of finite type over 𝔽_{p}, and let Y = holim_{n} Y_{n}. Then the bigraded groups \{E_{r}^{∗,∗}(\underline{Y})\}_{r} are the terms of a spectral sequence, with E_{2}-term

E_{2}^{s,t}(\underline{Y}) ≅ Ext_{𝒜}^{s,t}( colim_{n→−∞} H^{∗}(Y_{n}), 𝔽_{p}) ⟹ π_{t−s}(Y_{p}) ,

converging strongly to the homotopy groups of the p-completion of Y.

The difference between this statement and the statement in [11] lies in the hypothesis on the Y_{n}: we do not assume that Y_{n} is p-complete, and weaken the condition that Y_{n} should be of finite type over ℤ̂_{p} to the condition that H_{∗}(Y_{n}; 𝔽_{p}) should be of finite type. We refer to the spectral sequence \{E_{r}^{∗,∗}(\underline{Y})\}_{r} as the inverse limit of Adams spectral sequences associated to the tower {Y_{n}}_{n}.

Proof. For any bounded below spectrum B of finite type over 𝔽_{p}, the Adams spectral sequence converges strongly to π_{∗}(B_{p}). Since the E_{2}- and E_{∞}-terms of this spectral sequence are of finite type, the abelian groups π_{∗}(B_{p}) are compact and Hausdorff in the topology given by the Adams filtration. The category of compact Hausdorff abelian groups is an abelian category, as is the category of discrete abelian groups. The Pontryagin duality functor assigns to each abelian group G its character group Hom(G, 𝕋), where 𝕋 = S^{1} is the circle group. It induces a contravariant equivalence between the category of compact Hausdorff abelian groups and the category of discrete abelian groups. The functor taking a filtered diagram of discrete abelian groups to its colimit is well-known to be exact. It follows that the functor taking a filtered diagram of compact Hausdorff abelian groups to its inverse limit is also an exact functor. In particular, passing to filtered inverse limits commutes with the formation of kernels, images, cokernels and homology, in the abelian category of compact Hausdorff abelian groups. We now adapt the proof of [11, 7.1], using this version of the exactness of the inverse limit functor. First, we construct a double tower diagram (2.2) of spectra, where the n-th row is an Adams resolution of Y_{n}, such that each Z_{n,s} is a bounded below spectrum of finite type over 𝔽_{p}. The n-th row can be obtained in a functorial way by smashing Y_{n} with a fixed Adams resolution for the sphere spectrum S. The top row consists of the homotopy inverse limits Z_{s} = holim_{n} Z_{n,s} for all s ≥ 0.

(2.2)  \xymatrix{… \ar[r] & Z_{s} \ar[r] \ar[d] & … \ar[r] & Z_{0} = Y \ar[d] \\  & ⋮ \ar[d] &  & ⋮ \ar[d] \\ … \ar[r] & Z_{n−1,s} \ar[r] \ar[d] & … \ar[r] & Z_{n−1,0} = Y_{n−1} \ar[d]^{f_{n}} \\ … \ar[r] & Z_{n,s} \ar[r] & … \ar[r] & Z_{n,0} = Y_{n}}

By assumption, the homotopy cofiber K_{n,s} of the map Z_{n,s+1} → Z_{n,s} is a wedge sum of suspended copies of the Eilenberg–Mac Lane spectrum H = H𝔽_{p}, also bounded below and of finite type over 𝔽_{p}. The exact couple

\xymatrix{π_{∗}(Z_{n,s+1}) \ar[r]^{i} & π_{∗}(Z_{n,s}) \ar[d]^{j} \\  & π_{∗}(K_{n,s}) \ar@{-->}[ul]^{k}}

generates the Adams spectral sequence {E_{r}^{∗,∗}(Y_{n})}_{r}, with E_{1}^{s,t}(Y_{n}) = π_{t−s}(K_{n,s}). The dashed arrow has degree −1. Now consider the p-completion of diagram (2.2). The n-th row becomes

(2.3)  … ⟶ (Z_{n,s})_{p} ⟶ … ⟶ (Z_{n,0})_{p} = (Y_{n})_{p}

and the homotopy cofiber of (Z_{n,s+1})_{p} → (Z_{n,s})_{p} is the p-completion of K_{n,s}, which is just K_{n,s} again. There is therefore a second exact couple

(2.4)  \xymatrix{π_{∗}((Z_{n,s+1})_{p}) \ar[r]^{i} & π_{∗}((Z_{n,s})_{p}) \ar[d]^{j} \\  & π_{∗}(K_{n,s}) \ar@{-->}[ul]^{k}}

for each n, which generates the same spectral sequence as the first one. Furthermore, in the second exact couple all the (abelian) homotopy groups are compact Hausdorff, since the spectra Z_{n,s} and K_{n,s} are all bounded below and of finite type over 𝔽_{p}. The p-completion of the top row in (2.2) is the homotopy inverse limit over n of the p-completed rows (2.3). The exactness of filtered limits in the category of compact Hausdorff abelian groups now implies that there are isomorphisms π_{∗}((Z_{s})_{p}) ≅ lim_{n} π_{∗}((Z_{n,s})_{p}) for all s. Furthermore, the inverse limit over n of the exact couples (2.4) defines a third exact couple (of compact Hausdorff abelian groups)

(2.5)  \xymatrix{π_{∗}((Z_{s+1})_{p}) \ar[r]^{i} & π_{∗}((Z_{s})_{p}) \ar[d]^{j} \\  & lim_{n} π_{∗}(K_{n,s}) , \ar@{-->}[ul]^{k}}

which generates the spectral sequence \{E_{r}^{∗,∗}(\underline{Y})\}_{r} that we are after. Here E_{1}^{s,t}(\underline{Y}) ≅ lim_{n} E_{1}^{s,t}(Y_{n}), and by induction on r the same isomorphism holds for each E_{r}-term, since E_{r+1}^{∗,∗} is the homology of E_{r}^{∗,∗} with respect to the d_{r}- differentials, and we have seen that the formation of these limits commutes with homology. In particular, each abelian group E_{r}^{s,t}(\underline{Y}) is compact Hausdorff. The identification of the E_{2}-term for s = 0 amounts to the isomorphism

lim_{n} Hom_{𝒜} (H^{∗}(Y_{n}), N) ≅ Hom_{𝒜} (colim_{n} H^{∗}(Y_{n}), N)

for N = Σ^{t}𝔽_{p}. The general case follows, since we can compute Ext_{𝒜}^{s} by means of an injective resolution of 𝔽_{p}. We must now check the convergence of this spectral sequence, which we (and [11]) do following Boardman [3]. The Adams resolution for each Y_{n} is constructed so that

lim_{s} π_{∗}((Z_{n,s})_{p}) = Rlim_{s} π_{∗}((Z_{n,s})_{p}) = 0 .

These two conditions ensure that the Adams spectral sequence for Y_{n} converges conditionally [3, 5.10]. The standard interchange of limits isomorphism gives

lim_{s} π_{∗}((Z_{s})_{p}) ≅ lim_{n} lim_{s} π_{∗}((Z_{n,s})_{p}) = 0 .

Moreover, the exactness of the inverse limit functor in this case implies that the derived limit

Rlim_{s} π_{∗}((Z_{s})_{p}) = 0

vanishes, too. Hence the inverse limit Adams spectral sequence generated by (2.5) is conditionally convergent to π_{∗}(Y_{p}). This is a half-plane spectral sequence with entering differentials, in the sense of Boardman. For such spectral sequences, strong convergence follows from conditional convergence together with the vanishing of the groups

RE_{∞}^{s,t} = Rlim_{r} E_{r}^{s,t}(\underline{Y}) ,

see [3, 5.1, 7.1]. Again, the vanishing of this Rlim is ensured by the exactness of lim for the compact Hausdorff abelian groups E_{r}^{s,t}(\underline{Y}). □

2.2. Continuous (co-)homology. The spectral sequence in Proposition 2.2 is central to the proof of the Segal conjecture for groups of prime order and will be the foundation for the present work. Our work will, in analogy with Lin’s proof of the Segal conjecture, focus on the properties of the E_{2}-term of the above spectral sequence.

Definition 2.3. Let {Y_{n}}_{n} be a tower of spectra such that each Y_{n} is bounded below and of finite type over 𝔽_{p}, and let Y = holim_{n} Y_{n}. Define the continuous cohomology of Y as the colimit

H_{c}^{∗}(Y) = colim_{n→−∞} H^{∗}(Y_{n}) .

Dually, define the continuous homology of Y as the inverse limit

H_{∗}^{c}(Y) = lim_{n→−∞} H_{∗}(Y_{n}) .

Note that we choose to suppress from the notation the tower of which Y is a homotopy inverse limit, even if the continuous cohomology groups do depend on the choice of inverse system. For example, let p = 2 and let Y = S_{2} be the 2-completed sphere spectrum. Since Y is bounded below and of finite type over 𝔽_{2}, we may express Y by the constant tower of spectra. But by W. H. Lin’s theorem, S_{2}≃ holim_{n} ΣℝP_{n}^{∞}, where each ΣℝP_{n}^{∞} is also bounded below and of finite type over 𝔽_{2}. Now colim_{n} H^{∗}(ΣℝP_{n}^{∞}) = ΣP (x, x^{−1}) = R_{+}(𝔽_{2}) is much larger than H^{∗}(S_{2}) = 𝔽_{2}. By the universal coefficient theorem and our finite type assumptions, the 𝔽_{p}- linear dual of H_{c}^{∗}(Y) is naturally isomorphic to H_{∗}^{c}(Y). The continuous homology of Y will often not be of finite type, so its dual is in general not isomorphic to the continuous cohomology. However, if we take into account the linear topology on the inverse limit, given by the kernel filtration induced from the tower, we do get that the continuous dual of the continuous homology is isomorphic to the continuous cohomology. We discuss this in §2.4. Note that the continuous cohomology is a direct limit of bounded below 𝒜 - modules. The direct limit might of course not be bounded below, but we do get a natural 𝒜 -module structure on H_{c}^{∗}(Y) in the category of all 𝒜 -modules. Dually, the continuous homology is an inverse limit of bounded below 𝒜_{∗}- comodules, but the inverse limit might be neither bounded below nor an 𝒜_{∗}- comodule in the usual, algebraic, sense. Instead we get a completed coaction of 𝒜_{∗}

H_{∗}^{c}(Y) ⟶ 𝒜_{∗} ˆ⊗ H_{∗}^{c}(Y) ,

where ˆ⊗ is the tensor product completed with respect to the above-mentioned linear topology on the continuous homology. We discuss this in §2.5.

2.3. Filtrations. For every n ∈ ℤ, let A^{n} be a graded 𝔽_{p}-vector space and assume that these vector spaces fit into a sequence

(2.6)  0 ⟶ ⋯ ⟶ A^{n} ⟶ A^{n−1} ⟶ ⋯ ⟶ A^{−∞}

with trivial inverse limit, and colimit denoted by A^{−∞}. We assume further that each A^{n} is of finite type. Let A_{n} = Hom(A^{n}, 𝔽_{p}) be the dual of A^{n}. The diagram above dualizes to a sequence

(2.7)  A_{−∞} ⟶ ⋯ ⟶ A_{n−1} ⟶ A_{n} ⟶ ⋯ ⟶ 0

with inverse limit

A_{−∞} = lim_{n} A_{n} = lim_{n} Hom(A^{n}, 𝔽_{p}) ≅ Hom(colim_{n} A^{n}, 𝔽_{p}) = Hom(A^{−∞}, 𝔽_{p}) isomorphic to the dual of A^{−∞}, and trivial colimit. The last fact follows from the assumption that A^{n} is finite dimensional in each degree. Indeed,

lim_{n} A^{n} ≅ lim_{n} Hom(A_{n}, 𝔽_{p}) ≅ Hom(colim_{n} A_{n}, 𝔽_{p})

and thus lim_{n} A^{n} = 0 implies that colim_{n} A_{n} is trivial, since the latter injects into its double dual. Furthermore, the derived inverse limits Rlim_{n} A^{n} and Rlim_{n} A_{n} are zero, again because A^{n} and A_{n} are degreewise finite. Adapting Boardman’s notation [3, 5.4], we define filtrations of the colimit of (2.6) and the inverse limit of (2.7) using the corresponding sequential limit systems.

Definition 2.4. For each n ∈ ℤ, let

F^{n}A^{−∞} = im(A^{n} → A^{−∞})

and

F_{n}A_{−∞} = ker(A_{−∞} → A_{n}) .

Then

(2.8)  ⋯ ⊂ F^{n}A^{−∞} ⊂ F^{n−1}A^{−∞} ⊂ ⋯ ⊂ A^{−∞}

and

(2.9)  ⋯ ⊂ F_{n−1}A_{−∞} ⊂ F_{n}A_{−∞} ⊂ ⋯ ⊂ A_{−∞}

define a decreasing (resp. increasing) sequence of subspaces of A^{−∞} (resp. A_{−∞}).

The filtration (2.8) clearly exhausts A^{−∞}, in the terminology of Boardman [3, 2.1]. Since each A^{n} and ker(A^{n} → A^{−∞}) is of finite type, the right derived limits Rlim_{n} A^{n} and Rlim_{n} ker(A^{n} → A^{−∞}) are both zero. By assumption lim_{n} A^{n} = 0, hence both lim_{n} F^{n}A^{−∞} and Rlim_{n} F^{n}A^{−∞} vanish. This means that the filtration (2.8) is Hausdorff and complete, also in the sense of Boardman [3, 2.1], and that the canonical map A^{−∞} → lim_{n}(A^{−∞}/F^{n}A^{−∞}) is an isomorphism. Completeness is equivalent to saying that Cauchy sequences converge in the linear topology given by the filtration. That the filtration is Hausdorff is saying that Cauchy sequences have unique limits. For the filtration (2.9), the proof of [3, 5.4(b)]) shows, without any hypotheses, that the filtration is Hausdorff and complete. It also shows that the filtration is exhaustive, since the colimit of (2.7) is trivial. We collect these facts in the following lemma.

Lemma 2.5. Assume that the inverse limit lim_{n} A^{n} in (2.6) is trivial and that each A^{n} is of finite type. Then both filtrations given in Definition 2.4 (of colim_{n} A^{n} = A^{−∞} resp. its dual A_{−∞}) are exhaustive, Hausdorff and complete.

2.4. Dualization. The dual of the inverse limit A_{−∞} of (2.7) is the double dual of the colimit A^{−∞} of (2.6). It contains this colimit in a canonical way, but is often strictly bigger, since A^{−∞} need not be of finite type. To remedy this, we take into account the linear topology on the limit induced by the inverse system, and dualize by considering the continuous 𝔽_{p}-linear dual. In this topology on A_{−∞}, an open neighborhood basis of the origin is given by the collection of subspaces {F_{n}A_{−∞}}_{n}. A continuous homomorphism A_{−∞} → 𝔽_{p} is thus an 𝔽_{p}-linear function whose kernel contains F_{n}A_{−∞} for some n. The set of these forms an 𝔽_{p}-vector space Hom^{c}(A_{−∞}, 𝔽_{p}), which we call the continuous dual of A_{−∞}.

Lemma 2.6. There is a natural isomorphism

Hom(A^{−∞}, 𝔽_{p}) ≅ A_{−∞} .

Give A_{−∞} the linear topology induced by the system of neighborhoods {F_{n}A_{−∞}}_{n}. Then there is a natural isomorphism

Hom^{c}(A_{−∞}, 𝔽_{p}) ≅ A^{−∞} .

Proof. The first isomorphism has already been explained. For the second, we wish to compute

Hom^{c}(A_{−∞}, 𝔽_{p}) ≅ colim_{n} Hom(A_{−∞}/F_{n}A_{−∞}, 𝔽_{p}) .

The dual of the image F^{n}A^{−∞} = im(A^{n} → A^{−∞}) is the image

(2.10)  Hom(F^{n}A^{−∞}, 𝔽_{p}) ≅ im(A_{−∞} → A_{n}) ≅ A_{−∞}/F_{n}A_{−∞} ,

and F^{n}A^{−∞} is of finite type, so the canonical homomorphism

F^{n}A^{−∞} ⟶^{≅} Hom(A_{−∞}/F_{n}A_{−∞}, 𝔽_{p})

into its double dual is an isomorphism. Passing to the colimit as n → −∞ we get the desired isomorphism, since colim_{n} F^{n}A^{−∞} ≅ A^{−∞}. □

2.5. Limits of 𝒜_{∗}-comodules. Until now, the objects of our discussion have been graded vector spaces over 𝔽_{p}. We will now add more structure, and assume that (2.6) is a diagram of modules over the Steenrod algebra 𝒜. It follows that the finite terms A_{n} in the dual tower (2.7) are comodules under the dual Steenrod algebra 𝒜_{∗}. We need to discuss in what sense these comodule structures carry over to the inverse limit A_{−∞}. Let M_{∗} be a graded vector space, with a linear topology given by a system {U_{α}}_{α} of open neighborhoods, with each U_{α} a graded subspace of M_{∗}. We say that M_{∗} is complete Hausdorff if the canonical homomorphism M_{∗} → lim_{α}(M_{∗}/U_{α}) is an isomorphism. Let V_{∗} be a graded vector space, bounded below and given the discrete topology. By the completed tensor product V_{∗} ˆ⊗ M_{∗} we mean the limit lim_{α}(V_{∗} ⊗ (M_{∗}/U_{α})), with the linear topology given by the kernels of the surjections V_{∗} ˆ⊗ M_{∗} → V_{∗} ⊗ (M_{∗}/U_{α}). The completed tensor product is complete Hausdorff by construction. Given a second graded vector space W_{∗}, discrete and bounded below, there is a canonical isomorphism (V_{∗} ⊗ W_{∗}) ˆ⊗ M_{∗} ≅ V_{∗} ˆ⊗ (W_{∗} ˆ⊗ M_{∗}).

Definition 2.7. Let M_{∗} be a complete Hausdorff graded 𝔽_{p}-vector space. We say that M_{∗} is a complete 𝒜_{∗}-comodule if there is a continuous graded homomorphism ν: M_{∗} → 𝒜_{∗} ˆ⊗ M_{∗} such that the diagrams

\xymatrix{M_{∗} \ar[r]^{ν} \ar[dr]_{≅} & 𝒜_{∗} ˆ⊗ M_{∗} \ar[d]^{ϵˆ⊗1} \\  & 𝔽_{p} ˆ⊗ M_{∗}}

and

(2.11)  \xymatrix{M_{∗} \ar[r]^{ν} \ar[d]_{ν} & 𝒜_{∗} ˆ⊗ M_{∗} \ar@{-->}[dr]^{ψ ˆ⊗1} \\ 𝒜_{∗} ˆ⊗ M_{∗} \ar[r]^{1ˆ⊗ν} & 𝒜_{∗} ˆ⊗ (𝒜_{∗} ˆ⊗ M_{∗}) \ar[r]^{≅} & (𝒜_{∗} ⊗ 𝒜_{∗}) ˆ⊗ M_{∗}}

commute. Here ϵ: 𝒜_{∗} → 𝔽_{p} and ψ: 𝒜_{∗} → 𝒜_{∗} ⊗ 𝒜_{∗} denote the counit and coproduct in the dual Steenrod algebra, respectively. Let N_{∗} be another complete 𝒜_{∗}-comodule and let f: N_{∗} → M_{∗} be a continuous graded homomorphism. Then f ∈ Hom_{𝒜_{∗}}^{c} (N_{∗}, M_{∗}) if the diagram

\xymatrix{N_{∗} \ar[r]^{ν} \ar[d]_{f} & 𝒜_{∗} ˆ⊗ N_{∗} \ar[d]^{1ˆ⊗f} \\ M_{∗} \ar[r]^{ν} & 𝒜_{∗} ˆ⊗ M_{∗}}

commutes. Hence there is an equalizer diagram (2.12)

\xymatrix{Hom_{𝒜_{∗}}^{c} (N_{∗}, M_{∗}) \ar[r] & Hom^{c}(N_{∗}, M_{∗}) \ar[r]^{f↦(1ˆ⊗f)∘ν} \ar[r]_{f↦ν∘f} & Hom^{c}(N_{∗}, 𝒜_{∗} ˆ⊗ M_{∗}) .}

Lemma 2.8. Suppose given a sequence of graded 𝔽_{p}-vector spaces, as in (2.6), with each A^{n} bounded below and of finite type. Suppose also that A^{n} is an 𝒜 -module and that A^{n} → A^{n−1} is 𝒜 -linear, for each finite n. Then, with notation as above, A^{−∞} is an 𝒜 -module, and the topological 𝔽_{p}-vector space A_{−∞} is a complete 𝒜_{∗}-comodule.

Proof. The category of 𝒜 -modules is closed under direct limits, so the first claim of the lemma is immediate. For each n we get a commutative diagram

[figure: 𝒜 ⊗ A^{n}; λ^{n}; 𝒜 ⊗ F^{n}A^{−∞}; 𝒜 ⊗ A^{−∞}; λ; A^{n}; F^{n}A^{−∞}; A^{−∞},]

where the vertical arrows are the 𝒜 -module action maps. For every finite n, the dual of the 𝒜 -module action map λ^{n}: 𝒜 ⊗ A^{n} → A^{n} defines an 𝒜_{∗}-comodule coaction map ν_{n}: A_{n} = Hom(A^{n}, 𝔽_{p}) → Hom(𝒜 ⊗ A^{n}, 𝔽_{p}) ≅ Hom(𝒜 , 𝔽_{p}) ⊗ Hom(A^{n}, 𝔽_{p}) = 𝒜_{∗} ⊗ A_{n}, where the middle isomorphism uses that 𝒜 and A^{n} are bounded below and of finite type over 𝔽_{p}. Similarly, the dual of the diagram above gives a commutative diagram

[figure: 𝒜_{∗} ⊗ A_{n}; 𝒜_{∗} ⊗ A_{−∞}/F_{n}A_{−∞}; Hom(𝒜 ⊗ A^{−∞}, 𝔽_{p}); A_{n}; ν_{n}; A_{−∞}/F_{n}A_{−∞}; A_{−∞},]

where we use the identification from (2.10). Passing to limits over n, we get the diagram

\xymatrix{lim_{n} (𝒜_{∗} ⊗ A_{n}) & 𝒜_{∗} ˆ⊗ A_{−∞} \ar[l]_{≅} & Hom(𝒜 ⊗ A^{−∞}, 𝔽_{p}) \ar[l]_{≅} \\ lim_{n} A_{n} \ar[u]^{lim_{n} ν_{n}} & A_{−∞} \ar[l]_{≅} \ar[u]_{ν} & A_{−∞}. \ar[l]_{=} \ar[u]_{Hom(λ,𝔽_{p})}}

The middle vertical coaction map ν is continuous, as it is realized as the inverse limit of a homomorphism of towers. It is clear that the upper left hand horizontal map is injective, but we claim that it is also surjective. To see this, let Z_{n} be the cokernel of A_{−∞}/F_{n}A_{−∞} ↪ A_{n}. We know from Lemma 2.5 that lim_{n}(A_{−∞}/F_{n}A_{−∞}) ≅ lim_{n} A_{n} and Rlim_{n}(A_{−∞}/F_{n}A_{−∞}) = 0, so lim_{n} Z_{n} = 0. This implies that lim_{n}(𝒜_{∗} ⊗ Z_{n}) = 0, since there are natural injective maps 𝒜_{∗} ⊗ Z_{n} ↪ Hom(𝒜 , Z_{n}), and lim_{n}(𝒜_{∗} ⊗ Z_{n}) ↪ lim_{n} Hom(𝒜 , Z_{n}) ≅ Hom(𝒜 , lim_{n} Z_{n}) = 0. Now 𝒜_{∗} ⊗ Z_{n} is the cokernel of 𝒜_{∗} ⊗A_{−∞}/F_{n}A_{−∞} ↪ 𝒜_{∗} ⊗A_{n}, hence in the limit 𝒜_{∗} ˆ⊗A_{−∞} → lim_{n}(𝒜_{∗} ⊗A_{n}) is surjective. The commutativity of the diagrams in Definition 2.7 is immediate since they are obtained as the inverse limits of the corresponding diagrams involving A_{n} and ν_{n}. Thus A_{−∞} is a complete 𝒜_{∗}-comodule. □

Corollary 2.9. Let {Y_{n}}_{n} be a tower of spectra as in (2.1), each bounded below and of finite type over 𝔽_{p}, with homotopy inverse limit Y. Then the continuous cohomology H_{c}^{∗}(Y) = colim_{n} H^{∗}(Y_{n}) is an 𝒜 -module, the continuous homology H_{∗}^{c}(Y) = lim_{n} H_{∗}(Y_{n}) is a complete 𝒜_{∗}-comodule, and there are natural isomorphisms Hom(H_{c}^{∗}(Y), 𝔽_{p}) ≅ H_{∗}^{c}(Y) and Hom^{c}(H_{∗}^{c}(Y), 𝔽_{p}) ≅ H_{c}^{∗}(Y), in the respective categories. □

3. The algebraic Singer constructions

Classically, the algebraic Singer construction is an endofunctor on the category of modules over the Steenrod algebra. In §3.1 we recall its definition, and a key property proved by Adams, Gunawardena and Miller. We then dualize the construction in §3.2. Later, we will see how the algebraic Singer construction arises in its cohomological (resp. homological) form as the continuous cohomology (resp. continuous homology) of a certain tower of truncated Tate spectra. This tower of spectra induces a natural filtration on the Singer construction. We introduce this filtration in purely algebraic terms in the present section, and will show in §5.4 that the algebraic and topological definitions agree.

3.1. The cohomological Singer construction.

Definition 3.1. Let M be an 𝒜 -module. The Singer construction R_{+}(M) on M is a graded 𝒜 -module given additively by the formulas

Σ^{−1}R_{+}(M) = P (x, x^{−1}) ⊗ M

for p = 2, and

Σ^{−1}R_{+}(M) = E(x) ⊗ P (y, y^{−1}) ⊗ M

for p odd. Here deg(x) = 1, deg(y) = 2, and Σ^{−1} denotes desuspension by one degree. The action of the Steenrod algebra is given, for r ∈ ℤ and a ∈ M, by the formula

(3.1)  Sq^{s}(x^{r} ⊗ a) = ∑_{j} \binom{r − j}{s − 2j} x^{r+s−j} ⊗ Sq^{j}(a)

for p = 2, and the formulas

P^{s}(y^{r} ⊗ a) = ∑_{j} \binom{r − (p − 1)j}{s − pj} y^{r+(p−1)(s−j)} ⊗ P^{j}(a)

+ ∑_{j} \binom{r − (p − 1)j − 1}{s − pj − 1} xy^{r+(p−1)(s−j)−1} ⊗ β P^{j}(a)

P^{s}(xy^{r−1} ⊗ a) = ∑_{j} \binom{r − (p − 1)j − 1}{s − pj} xy^{r+(p−1)(s−j)−1} ⊗ P^{j}(a)

and

β(y^{r} ⊗ a) = 0
β(xy^{r−1} ⊗ a) = y^{r} ⊗ a

for p odd.

This is the form of the Singer construction that is related to the cyclic group C_{p}. The cohomology of the classifying space of this group is H^{∗}(BC_{p}) ≅ E(x) ⊗ P (y) for p odd, with deg(x) = 1, deg(y) = 2 and β(x) = y, as above. The natural 𝒜 -module structure on H^{∗}(BC_{p}) extends to the localization H^{∗}(BC_{p})[y^{−1}] = E(x) ⊗ P (y, y^{−1}), and letting M = 𝔽_{p} we get that Σ^{−1}R_{+}(𝔽_{p}) is isomorphic to H^{∗}(BC_{p})[y^{−1}] as an 𝒜 -module. The case p = 2 is similar. When p odd there is a second form of the Singer construction, related to the symmetric group Σ_{p}. Following [19, p. 272] we identify H^{∗}(BΣ_{p}) with the subalgebra E(u)⊗P (v) of H^{∗}(BC_{p}) generated by u = −xy^{p−2} and v = −y^{p−1}, with deg(u) = 2p − 3 and deg(v) = 2p − 2. The smaller form of the Singer construction then corresponds to the direct summand E(u) ⊗ P (v, v^{−1}) ⊗ M of index (p − 1) in E(x) ⊗ P (y, y^{−1}) ⊗ M. Explicit formulas for the action of the Steenrod operations on the smaller form of the Singer construction are given in [26, (3.2)], [19, §2] and [9, p. 47]. In our work, we are only concerned with the version of the Singer construction related to the group C_{p}. The exact form of the formulas in Definition 3.1 is justified by Theorem 5.2 below.

3.1.1. The cohomological ϵ-map. An important property of R_{+}(M) is that there exists a natural homomorphism ϵ: R_{+}(M) → M of 𝒜 -modules. In Singer’s original definition for p = 2, the map is given by the formula

(3.2)  ϵ(Σx^{r−1} ⊗ a) = \begin{cases}Sq^{r}(a) & for r ≥ 0, \\ 0 & for r < 0. \end{cases}

For p odd, the 𝒜 -submodule spanned by elements of the form Σy^{(p−1)r} ⊗ a or Σxy^{(p−1)r−1} ⊗ a is a direct summand in R_{+}(M). The homomorphism ϵ is given by first projecting onto this direct summand and then composing with the map

\begin{matrix}Σy^{(p−1)r} ⊗ a & ⟼ −(−1)^{r}β P^{r}(a) \\ Σxy^{(p−1)r−1} ⊗ a & ⟼ (−1)^{r} P^{r}(a) \end{matrix} ; (3.3)

for r ≥ 0, still mapping to 0 for r < 0. See [9, p. 50]. It is clear that ϵ is surjective. We recall the key property of ϵ. Adams, Gunawardena and Miller [1] make the following definition.

Definition 3.2. An 𝒜 -module homomorphism L → M is a Tor-equivalence if the induced map

(3.4)  Tor_{∗,∗}^{𝒜}(𝔽_{p}, L) ⟶ Tor_{∗,∗}^{𝒜}(𝔽_{p}, M)

is an isomorphism.

The relevance of this condition is:

Proposition 3.3 ([1, 1.2]). If L → M is a Tor-equivalence, then for any 𝒜 -module N that is bounded below and of finite type the induced map

(3.5)  Ext_{𝒜}^{∗,∗}(M, N) ⟶ Ext_{𝒜}^{∗,∗}(L, N)

is an isomorphism.

Here is their key result, proved in [1, 1.3].

Theorem 3.4 (Gunawardena, Miller). The Singer homomorphism

ϵ: R_{+}(M) → M

is a Tor-equivalence.

We will later encounter instances of 𝒜 -module homomorphisms R_{+}(M) → N induced by maps of spectra. It is often possible to determine those homomorphisms by the following corollary.

Corollary 3.5. Let M, N be 𝒜 -modules such that N is bounded below and of finite type. Then

ϵ^{∗}: Hom_{𝒜} (M, N) ⟶ Hom_{𝒜} (R_{+}(M), N)

is an isomorphism, so any 𝒜 -linear homomorphism f: R_{+}(M) → N factors as g ∘ ϵ for a unique 𝒜 -linear homomorphism g: M → N:

\xymatrix{R_{+}(M) \ar[rr]^{ϵ} \ar[dr]_{f} &  & M \ar[dl]^{g} \\  & N}

Proof. This is clear from Theorem 3.4 and Proposition 3.3. □

Remark 3.6. A special case of this occurs when M = N is a non-trivial cyclic 𝒜 -module. Then

𝔽_{p} ≅ Hom_{𝒜} (M, M) ≅ Hom_{𝒜} (R_{+}(M), M) ,

so any 𝒜 -linear homomorphism R_{+}(M) → M is equal to a scalar multiple of ϵ.

3.2. The homological Singer construction. Before we define the homological version of the Singer construction on an 𝒜_{∗}-comodule M_{∗}, we need to discuss a natural filtration on the cohomological Singer construction. For a bounded below 𝒜 -module M of finite type over 𝔽_{p}, let F^{n}R_{+}(M) be equal to

𝔽_{2}{Σx^{r} ⊗ a | r ∈ ℤ, deg(a) = q, 1 + r − q ≥ n}

for p = 2, and

𝔽_{p}{Σx^{i}y^{r} ⊗ a | i ∈ {0, 1}, r ∈ ℤ, deg(a) = q, 1 + i + 2r − (p − 1)q ≥ n}

for p odd. In each case a runs through an 𝔽_{p}-basis for M. Then

(3.6)  ⋯ ⊂ F^{n}R_{+}(M) ⊂ F^{n−1}R_{+}(M) ⊂ ⋯ ⊂ R_{+}(M)

is an exhaustive filtration of R_{+}(M), which is clearly Hausdorff. Because M is bounded below and of finite type, each F^{n}R_{+}(M) is bounded below and of finite type, so Rlim_{n} F^{n}R_{+}(M) is trivial. Hence the filtration is complete. For reasons made clear in Corollary 5.16, we will refer to this filtration as the Tate filtration. When M is the cohomology of a bounded below spectrum of finite type over 𝔽_{p}, we will see how (3.6) is induced from topology. In this case, it will be immediate that the filtration is one of 𝒜 -modules. For a general 𝒜 -module M, this can be checked directly using the explicit formulas in Definition 3.1. We are now in the situation discussed in the previous section, with A^{n} = F^{n}R_{+}(M) and A^{−∞} = R_{+}(M). Letting F^{n}R_{+}(M)_{∗} = Hom(F^{n}R_{+}(M), 𝔽_{p}) = A_{n} we get an inverse system

(3.7)  ⋯ ⟶ F^{n−1}R_{+}(M)_{∗} ⟶ F^{n}R_{+}(M)_{∗} ⟶ ⋯

as in (2.7), dual to the direct system (3.6). We are interested in the inverse limit A_{−∞} = lim_{n} A_{n}, with the linear topology given by this tower of surjections. Recall Definition 2.7 of a complete 𝒜_{∗}-comodule.

Definition 3.7. Let M_{∗} be a bounded below 𝒜_{∗}-comodule of finite type. Its dual M = Hom(M_{∗}, 𝔽_{p}) is a bounded below 𝒜 -module of finite type, and M_{∗} ≅ Hom(M, 𝔽_{p}). We define the homological Singer construction on M_{∗} to be the complete 𝒜_{∗}-comodule given by

R_{+}(M_{∗}) = Hom(R_{+}(M), 𝔽_{p}) .

It is isomorphic to the inverse limit lim_{n} F^{n}R_{+}(M)_{∗}.

A more explicit description can be given. For p = 2 the 𝔽_{2}-linear dual of

Ĥ_{−∗}(C_{2}; 𝔽_{2}) ≅ ΣH^{∗}(C_{2}; 𝔽_{2})[x^{−1}] = ΣP (x, x^{−1})

is isomorphic to the ring of Laurent polynomials Ĥ^{−∗}(C_{2}; 𝔽_{2}) = P (u, u^{−1}), where deg(u) = −1 and u^{−r} is dual to Σx^{r−1}. For p odd, the 𝔽_{p}-linear dual of

Ĥ_{−∗}(C_{p}; 𝔽_{p}) ≅ ΣH^{∗}(C_{p}; 𝔽_{p})[y^{−1}] = ΣE(x) ⊗ P (y, y^{−1})

is isomorphic to Ĥ^{−∗}(C_{p}; 𝔽_{p}) = E(u) ⊗ P (t, t^{−1}), where deg(u) = −1, deg(t) = −2 and u^{1−i}t^{−r} is dual to Σx^{i}y^{r−1}. These notations are compatible with those from [5]. We get an identification of F^{n}R_{+}(M)_{∗} with

𝔽_{2}{u^{r} ⊗ α | r ∈ ℤ, deg(α) = q, r + q ≤ −n}

for p = 2, and with

𝔽_{p}{u^{i}t^{r} ⊗ α | i ∈ {0, 1}, r ∈ ℤ, deg(α) = q, i + 2r + (p − 1)q ≤ −n}

for p odd. In each case α ranges over an 𝔽_{p}-basis for M_{∗}. The maps of (3.7) are given by the obvious projections. Thus, R_{+}(M_{∗}) is isomorphic to the graded vector space of formal series

∑_{r=−∞}^{∞} u^{r} ⊗ α_{r}

for p = 2, and

∑_{r=−∞}^{∞} t^{r} ⊗ α_{0,r} + ∑_{r=−∞}^{∞} ut^{r} ⊗ α_{1,r}

for p odd. In each of these sums r is bounded below, but not above, since M_{∗} is bounded below. Using the linear topology on R_{+}(M_{∗}) given by the kernel filtration coming from (3.7), we may reformulate this as follows: Let

Λ = Ĥ^{−∗}(C_{p}; 𝔽_{p}) = \begin{cases}P (u, u^{−1}) & for p = 2, \\ E(u) ⊗ P (t, t^{−1}) & for p odd. \end{cases}

Consider Λ⊗M_{∗} ⊂ R_{+}(M_{∗}). For every n the composition Λ⊗M_{∗} ⊂ R_{+}(M_{∗}) → F^{n}R_{+}(M)_{∗} is surjective, so the completed tensor product Λ ˆ⊗M_{∗} (for the linear topology on Λ derived from the grading) is canonically isomorphic to R_{+}(M_{∗}).

3.2.1. The homological ϵ_{∗}-map. Let

ϵ_{∗}: M_{∗} ⟶ R_{+}(M_{∗})

be the dual of ϵ: R_{+}(M) → M. Then ϵ_{∗} is a continuous homomorphism of complete 𝒜_{∗}-comodules. Continuity is trivially satisfied since the source of ϵ_{∗} has the discrete topology. Dualizing (3.2) and (3.3), we see that ϵ_{∗} is given by the formulas

(3.8)  ϵ_{∗}(α) = ∑_{r=0}^{∞} u^{−r} ⊗ Sq_{∗}^{r}(α)

for p = 2, and

(3.9) ϵ_{∗}(α) = ∑_{r=0}^{∞}(−1)^{r}t^{−(p−1)r} ⊗ P_{∗}^{r}(α) − ∑_{r=0}^{∞}(−1)^{r}ut^{−(p−1)r−1} ⊗ (β P^{r})_{∗}(α)

for p odd. This expression may be compared with [1, (3.6)]. It is clear that ϵ_{∗} is injective.

Lemma 3.8. Let M and N be bounded below 𝒜 -modules of finite type, and let M_{∗} and N_{∗} be the dual 𝒜_{∗}-comodules. Then

ϵ_{∗}: Hom_{𝒜_{∗}} (N_{∗}, M_{∗}) ⟶ Hom_{𝒜_{∗}}^{c} (N_{∗}, R_{+}(M_{∗}))

is an isomorphism, so any continuous 𝒜_{∗}-comodule homomorphism f_{∗}: N_{∗} → R_{+}(M_{∗}) factors as f_{∗} = ϵ_{∗} ∘ g_{∗} for a unique 𝒜_{∗}-comodule homomorphism g_{∗}: N_{∗} → M_{∗}.

Proof. Notice that Hom_{𝒜_{∗}} (N_{∗}, M_{∗}) = Hom_{𝒜_{∗}}^{c} (N_{∗}, M_{∗}) and 𝒜_{∗}⊗N_{∗} = 𝒜_{∗} ˆ⊗N_{∗}, since M_{∗} and N_{∗} are discrete. Applying Hom(−, 𝔽_{p}) to a commutative square

\xymatrix{𝒜 ⊗ R_{+}(M) \ar[r]^{λ} \ar[d]_{1⊗f} & R_{+}(M) \ar[d]^{f} \\ 𝒜 ⊗ N \ar[r]^{λ} & N}

we get a commutative square

\xymatrix{𝒜_{∗} ˆ⊗ R_{+}(M_{∗}) & R_{+}(M_{∗}) \ar[l]_{ν} \\ 𝒜_{∗} ⊗ N_{∗} \ar[u]^{1ˆ⊗f_{∗}} & N_{∗} \ar[l]_{ν} \ar[u]_{f_{∗}}}

of continuous homomorphisms, where R_{+}(M_{∗}) and 𝒜_{∗} ˆ⊗R_{+}(M_{∗}) have the limit topologies, while N_{∗} and 𝒜_{∗} ⊗ N_{∗} are discrete. Applying Hom^{c}(−, 𝔽_{p}) to the latter square we recover the first, by Lemma 2.6. Hence the right hand vertical map in the commutative square

\xymatrix{Hom_{𝒜} (M, N) \ar[r]^{ϵ^{∗}}_{≅} \ar[d]_{≅} & Hom_{𝒜} (R_{+}(M), N) \ar[d]^{≅} \\ Hom_{𝒜_{∗}} (N_{∗}, M_{∗}) \ar[r]^{ϵ_{∗}} & Hom_{𝒜_{∗}}^{c} (N_{∗}, R_{+}(M_{∗}))}

is an isomorphism. It is easy to see that the left hand vertical map is an isomorphism, and the upper horizontal map is an isomorphism by Corollary 3.5.

□

3.2.2. Various remarks on the homological Singer construction. The following remarks are not necessary for our immediate applications, but we include them to shed some light on the coaction ν: R_{+}(M_{∗}) → 𝒜_{∗} ˆ⊗ R_{+}(M_{∗}) and the dual Singer map ϵ_{∗}: M_{∗} → R_{+}(M_{∗}), and their relations to the completions introduced so far. Dualizing (3.1), we get that the dual Steenrod operations on classes u^{r} ⊗ α in Λ ⊗ M_{∗} ⊂ R_{+}(M_{∗}) are given by

(3.10)  Sq_{∗}^{s}(u^{r} ⊗ α) = ∑_{j} \binom{−r − s − 1}{s − 2j}u^{r+s−j} ⊗ Sq_{∗}^{j}(α)

for p = 2, and similarly for p odd. This sum is finite, since M_{∗} is assumed to be bounded below, so we have the following commutative diagram:

(3.11)  [figure: R_{+}(M_{∗}); ν; 𝒜_{∗} ˆ⊗ R_{+}(M_{∗}); Λ ⊗ M_{∗}; 𝒜_{∗} ˆ⊗ (Λ ⊗ M_{∗})]

Two remarks are in order. First, Λ ⊗ M_{∗} is not complete with respect to the subspace topology from R_{+}(M_{∗}). Hence Λ⊗M_{∗} is not a complete 𝒜_{∗}-comodule in the sense explained above. Second, there are elements u^{r} ⊗α in Λ⊗M_{∗} with the property that Sq_{∗}^{s}(u^{r} ⊗ α) is nonzero for infinitely many s, and similarly for p odd. For example, Sq_{∗}^{s}(u^{−1} ⊗ α) contains the term

\binom{−s}{s}u^{s−1} ⊗ α = \binom{2s − 1}{s}u^{s−1} ⊗ α

for j = 0, according to (3.10). This equals u^{s−1} ⊗ α whenever s = 2^{e} is a power of 2, so ν(u^{−1} ⊗ α) is an infinite sum. Hence Λ ⊗ M_{∗} is not an algebraic 𝒜_{∗}-comodule, either. We will now identify the image of the homological version of the Singer map

ϵ_{∗}: M_{∗} ⟶ R_{+}(M_{∗})

with the maximal algebraic 𝒜_{∗}-comodule contained in R_{+}(M_{∗}).

Definition 3.9. Given a complete 𝒜_{∗}-comodule N_{∗}, let N_{∗}^{alg} ⊆ N_{∗} be given by the pullback

[figure: N_{∗}^{alg}; ν|N_{∗}^{alg}; 𝒜_{∗} ⊗ N_{∗}; N_{∗}; ν; 𝒜_{∗} ˆ⊗ N_{∗}]

in graded 𝔽_{p}-vector spaces. In other words, N_{∗}^{alg} consists of the α ∈ N_{∗} whose coaction ν(α) = ∑_{I} Sq_{∗}^{I} ⊗α_{I} (in the notation for p = 2) is a finite sum, rather than a formal infinite sum. Here I runs over the admissible sequences, so that {Sq_{∗}^{I}}_{I} is a basis for 𝒜_{∗}, and α_{I} = Sq_{∗}^{I}(α).

Lemma 3.10. The restricted coaction map ν|N_{∗}^{alg} factors (uniquely) through the inclusion 𝒜_{∗} ⊗ N_{∗}^{alg} ⊆ 𝒜_{∗} ⊗ N_{∗}, hence defines a map

ν^{alg} : N_{∗}^{alg} ⟶ 𝒜_{∗} ⊗ N_{∗}^{alg}

that makes N_{∗}^{alg} an 𝒜_{∗}-comodule in the algebraic sense.

Proof. The composite

\xymatrix{N_{∗}^{alg} \ar[r]^{ν|N_{∗}^{alg}} & 𝒜_{∗} ⊗ N_{∗} \ar[r]^{1⊗ν} & 𝒜_{∗} ⊗ (𝒜_{∗} ˆ⊗ N_{∗})}

factors as

\xymatrix{N_{∗}^{alg} \ar[r]^{ν|N_{∗}^{alg}} & 𝒜_{∗} ⊗ N_{∗} \ar[r]^{ψ⊗1} & 𝒜_{∗} ⊗ 𝒜_{∗} ⊗ N_{∗} ⊆ 𝒜_{∗} ⊗ (𝒜_{∗} ˆ⊗ N_{∗})}

by coassociativity (2.11) of the complete coaction. Hence ν|N_{∗}^{alg} factors through the pullback 𝒜_{∗} ⊗ N_{∗}^{alg} in

[figure: 𝒜_{∗} ⊗ N_{∗}^{alg}; 1⊗ν|N_{∗}^{alg}; 𝒜_{∗} ⊗ 𝒜_{∗} ⊗ N_{∗}; 𝒜_{∗} ⊗ N_{∗}; 1⊗ν; 𝒜_{∗} ⊗ (𝒜_{∗} ˆ⊗ N_{∗}) .]

Algebraic counitality and coassociativity of the lifted map ν^{alg} follow from the corresponding properties of ν displayed in Definition 2.7. □

The following identification stems from a conversation with M. Bökstedt.

Proposition 3.11. The image of the injective homomorphism ϵ_{∗}: M_{∗} → R_{+}(M_{∗}) equals the maximal algebraic sub 𝒜_{∗}-comodule R_{+}(M_{∗})^{alg} ⊂ R_{+}(M_{∗}).

Proof. Let L_{∗} be any algebraic 𝒜_{∗}-comodule. Given any α ∈ L_{∗}, with coaction ν(α) = ∑_{I} Sq_{∗}^{I} ⊗α_{I}, let ⟨α⟩ ⊆ L_{∗} be the graded vector subspace spanned by the α_{I} = Sq_{∗}^{I}(α). Here we are using the notation appropriate for p = 2; the case p odd is completely similar. Since ν(α) is a finite sum, ⟨α⟩ is a finite dimensional subspace. Furthermore, it is a sub 𝒜_{∗}-comodule, since ν(α_{I}) = ∑_{J} Sq_{∗}^{J} ⊗ Sq_{∗}^{J}(α_{I}) and Sq_{∗}^{J}(α_{I}) = (Sq^{I} Sq^{J})_{∗}(α) is a finite sum of terms Sq_{∗}^{K}(α) = α_{K}. Now consider the case L_{∗} = R_{+}(M_{∗})^{alg}. It is clear that ϵ_{∗}(M_{∗}) ⊆ R_{+}(M_{∗})^{alg}, since M_{∗} is an algebraic 𝒜_{∗}-comodule and ϵ_{∗} respects the coaction. Let α ∈ R_{+}(M_{∗})^{alg} be any element, and consider the linear span

N_{∗} = ϵ_{∗}(M_{∗}) + ⟨α⟩ ⊆ R_{+}(M_{∗})^{alg} .

It is bounded below and of finite type, so by Lemma 3.8 there is a unique lift g_{∗}

[figure: M_{∗}; ϵ_{∗}; R_{+}(M_{∗}); N_{∗}; g_{∗}; f_{∗}]

of the inclusion f_{∗}: N_{∗} → R_{+}(M_{∗}). Hence N_{∗} ⊆ ϵ_{∗}(M_{∗}), so in fact α ∈ ϵ_{∗}(M_{∗}). □

4. The Tate construction

We recall the Tate construction of Greenlees [12], and its relation with homotopy orbit and homotopy fixed point spectra. We then show how it can be expressed as the homotopy inverse limit of bounded below spectra, in two equivalent ways. This lets us make sense of the continuous (co-)homology groups of the Tate construction. We then describe the homological Tate spectral sequences. There are two types, one converging to the continuous homology of the Tate construction and one converging to the continuous cohomology. The terms of these spectral sequences will be linearly dual to each other, but, as already noted in §2.2, their target groups will only be dual in a topologized sense. The main properties of these spectral sequences are summarized in Propositions 4.14, 4.15 and 4.17.

4.1. Equivariant spectra and various fixed point constructions. We review some notions from stable equivariant homotopy theory, in the framework of Lewis–May spectra [18]. Let G be a compact Lie group, quite possibly finite, and let 𝒰 be a complete G-universe. We fix an identification 𝒰^{G} = ℝ^{∞}, and write i: ℝ^{∞} → 𝒰 for the inclusion. Let G𝒮 𝒰 be the category of genuine G-spectra, and let G𝒮 ℝ^{∞} be the category of naive G-spectra. Similarly, let 𝒮 ℝ^{∞} be the category of (nonequivariant) spectra. The restriction of universe functor i^{∗}: G𝒮 𝒰 → G𝒮 ℝ^{∞} has a left adjoint, the extension of universe functor i_{∗}: G𝒮 ℝ^{∞} → G𝒮 𝒰 , see [18, §II.1]. The functor 𝒮 ℝ^{∞} → G𝒮 ℝ^{∞}, giving a spectrum the trivial G-action, has a left adjoint taking a naive G-spectrum Y to the orbit spectrum Y/G, as well as a right adjoint taking Y to the fixed point spectrum Y^{G}. For a genuine G-spectrum X, the orbit spectrum X/G = (i^{∗}X)/G and fixed point spectrum X^{G} = (i^{∗}X)^{G} are defined by first restricting to the underlying naive G-spectra.

Let EG be a free, contractible G-CW complex. Let c: EG_{+} → S^{0} be the collapse map that sends EG to the non-base point of S^{0}, and let ẼG be its mapping cone, so that we have a (homotopy) cofiber sequence

(4.1)  EG_{+} ⟶^{c} S^{0} ⟶ ẼG

of based G-CW complexes. The n-skeleton ẼG^{(n)} of ẼG is then the mapping cone of the restricted collapse map EG_{+}^{(n−1)} → S^{0}, for each n ≥ 0. We may and will assume that each skeleton EG^{(n−1)} is a finite G-CW complex.

Definition 4.1. For each naive G-spectrum Y let Y_{hG} = (EG_{+} ∧Y)/G be the homotopy orbit spectrum, and let Y^{hG} = F (EG_{+}, Y)^{G} be the homotopy fixed point spectrum. For each genuine G-spectrum X let

X_{hG} = (EG_{+} ∧ i^{∗}X)/G = (i^{∗}X)_{hG}

and

X^{hG} = F (EG_{+}, X)^{G} = (i^{∗}X)^{hG}

be defined by first restricting to the G-trivial universe. Furthermore, let

X^{tG} = [ẼG ∧ F (EG_{+}, X)]^{G}

be the Tate construction on X. This is the spectrum denoted ℍ̂(G, X) by Bökstedt and Madsen [5] and t_{G}(X)^{G} by Greenlees and May [14].

The Segal conjecture is concerned with the map Γ: X^{G} → X^{hG} induced by F (c, 1): X ≅ F (S^{0}, X) → F (EG_{+}, X) by passing to fixed points. By smashing the cofiber sequence (4.1) with F (c, 1) and passing to G-fixed points, we can embed this map in the following diagram, consisting of two horizontal cofiber sequences:

\xymatrix{[EG_{+} ∧ X]^{G} \ar[r] \ar[d]_{≃} & X^{G} \ar[r] \ar[d]^{Γ} & [ẼG ∧ X]^{G} \ar[d]^{Γ̂} \\ [EG_{+} ∧ F (EG_{+}, X)]^{G} \ar[r] & F (EG_{+}, X)^{G} \ar[r] & [ẼG ∧ F (EG_{+}, X)]^{G}}

The adjunction counit ϵ: i_{∗}i^{∗}X → X and the map F (c, 1) are both G-maps and non-equivariant equivalences. By the G-Whitehead theorem, both maps

1 ∧ ϵ: i_{∗}(EG_{+} ∧ i^{∗}X) = EG_{+} ∧ i_{∗}i^{∗}X ⟶ EG_{+} ∧ X

and

1 ∧ F (c, 1): EG_{+} ∧ X ⟶ EG_{+} ∧ F (EG_{+}, X)

are genuine G-equivalences. Hence we have the equivalence indicated on the left. Furthermore, there is an Adams transfer equivalence

(4.2)  τ̃: (Σ^{ad G}EG_{+} ∧ i^{∗}X)/G ⟶^{≃} [i_{∗}(EG_{+} ∧ i^{∗}X)]^{G} ,

where ad G denotes the adjoint representation of G. See [18, §II.2] and [14, Part I] for further details.

In the cases of interest to us, when G is discrete or abelian, the adjoint representation is trivial so that Σ^{ad G} = Σ^{dim G}. Hence we may rewrite the diagram above as the following norm–restriction diagram

(4.3)  \xymatrix{Σ^{dim G}X_{hG} \ar[r]^{N} \ar[d]_{=} & X^{G} \ar[r]^{R} \ar[d]^{Γ} & [ẼG ∧ X]^{G} \ar[d]^{Γ̂} \\ Σ^{dim G}X_{hG} \ar[r]^{N^{h}} & X^{hG} \ar[r]^{R^{h}} & X^{tG}}

for any genuine G-spectrum X. We note that the adjunction counit ϵ: i_{∗}i^{∗}X → X induces equivalences (i_{∗}i^{∗}X)_{hG} ≃ X_{hG} and (i_{∗}i^{∗}X)^{hG} ≃ X^{hG}, hence (i_{∗}i^{∗}X)^{tG} ≃ X^{tG}, so the Tate construction on X only depends on the naive G-spectrum underlying X. The spectra in the lower row have been studied by means of spectral sequences converging to their homotopy groups, e.g. in [5], [15], [23], [2] and [16]. These spectral sequences arise in the case of the homotopy orbit and fixed point spectra by choosing a filtration of EG, and by a filtration of ẼG introduced by Greenlees [12] in the case of the Tate spectrum X^{tG}. We shall instead be concerned with the spectral sequences that arise by applying homology in place of homotopy.

4.2. Tate cohomology and the Greenlees filtration of ẼG. We recall the definition of the Tate cohomology groups from [10, §XII.3], and the associated Tate homology groups. Let G be a finite group, let 𝔽_{p}G = 𝔽_{p}[G] be its group algebra, and let (P_{∗}, d_{∗}) be a complete resolution of the trivial 𝔽_{p}G-module 𝔽_{p} by free 𝔽_{p}G-modules. This is a commutative diagram

[figure: ⋯; P_{1}; d_{1}; P_{0}; d_{0}; P_{−1}; d_{−1}; P_{−2}; ⋯; 𝔽_{p}]

of 𝔽_{p}G-modules, where the P_{n}’s are free and the horizontal sequence is exact. The image of d_{0} is identified with 𝔽_{p}, as indicated.

Definition 4.2. Given an 𝔽_{p}G-module M the Tate cohomology and Tate homology groups are defined by

Ĥ^{n}(G; M) = H^{n}(Hom_{𝔽_{p}G}(P_{∗}, M))

and

Ĥ_{n}(G; M) = H_{n}(P_{∗} ⊗_{𝔽_{p}G} M) ,

respectively, where (P_{∗}, d_{∗}) is a complete 𝔽_{p}G-resolution. (To form the balanced tensor product, we turn P_{∗} into a complex of right 𝔽_{p}G-modules by means of the group inverse.) These groups are independent of the chosen complete 𝔽_{p}G-resolution, and there are isomorphisms

Ĥ^{n}(G; M) ≅ Ĥ_{−n−1}(G; M)

and

Hom(Ĥ_{n}(G; M), 𝔽_{p}) ≅ Ĥ^{n}(G; Hom(M, 𝔽_{p}))

for all integers n. Note that we do not follow the shifted grading convention for Tate homology given in [14, 11.2].

The topological analogue of a complete resolution is a bi-infinite filtration of ẼG, in the category of G-spectra, which was introduced by Greenlees [12]. We recall the details of the construction. For brevity we shall not distinguish notationally between a based G-CW complex and its suspension G-CW spectrum. For integers n ≥ 0 we let Ẽ_{n} = ẼG^{(n)} be (the suspension spectrum of) the n-skeleton of ẼG, while Ẽ_{−n} = D(Ẽ_{n}) = F (ẼG^{(n)}, S) is its functional dual. These definitions agree for n = 0, as Ẽ_{0} = S is the sphere spectrum. Splicing the skeleton filtration of ẼG with its functional dual, we get the following diagram

(4.4) D(ẼG) ⟶ … ⟶ Ẽ_{−1} ⟶ Ẽ_{0} = S ⟶ Ẽ_{1} ⟶ Ẽ_{2} ⟶ … ⟶ ẼG ,

which we call the Greenlees filtration. Both ẼG ≃ hocolim_{n} Ẽ_{n} and D(ẼG) ≃ holim_{n} Ẽ_{n} are non-equivariantly contractible. Applying homology to the non-limiting terms of this filtration (i.e., the Ẽ_{n} for n ∈ ℤ) gives a spectral sequence with E_{s,t}^{1} = H_{s+t}(Ẽ_{s}/Ẽ_{s−1}) that converges to H_{∗}(ẼG, D(ẼG)) = 0. It is concentrated on the horizontal axis, since Ẽ_{n}/Ẽ_{n−1} is a finite wedge sum of G-free n-sphere spectra G_{+} ∧ S^{n} for each integer n. Hence the spectral sequence collapses at the E^{2}-term, and we get a long exact sequence

(4.5) [figure: …; H_{2}(Ẽ_{2}/Ẽ_{1}); d_{2,0}^{1}; H_{1}(Ẽ_{1}/Ẽ_{0}); d_{1,0}^{1}; H_{0}(Ẽ_{0}/Ẽ_{−1}); …; H_{0}(S)]

of finitely generated free 𝔽_{p}G-modules. Letting

P_{n} = H_{n+1}(Ẽ_{n+1}/Ẽ_{n})

and d_{n} = d_{n+1,0}^{1} for all integers n yields a complete resolution (P_{∗}, d_{∗}) of 𝔽_{p} = H_{0}(S).

4.3. Continuous homology of the Tate construction. Let G be a finite group and let X be a genuine G-spectrum. By means of the Greenlees filtration, we may filter the Tate construction X^{tG} by a tower of spectra.

Definition 4.3. For each integer n let ẼG/Ẽ_{n−1} be the homotopy cofiber of the map Ẽ_{n−1} → ẼG, and define

X^{tG}[−∞, n−1] = [Ẽ_{n−1} ∧ F (EG_{+}, X)]^{G}
X^{tG}[n] = X^{tG}[n, ∞] = [ẼG/Ẽ_{n−1} ∧ F (EG_{+}, X)]^{G} .

Smashing the cofiber sequence Ẽ_{n−1} → ẼG → ẼG/Ẽ_{n−1} with F (EG_{+}, X) and taking G-fixed points, we get a cofiber sequence

X^{tG}[−∞, n−1] ⟶ X^{tG} ⟶ X^{tG}[n, ∞]

for each integer n. The maps Ẽ_{n−1} → Ẽ_{n} in the Greenlees filtration (4.4) induce maps between these cofiber sequences, which combine to the non-limiting terms of the following horizontal tower of vertical cofiber sequences: (4.6)

\xymatrix{∗ \ar[r] \ar[d] & … \ar[r] & X^{tG}[−∞, n−1] \ar[r] \ar[d] & X^{tG}[−∞, n] \ar[r] \ar[d] & … \ar[r] & X^{tG} \ar[d]^{=} \\ X^{tG} \ar[r]^{=} \ar[d]_{=} & … \ar[r]^{=} & X^{tG} \ar[r]^{=} \ar[d] & X^{tG} \ar[r]^{=} \ar[d] & … \ar[r]^{=} & X^{tG} \ar[d] \\ X^{tG} \ar[r] & … \ar[r] & X^{tG}[n, ∞] \ar[r] & X^{tG}[n+1, ∞] \ar[r] & … \ar[r] & ∗}

Lemma 4.4. Let X be a G-spectrum. Then

holim_{n→−∞} X^{tG}[−∞, n] ≃ ∗  and  hocolim_{n→∞} X^{tG}[n, ∞] ≃ ∗

so

holim_{n→−∞} X^{tG}[n, ∞] ≃ X^{tG}  and  hocolim_{n→∞} X^{tG}[−∞, n] ≃ X^{tG} .

Proof. For negative n, Ẽ_{n} = D(Ẽ_{m}) for m = −n, and there is a G-equivariant equivalence ν: D(Ẽ_{m})∧Z ⟶^{≃} F (Ẽ_{m}, Z) for any G-spectrum Z, since the finite G-CW spectrum Ẽ_{m} is dualizable [18, III.2.8]. Hence

holim_{n→−∞} X^{tG}[−∞, n] = holim_{n→−∞} [Ẽ_{n} ∧ F (EG_{+}, X)]^{G}

= holim_{m→∞} [D(Ẽ_{m}) ∧ F (EG_{+}, X)]^{G} ≃ holim_{m→∞} F (Ẽ_{m} ∧ EG_{+}, X)^{G}

≅ F (hocolim_{m→∞} Ẽ_{m} ∧ EG_{+}, X)^{G} ≃ F (ẼG ∧ EG_{+}, X)^{G} ,

which is contractible because ẼG ∧ EG_{+} is G-equivariantly contractible. For the second claim we use that ẼG/Ẽ_{n} is a free G-CW spectrum. Indeed, for n ≥ 0, ẼG/Ẽ_{n} ≃ Σ(EG/EG^{(n−1)}). Thus, by the G-Whitehead theorem and the Adams transfer equivalence (4.2) we have

hocolim_{n→∞} X^{tG}[n+1, ∞] = hocolim_{n→∞} [ẼG/Ẽ_{n} ∧ F (EG_{+}, X)]^{G}

≃ hocolim_{n→∞} [ẼG/Ẽ_{n} ∧ X]^{G} ≃ hocolim_{n→∞} [ẼG/Ẽ_{n} ∧ i_{∗}i^{∗}X]^{G}

≃ hocolim_{n→∞} (ẼG/Ẽ_{n} ∧ i^{∗}X)/G ≅ (hocolim_{n→∞} ẼG/Ẽ_{n} ∧ i^{∗}X)/G ,

which is contractible since hocolim_{n→∞} (ẼG/Ẽ_{n}) is G-equivariantly contractible. The remaining claims follow, since the homotopy limit of a fiber sequence is a fiber sequence, and the homotopy colimit of a cofiber sequence is a cofiber sequence. □

Hereafter we abbreviate X^{tG}[n, ∞] to X^{tG}[n]. Then the lower horizontal tower {X^{tG}[n]}_{n} in (4.6) becomes

(4.7)  X^{tG} ⟶ … ⟶ X^{tG}[n] ⟶ X^{tG}[n+1] ⟶ … ⟶ ∗ ,

which we will refer to as the Tate tower. The following two lemmas should be compared with the sequences (2.6) and (2.7).

Lemma 4.5. Let X be a G-spectrum. Then

lim_{n→∞} H^{∗}(X^{tG}[n]) = Rlim_{n→∞} H^{∗}(X^{tG}[n]) = colim_{n→∞} H_{∗}(X^{tG}[n]) = 0 .

Proof. In cohomology, we have a Milnor lim-Rlim short exact sequence

0 ⟶ Rlim_{n} H^{∗−1}(X^{tG}[n]) ⟶ H^{∗}(hocolim_{n} X^{tG}[n]) ⟶ lim_{n} H^{∗}(X^{tG}[n]) ⟶ 0 .

By Lemma 4.4 the middle term is zero, hence so are the other two terms. In homology, we have the isomorphism

colim_{n→∞} H_{∗}(X^{tG}[n]) ≅ H_{∗}(hocolim_{n→∞} X^{tG}[n]) .

By the same lemma the right hand side is zero. □

Lemma 4.6. Suppose that X is bounded below and of finite type over 𝔽_{p}. Then each spectrum X^{tG}[n] is bounded below and of finite type over 𝔽_{p}. Hence

Rlim_{n→−∞} H_{∗}(X^{tG}[n]) = 0 .

Proof. Let X^{tG}[n, m] = [Ẽ_{m}/Ẽ_{n−1} ∧ F (EG_{+}, X)]^{G}. For m ≥ n there is a cofiber sequence

X^{tG}[n, m−1] ⟶ X^{tG}[n, m] ⟶ ⋁ Σ^{m}X ,

with one copy of Σ^{m}X in the wedge sum for each of the finitely many G-free m-cells in ẼG. Since the connectivity of Σ^{m}X grows to infinity with m, the first claim of the lemma follows by induction on m. The derived limit of any tower of finite groups is zero, which gives the second conclusion. □

We use the Tate tower (4.7) to define the continuous (co-)homology of X^{tG}, as in Definition 2.3 and Corollary 2.9.

Definition 4.7. Let G be a finite group and X a G-spectrum whose underlying non-equivariant spectrum is bounded below and of finite type over 𝔽_{p}. By the continuous cohomology of X^{tG} we mean the 𝒜 -module

H_{c}^{∗}(X^{tG}) = colim_{n→−∞} H^{∗}(X^{tG}[n]) .

By the continuous homology of X^{tG} we mean the complete 𝒜_{∗}-comodule

H_{∗}^{c}(X^{tG}) = lim_{n→−∞} H_{∗}(X^{tG}[n]) .

There is a natural isomorphism Hom(H_{c}^{∗}(X^{tG}), 𝔽_{p}) ≅ H_{∗}^{c}(X^{tG}) of complete 𝒜_{∗}-comodules, as well as a natural isomorphism Hom^{c}(H_{∗}^{c}(X^{tG}), 𝔽_{p}) ≅ H_{c}^{∗}(X^{tG}) of 𝒜 -modules.

Remark 4.8. Different G-CW structures on EG will give rise to different Greenlees filtrations and Tate towers, but by cellular approximation any two choices give pro-isomorphic towers in homology. The continuous homology, as a complete 𝒜_{∗}-comodule, is therefore independent of the choice. Likewise for continuous cohomology.

We end this subsection by giving a reformulation of the Tate construction, known as Warwick duality [13, §4], which will be used in §5.2 when making a topological model for the Singer construction.

Proposition 4.9 ([14, 2.6]). There is a natural chain of equivalences

ΣF (ẼG, EG_{+} ∧ X)^{G} ≃ [ẼG ∧ F (EG_{+}, X)]^{G} = X^{tG} .

Proof. We have a commutative diagram of G-spectra

(4.8)  \xymatrix{EG_{+} ∧ X \ar[r]^{F (c,1∧1)} & F (EG_{+}, EG_{+} ∧ X) \\ EG_{+} ∧ EG_{+} ∧ X \ar[u]^{c∧1∧1}_{≃} \ar[r] \ar[d]_{1∧F (c,c∧1)}^{≃} & F (EG_{+}, EG_{+} ∧ X) \ar[u]_{=} \ar[d]^{F (1,c∧1)}_{≃} \\ EG_{+} ∧ F (EG_{+}, X) \ar[r]^{c∧1} & F (EG_{+}, X) .}

The maps labeled ≃ are G-equivalences. The proposition follows by taking horizontal homotopy cofibers and fixed points. □

We now strengthen this to a statement about towers.

Lemma 4.10. For each integer m there is a natural chain of equivalences

ΣF (Ẽ_{m}, EG_{+} ∧ X)^{G} ≃ [ẼG/Ẽ_{−m} ∧ F (EG_{+}, X)]^{G} = X^{tG}[1−m]

connecting the tower

ΣF (ẼG, EG_{+} ∧ X)^{G} → … ⟶ ΣF (Ẽ_{m+1}, EG_{+} ∧ X)^{G}

⟶ ΣF (Ẽ_{m}, EG_{+} ∧ X)^{G} ⟶ …

to the Tate tower

X^{tG} ⟶ … ⟶ X^{tG}[−m] ⟶ X^{tG}[1−m] ⟶ ….

Proof. First observe that there is a natural chain of G-homotopy equivalences

\xymatrix{ΣF (Ẽ_{m}, EG_{+} ∧ X) \\ ΣD(Ẽ_{m}) ∧ EG_{+} ∧ X \ar[u]_{ν}^{≃} \ar[d]_{≃}^{1∧1∧F (c,1)} \\ ΣD(Ẽ_{m}) ∧ EG_{+} ∧ F (EG_{+}, X) \\ ΣẼ_{−m} ∧ EG_{+} ∧ F (EG_{+}, X) . \ar[u]^{≃}}

The upper map ν is an equivalence because Ẽ_{m} is dualizable. The middle map is an equivalence because EG_{+} is G-free and F (c, 1) is a non-equivariant equivalence. The lower map is the identity when m ≥ 0 and the map induced by the natural equivalence Ẽ_{−m} → D(D(Ẽ_{−m})) = D(Ẽ_{m}) when m ≤ 0. The latter two maps agree for m = 0, since Ẽ_{0} = S. We proceed to prove that there is a chain of G-equivalences

ΣẼ_{−m} ∧ EG_{+} ≃ ẼG/Ẽ_{−m} ,

natural for varying m ∈ ℤ. This will prove the lemma after passage to G-fixed points. For m ≥ 0, let c_{m}: EG_{+}^{(m−1)} → S^{0} be the restricted collapse map. Consider the following commutative diagram of G-spectra:

\xymatrix{∗ \ar[d] \ar[r] & D(Ẽ_{m}) \ar[r]^{=} \ar[d] & Ẽ_{−m} \ar[d] \\ EG_{+} \ar[r]^{c} \ar[d]_{=} & S \ar[r] \ar[d]^{Dc_{m}} & ẼG \ar[d] \\ EG_{+} \ar[r]^{Dc_{m}∘c} & D(EG_{+}^{(m−1)}) \ar[r] & ẼG/Ẽ_{−m} \\ EG_{+} \ar[r]^{Dc_{m}∧1} \ar[u]^{=} & D(EG_{+}^{(m−1)}) ∧ EG_{+} \ar[r] \ar[u]_{1∧c}^{≃} & ΣẼ_{−m} ∧ EG_{+} \ar[u]^{≃}}

All four rows are cofiber sequences. If we disregard the lower row, then all three remaining columns are also cofiber sequences. Hence, the homotopy cofiber of Dc_{m} ∘ c is equivalent to ẼG/Ẽ_{−m}, and we have a G-equivalence ΣẼ_{−m} ∧ EG_{+} ⟶^{≃} ẼG/Ẽ_{−m}, natural for varying m ≥ 0.

For m ≤ 0 let n = −m ≥ 0, let i_{n}: EG_{+}^{(n−1)} → EG_{+} be the inclusion, let δ_{n}: EG_{+}^{(n−1)} → EG_{+}^{(n−1)} ∧ EG_{+} be the diagonal, and let c_{n}: EG_{+}^{(n−1)} → S^{0} be as above. Consider the following commutative diagram of G-spectra:

\xymatrix{S \ar[d] \ar[r]^{=} & S \ar[r] \ar[d] & ∗ \ar[d] \\ Ẽ_{n} \ar[r] \ar[d] & ẼG \ar[r] \ar[d] & ẼG/Ẽ_{n} \ar[d]^{≃} \\ ΣEG_{+}^{(n−1)} \ar[r]^{Σi_{n}} \ar[d]_{≃}^{Σδ_{n}} & ΣEG_{+} \ar[r] \ar[d]_{=} & Σ(EG/EG^{(n−1)}) \ar[d]^{≃} \\ ΣEG_{+}^{(n−1)} ∧ EG_{+} \ar[r]^{Σc_{n}∧1} & ΣEG_{+} \ar[r] & ΣẼ_{n} ∧ EG_{+}}

Again, all four rows are cofiber sequences. If we disregard the lower row, then all three remaining columns are also cofiber sequences. Hence, the middle vertical map on the right is a G-equivalence, and we have a G-equivalence ẼG/Ẽ_{−m} ⟶^{≃} ΣẼ_{−m} ∧ EG_{+}, natural for varying m ≤ 0. The direction of the G-equivalence changes when we pass from m ≥ 0 to m ≤ 0, but for m = 0 the two equivalences are mutually inverse identifications of ΣEG_{+} and ẼG/S. Hence we can connect ΣẼ_{−m}∧EG_{+} to ẼG/Ẽ_{−m} by a chain of two G-equivalences, where the first uses the maps ΣẼ_{−m}∧EG_{+} → ẼG/Ẽ_{−m} for m ≥ 0 and identity maps for m < 0, and the second uses identity maps for m ≥ 0 and the maps ẼG/Ẽ_{−m} → ΣẼ_{−m} ∧ EG_{+} for m < 0. These are then compatible for all m ∈ ℤ. □

Corollary 4.11. The continuous (co-)homology of X^{tG} may be computed from the tower

X^{tG} ⟶ … ⟶ ΣF (Ẽ_{m+1}, EG_{+} ∧ X)^{G} ⟶ ΣF (Ẽ_{m}, EG_{+} ∧ X)^{G} ⟶ …

as

H_{c}^{∗}(X^{tG}) ≅ colim_{m→∞} ΣH^{∗}(F (Ẽ_{m}, EG_{+} ∧ X)^{G})

and

H_{∗}^{c}(X^{tG}) ≅ lim_{m→∞} ΣH_{∗}(F (Ẽ_{m}, EG_{+} ∧ X)^{G}) .

4.4. The (co-)homological Tate spectral sequences. Let G be a finite group and X a G-spectrum. The cofiber sequence Ẽ_{s}/Ẽ_{s−1} → ẼG/Ẽ_{s−1} → ẼG/Ẽ_{s} induces a cofiber sequence

[Ẽ_{s}/Ẽ_{s−1} ∧ F (EG_{+}, X)]^{G} ⟶ X^{tG}[s] ⟶^{i} X^{tG}[s+1]

for every integer s. The left hand term is equivalent to

[Ẽ_{s}/Ẽ_{s−1} ∧ X]^{G} ≃ (Ẽ_{s}/Ẽ_{s−1} ∧ i^{∗}X)/G since Ẽ_{s}/Ẽ_{s−1} is G-free. Applying cohomology, we get an exact couple of 𝒜 -modules

(4.9)  \xymatrix{A^{s+1,∗} \ar[r]^{i} & A^{s,∗} \ar[d] \\  & Ê_{1}^{s,∗} \ar@{-->}[ul]}

with

A^{s,t} = H^{s+t}(X^{tG}[s])  and  Ê_{1}^{s,t} = H^{s+t}((Ẽ_{s}/Ẽ_{s−1} ∧ i^{∗}X)/G) .

By Lemma 4.5, lim_{s} A^{s} = Rlim_{s} A^{s} = 0, so this spectral sequence converges conditionally to the colimit H_{c}^{∗}(X^{tG}), in the first sense of [3, 5.10]. Applying homology instead, we get an exact couple of algebraic 𝒜_{∗}-comodules

(4.10)  \xymatrix{A_{s,∗} \ar[r]^{i} & A_{s+1,∗} \ar@{-->}[dl] \\ Ê_{s,∗}^{1} \ar[u]}

with

A_{s,t} = H_{s+t}(X^{tG}[s])  and  Ê_{s,t}^{1} = H_{s+t}((Ẽ_{s}/Ẽ_{s−1} ∧ i^{∗}X)/G) .

By Lemma 4.5, colim_{s} A_{s} = 0, so this spectral sequence converges conditionally to the limit H_{∗}^{c}(X^{tG}), in the second sense of [3, 5.10]. We can rewrite the Ê^{1}-term as

Ê_{s,t}^{1} ≅ H_{s}(Ẽ_{s}/Ẽ_{s−1}) ⊗_{𝔽_{p}G} H_{t}(X) = P_{s−1} ⊗_{𝔽_{p}G} H_{t}(X) ,

and the d^{1}-differential is induced by the differential in the complete resolution (P_{∗}, d_{∗}), so

Ê_{s,t}^{2} ≅ Ĥ_{s−1}(G; H_{t}(X)) ≅ Ĥ^{−s}(G; H_{t}(X)) .

Dually, the Ê_{1}-term is

Ê_{1}^{s,t} ≅ Hom(P_{s−1} ⊗_{𝔽_{p}G} H_{t}(X), 𝔽_{p}) ≅ Hom_{𝔽_{p}G}(P_{s−1}, H^{t}(X))

and

Ê_{2}^{s,t} ≅ Ĥ^{s−1}(G; H^{t}(X)) ≅ Ĥ_{−s}(G; H^{t}(X)) .

Definition 4.12. Let G be a finite group and X a G-spectrum. The cohomological Tate spectral sequence of X is the conditionally convergent spectral sequence

Ê_{2}^{s,t} = Ĥ_{−s}(G; H^{t}(X)) ⟹ H_{c}^{s+t}(X)

associated with the exact couple of 𝒜 -modules (4.9). Dually, the homological Tate spectral sequence of X is the conditionally convergent spectral sequence

Ê_{s,t}^{2} = Ĥ^{−s}(G; H_{t}(X)) ⟹ H_{s+t}^{c}(X)

associated with the exact couple of 𝒜_{∗}-comodules (4.10).

Remark 4.13. There is a natural isomorphism Ê_{r}^{s,t} ≅ Hom(Ê_{s,t}^{r}, 𝔽_{p}) for all finite r, s and t, so that the d_{r}-differential d_{r}^{s,t}: Ê_{r}^{s,t} → Ê_{r}^{s+r,t−r+1} is the linear dual of the d^{r}-differential d_{s+r,t−r+1}^{r}: Ê_{s+r,t−r+1}^{r} → Ê_{s,t}^{r}. In this sense the cohomological Tate spectral sequence is dual to the homological Tate spectral sequence.

To get strong convergence, we need X to be bounded below in the cohomological case, and that X is bounded below and of finite type over 𝔽_{p} in the homological case.

Proposition 4.14. Let G be a finite group and X a G-spectrum. Assume that X is bounded below. Then X^{tG} is the homotopy inverse limit of a tower {X^{tG}[s]}_{s} of bounded below spectra, and the cohomological Tate spectral sequence

Ê_{2}^{s,t}(X) = Ĥ_{−s}(G; H^{t}(X)) ⟹ H_{c}^{s+t}(X^{tG})

converges strongly to the continuous cohomology of X^{tG} as an 𝒜 -module.

Proof. When H^{∗}(X) is bounded below, the cohomological Tate spectral sequence has exiting differentials in the sense of Boardman, so the spectral sequence is automatically strongly convergent by [3, 6.1]. In other words, the filtration of H_{c}^{s+t}(X^{tG}) by the sub 𝒜 -modules

F^{s}H_{c}^{∗}(X^{tG}) = im(H^{∗}(X^{tG}[s]) → H_{c}^{∗}(X^{tG}))

is exhaustive, complete and Hausdorff, and there are 𝒜 -module isomorphisms

F^{s}H_{c}^{∗}(X^{tG})/F^{s+1}H_{c}^{∗}(X^{tG}) ≅ Ê_{∞}^{s,∗} .

□

Proposition 4.15. Let G be a finite group and X a G-spectrum. Assume that X is bounded below and of finite type over 𝔽_{p}. Then X^{tG} is the homotopy inverse limit of a tower {X^{tG}[s]}_{s} of bounded below spectra of finite type over 𝔽_{p}, and the homological Tate spectral sequence

Ê_{s,t}^{2}(X) = Ĥ^{−s}(G; H_{t}(X)) ⟹ H_{s+t}^{c}(X^{tG})

converges strongly to the continuous homology of X^{tG} as a complete 𝒜_{∗}- comodule.

Proof. When H_{∗}(X) is bounded below, the homological Tate spectral sequence has entering differentials in the sense of Boardman. The derived limit RÊ_{∗,∗}^{∞} = Rlim_{r} Ê_{∗,∗}^{r} vanishes since H_{∗}(X), and thus Ê_{∗,∗}^{2}, is finite in each (bi-)degree. Hence the spectral sequence is strongly convergent by [3, 7.4]. In other words, the filtration of H_{s+t}^{c}(X^{tG}) by the complete sub 𝒜_{∗}-comodules

F_{s}H_{∗}^{c}(X^{tG}) = ker(H_{∗}^{c}(X^{tG}) → H_{∗}^{c}(X^{tG}[s]))

is exhaustive, complete and Hausdorff, and there are algebraic 𝒜_{∗}-comodule isomorphisms

F_{s+1}H_{∗}^{c}(X^{tG})/F_{s}H_{∗}^{c}(X^{tG}) ≅ Ê_{s,∗}^{∞} .

□

4.4.1. Homotopy vs. Homology. We used the Tate tower (4.7) to define our homological Tate spectral sequence, by applying homology with 𝔽_{p}-coefficients. When studying the homotopy groups of the Tate construction, it has been customary to apply π_{∗}(−) to the upper tower in (4.6): (4.11)

\xymatrix{∗ \ar[r] & … \ar[r] & X^{tG}[−∞, n−1] \ar[r] & X^{tG}[−∞, n] \ar[r] & … \ar[r] & X^{tG}}

Applying a homological functor to these two towers of spectra gives two different exact couples with isomorphic spectral sequences. If we are working with homotopy, we get two spectral sequences converging to the same groups. Using (4.11) yields a spectral sequence converging to the colimit

colim_{n} π_{∗}(X^{tG}[−∞, n]) ≅ π_{∗}(X^{tG}) ,

while using (4.7) yields an isomorphic spectral sequence converging to the inverse limit

(4.12)  lim_{n} π_{∗}(X^{tG}[n]) ≅ π_{∗}(X^{tG}) .

The latter isomorphism assumes that X is bounded below and of suitably finite type, so that the right derived limit Rlim_{n} π_{∗}(X^{tG}[n]) = 0. For example, it suffices if π_{∗}(X) is of finite type over ℤ or ℤ̂_{p}. When working with homology with 𝔽_{p}-coefficients, instead of homotopy groups, the failure of the isomorphism we made use of in (4.12) makes the situation more interesting. Applying H_{∗}(−) to the tower (4.11) will produce a sequence of homology groups whose inverse limit is not trivial in general. This means that the associated spectral sequence will not be conditionally convergent to the direct limit

colim_{n} H_{∗}(X^{tG}[−∞, n]) ≅ H_{∗}(X^{tG}) .

In fact, we have seen that the (isomorphic) homological Tate spectral sequence, arising from (4.7), converges strongly to

lim_{n} H_{∗}(X^{tG}[n]) = H_{∗}^{c}(X^{tG}) ,

which is only rarely isomorphic to H_{∗}(X^{tG}), since inverse limits and homology do not commute in general. We end this discussion by noticing that the continuous homology groups of the Tate construction on X can be thought of as the homotopy groups of the Tate construction on H ∧ X, where H = H𝔽_{p} is the Eilenberg–Mac Lane spectrum of 𝔽_{p}. In other words, continuous homology of X^{tG} is a special case of homotopy.

Proposition 4.16. For any bounded below G-spectrum X of finite type over 𝔽_{p} there is a natural isomorphism

π_{∗}(H ∧ X)^{tG} ≅ H_{∗}^{c}(X^{tG}) ,

where H has the trivial G-action.

Proof. For all integers m we have

(H ∧ X)^{tG}[1−m] ≃ ΣF (Ẽ_{m}, EG_{+} ∧ H ∧ X)^{G}
≃ H ∧ ΣF (Ẽ_{m}, EG_{+} ∧ X)^{G} ≃ H ∧ X^{tG}[1−m] .

The first and last equivalences follow from Lemma 4.10, while the middle equivalence follows from the fact that Ẽ_{m} is G-equivariantly dualizable. Thus, we have that π_{∗}(H ∧ X)^{tG}[n] ≅ H_{∗}(X^{tG}[n]) for all integers n. For a general G-spectrum X, we then have the following surjective maps:

(4.13)  π_{∗}(H ∧ X)^{tG} ⟶ lim_{n→−∞} π_{∗}(H ∧ X)^{tG}[n]

≅ lim_{n→−∞} H_{∗}(X^{tG}[n]) = H_{∗}^{c}(X^{tG}) .

Since X was assumed to be bounded below and of finite type over 𝔽_{p}, the groups in the first inverse limit system are all of finite type over 𝔽_{p}, so their Rlim vanishes and the map in (4.13) is an isomorphism. □

The previous proposition and discussion tells us that the continuous homology of X^{tG} can be considered both as the direct limit colim_{n} π_{∗}(H ∧ X)^{tG}[−∞, n] or as the inverse limit lim_{n} π_{∗}(H ∧ X)^{tG}[n, ∞]. In both cases, the filtration of the two groups given by their defining towers are the same.

4.5. Multiplicative structure. We now discuss how the treatment in [16, §4.3] of multiplicative structure in the homotopical Tate spectral sequence carries over to the homological Tate spectral sequence. Let X be a bounded below G-equivariant ring spectrum of finite type over 𝔽_{p}. We assume that the unit map η: S → X and the multiplication map μ: X ∧ X → X are equivariant with respect to the trivial G-action on S and the diagonal G-action on X ∧ X. By [14, 3.5], the homotopy cartesian square

(4.14)  \xymatrix{X^{G} \ar[r]^{R} \ar[d]_{Γ} & [ẼG ∧ X]^{G} \ar[d]^{Γ̂} \\ X^{hG} \ar[r]^{R^{h}} & X^{tG}}

in (4.3) is a diagram of ring spectra and ring spectrum maps. Up to homotopy there is a unique G-equivalence f: ẼG ∧ ẼG →^{≃} ẼG, compatible with the inclusion S^{0} → ẼG and the homeomorphism S^{0} ∧ S^{0} ≅ S^{0}. Using f, the multiplication map on X^{tG} is given by the composition

[ẼG ∧ F (EG_{+}, X)]^{G} ∧ [ẼG ∧ F (EG_{+}, X)]^{G}

[ẼG ∧ ẼG ∧ F (EG_{+} ∧ EG_{+}, X ∧ X)]^{G}
f∧F (Δ,μ)
[ẼG ∧ F (EG_{+}, X)]^{G} .

The other multiplication maps arise by replacing ẼG, EG_{+} or both by S^{0}. The unit map to X^{G} is adjoint to the G-map S → X, since S has the trivial G-action. The other unit maps arise by composition with the maps in (4.14). The non-equivariant ring spectrum structure on the Eilenberg–Mac Lane spectrum H makes H ∧ X a naively G-equivariant ring spectrum, so (H ∧ X)^{tG} is a ring spectrum. The induced graded ring structure on π_{∗}(H ∧ X)^{tG} then gives a graded ring structure on the continuous homology H_{∗}^{c}(X^{tG}), by the isomorphism of Proposition 4.16.

Proposition 4.17. Let G be a finite group and X a G-equivariant ring spectrum. Assume that X is bounded below and of finite type over 𝔽_{p}. Then the homological Tate spectral sequence

Ê_{s,t}^{2} = Ĥ^{−s}(G; H_{t}(X)) ⟹ H_{s+t}^{c}(X^{tG})

is a strongly convergent 𝒜_{∗}-comodule algebra spectral sequence, whose product at the Ê^{2}-term is given by the cup product in Tate cohomology and the Pontryagin product on H_{∗}(X).

Proof. Using the Greenlees filtration, we may filter (H ∧ X)^{tG} by the tower (4.11). This produces a homotopical Tate spectral sequence with Ê^{2}- term

Ê_{s,t}^{2} = Ĥ^{−s}(G; π_{t}(H ∧ X)) = Ĥ^{−s}(G; H_{t}(X)) ,

converging strongly to the homotopy π_{s+t}(H ∧ X)^{tG} ≅ H_{s+t}^{c}(X^{tG}). By the proof of Proposition 4.16 it is additively isomorphic to the homological Tate spectral sequence of Proposition 4.15. By [16, 4.3.5], it is also an algebra spectral sequence, with differentials being derivations with respect to the product. The Ê^{∞}-term is the associated graded of the multiplicative filtration of π_{s+t}(H ∧ X)^{tG} given by the images

im(π_{∗}(H ∧ X)^{tG}[−∞, s] → π_{∗}(H ∧ X)^{tG}) .

□

5. The topological Singer construction

5.1. Realizing the Singer construction as continuous cohomology. As observed by Miller, and presented by Bruner, May, McClure and Steinberger in [9, §II.5], there is a particular inverse system of spectra whose continuous cohomology realizes the Singer construction in the version related to the symmetric group Σ_{p}. We go through the adjustments needed to realize the version of the Singer construction related to the subgroup C_{p} generated by the cyclic permutation (1 2 ⋯ p). Let B be a non-equivariant spectrum that is bounded below and of finite type over 𝔽_{p}. For each subgroup G ⊆ Σ_{p} there is an extended power construction [9, §I.2]

D_{G}(B) = EG ⋉_{G} B^{(p)} ,

well defined in the stable homotopy category. Here B^{(p)} denotes the external pth smash power of B, and G permutes the p copies of B. It follows from [9, I.2.4] that D_{G}(B) is bounded below and of finite type over 𝔽_{p}. More precisely, we have the following calculation.

Lemma 5.1 ([9, I.2.3]). There is a natural isomorphism

H_{∗}(D_{G}(B)) ≅ H_{∗}(G; H_{∗}(B)^{⊗p}) ,

where G permutes the p copies of H_{∗}(B).

The p-fold diagonal map S^{1} → S^{1} ∧⋯∧S^{1} ≅ S^{p} induces maps Δ : ΣD_{G}(B) → D_{G}(ΣB) as in [9, §II.3]. Applied to desuspensions of B, these assemble to an inverse system

(5.1) … ⟶ Σ^{n+1}D_{G}(Σ^{−n−1}B) ⟶^{Σ^{n}Δ} Σ^{n}D_{G}(Σ^{−n}B) ⟶ … ⟶^{Δ} D_{G}(B)

in the stable homotopy category. This is a tower of bounded below spectra of finite type over 𝔽_{p}, so it makes sense to talk about its associated continuous cohomology. We now follow [9, §II.5], but focus on the case G = C_{p} instead of G = Σ_{p}. There is an additive isomorphism

H_{∗}(D_{C_{p}}(B)) ≅ 𝔽_{p}{e_{0} ⊗ α_{1} ⊗ ⋯ ⊗ α_{p}} ⊕ 𝔽_{p}{e_{j} ⊗ α^{⊗p} | j ≥ 0}

where the α_{i} and α range over a basis for H_{∗}(B), the α_{i} are not all equal, and only one representative is taken from each C_{p}-orbit of the tensors α_{1} ⊗⋯⊗α_{p}. The grading is determined by deg(e_{j}) = j. Dually, there is an isomorphism

H^{∗}(D_{C_{p}}(B)) ≅ 𝔽_{p}{w_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{p}} ⊕ 𝔽_{p}{w_{j} ⊗ a^{⊗p} | j ≥ 0}

where the a_{i} and a range over the dual basis for H^{∗}(B), and w_{j} ⊗ a^{⊗p} is dual to e_{j} ⊗ α^{⊗p} when a is dual to α. It follows that

(5.2) H^{∗}(Σ^{n}D_{C_{p}}(Σ^{−n}B)) ≅ 𝔽_{p}{Σ^{n}w_{0} ⊗ Σ^{−n}a_{1} ⊗ ⋯ ⊗ Σ^{−n}a_{p}}

⊕ 𝔽_{p}{Σ^{n}w_{j} ⊗ (Σ^{−n}a)^{⊗p} | j ≥ 0} .

By [9, II.5.6], the map Δ in (5.1) is given in cohomology as

Δ^{∗}(w_{j} ⊗ a^{⊗p}) = (−1)^{j+1}α(q) · Σw_{j+p−1} ⊗ (Σ^{−1}a)^{⊗p}

where deg(a) = q, m = (p − 1)/2 and α(q) = −(−1)^{mq}m!. For p = 2 the numerical coefficient should be read as 1. The other classes w_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{p} map to zero. It follows that

(5.3) (Σ^{n}Δ)^{∗}(Σ^{n}w_{j} ⊗ (Σ^{−n}a)^{⊗p})

= (−1)^{j+1}α(q−n) · Σ^{n+1}w_{j+p−1} ⊗ (Σ^{−n−1}a)^{⊗p} .

The action of the Steenrod algebra 𝒜 on H^{∗}(D_{C_{p}}(B)) is given by the Nishida relations. Together with the explicit formula above for the maps (Σ^{n}Δ)^{∗}, this determines the direct limit of cohomology groups as an 𝒜 -module. Miller observed that this direct limit can be described in closed form by the Singer construction on the 𝒜 -module H^{∗}(B), up to a single degree shift. We now give the C_{p}-equivariant extension of the Σ_{p}-equivariant case discussed in [9, II.5.1].

Theorem 5.2. For each spectrum B that is bounded below and of finite type over 𝔽_{p}, there is a natural isomorphism of 𝒜 -modules

ω: colim_{n→∞} H^{∗}(Σ^{n}D_{C_{p}}(Σ^{−n}B)) ⟶^{≅} Σ^{−1}R_{+}(H^{∗}(B)) .

Proof. For p = 2 the isomorphism is given by

ω(Σ^{n}w_{r+n} ⊗ (Σ^{−n}a)^{⊗2}) = x^{r+q} ⊗ a

where deg(a) = q. For p odd, the isomorphism is given by

ω(Σ^{n}w_{2(r+mn)} ⊗ (Σ^{−n}a)^{⊗p}) = \frac{(−1)^{q−n}}{ν(q−n)} · y^{r+mq} ⊗ a

and

ω(Σ^{n}w_{2(r+mn)−1} ⊗ (Σ^{−n}a)^{⊗p}) = \frac{(−1)^{q}}{ν(q−n)} · xy^{r+mq−1} ⊗ a ,

where deg(a) = q, m = (p − 1)/2 and ν(2j + ϵ) = (−1)^{j}(m!)^{ϵ} for ϵ ∈ {0, 1}. It follows from (5.3) and the relation α(q)ν(q − 1)^{−1} = ν(q)^{−1} that these homomorphisms are compatible under (Σ^{n}Δ)^{∗}. It is then clear from (5.2) that ω is an additive isomorphism. This isomorphism commutes with the Bockstein and Steenrod operations on the extended powers, described in [18, VIII.3.2] and [9, II.5.5], and the explicitly defined Bockstein and Steenrod operations on the Singer construction, given in Definition 3.1. Here are the details: For p = 2, the Steenrod reduced square Sq^{s} takes ω(Σ^{n}w_{r+n} ⊗ (Σ^{−n}a)^{⊗2}) = x^{r+q} ⊗ a to

∑_{j} \binom{r + q − j}{s − 2j}x^{r+q+s−j} ⊗ Sq^{j}(a) ,

by Definition 3.1, while ω takes

Sq^{s}(Σ^{n}w_{r+n} ⊗ (Σ^{−n}a)^{⊗2})

= ∑_{j} \binom{r + n + q − n − j}{s − 2j}Σ^{n}w_{r+n+s−2j} ⊗ (Σ^{−n} Sq^{j}(a))^{⊗2}

(given by [9, II.5.5(i)]) to

∑_{j} \binom{r + q − j}{s − 2j}x^{r+s−2j+q+j} ⊗ Sq^{j}(a) ,

and these expressions are equal. For p odd, the Bockstein operation β takes ω(Σ^{n}w_{2(r+mn)} ⊗ (Σ^{−n}a)^{⊗p}) = (−1)^{q−n}ν(q − n)^{−1} · y^{r+mq} ⊗ a to 0 by Definition 3.1. This equals the image under ω of β(Σ^{n}w_{2(r+mn)} ⊗ (Σ^{−n}a)^{⊗p}) = 0. The Steenrod reduced power P^{s} takes ω(Σ^{n}w_{2(r+mn)} ⊗ (Σ^{−n}a)^{⊗p}) to (−1)^{q−n}ν(q − n)^{−1} times the sum

∑_{j} \binom{r + mq − (p − 1)j}{s − pj}y^{r+mq+(p−1)(s−j)} ⊗ P^{j}(a)

+ ∑_{j} \binom{r + mq − (p − 1)j − 1}{s − pj − 1}xy^{r+mq+(p−1)(s−j)−1} ⊗ β P^{j}(a) ,

according to Definition 3.1. The homomorphism ω takes P^{s}(Σ^{n}w_{2(r+mn)} ⊗ (Σ^{−n}a)^{⊗p}), which equals the sum

∑_{j} \binom{r + mq − (p − 1)j}{s − pj}Σ^{n}w_{I} ⊗ (Σ^{−n} P^{j}(a))^{⊗p}

+ α(q − n) ∑_{j} \binom{r + mq − (p − 1)j − 1}{s − pj − 1}Σ^{n}w_{I−p} ⊗ (−1)^{n}(Σ^{−n}β P^{j}(a))^{⊗p}

where I = 2(r + mn) + 2(s − pj)(p − 1) (given by [9, II.5.5(ii)] and the fact that β anti-commutes with suspension), to

∑_{j} \binom{r + mq − (p − 1)j}{s − pj}\frac{(−1)^{q−n}}{ν(q − n)} · y^{J} ⊗ P^{j}(a)

+ α(q − n) ∑_{j} \binom{r + mq − (p − 1)j − 1}{s − pj − 1}\frac{(−1)^{q+1−n}}{ν(q + 1 − n)} · xy^{J−1} ⊗ β P^{j}(a)

where J = r + (s − j)(p − 1) + mq. These expressions are equal because of the relation α(q − n)ν(q − n − 1)^{−1} = ν(q − n)^{−1} mentioned above, and the fact that ν(q − n − 1) = −ν(q + 1 − n). Also for p odd, β takes ω(Σ^{n}w_{2(r+mn)−1} ⊗ (Σ^{−n}a)^{⊗p}) = (−1)^{q}ν(q − n)^{−1} · xy^{r+mq−1} ⊗ a to (−1)^{q}ν(q − n)^{−1} · y^{r+mq} ⊗ a by Definition 3.1. This equals the image under ω of β(Σ^{n}w_{2(r+mn)−1} ⊗ (Σ^{−n}a)^{⊗p}) = (−1)^{n}Σ^{n}w_{2(r+mn)} ⊗ (Σ^{−n}a)^{⊗p} (given in [18, VIII.3.2(iii)]). Finally, P^{s} takes ω(Σ^{n}w_{2(r+mn)−1} ⊗ (Σ^{−n}a)^{⊗p}) to (−1)^{q}ν(q − n)^{−1} times the sum

∑_{j} \binom{r + mq − (p − 1)j − 1}{s − pj}xy^{r+mq+(p−1)(s−j)−1} ⊗ P^{j}(a) ,

according to Definition 3.1, while ω takes P^{s}(Σ^{n}w_{2(r+mn)−1}⊗(Σ^{−n}a)^{⊗p}), which equals the sum

∑_{j} \binom{r + mq − 1 − (p − 1)j}{s − pj}Σ^{n}w_{2(r+mn)−1+2(s−pj)(p−1)} ⊗ (Σ^{−n} P^{j}(a))^{⊗p}

(given by [9, II.5.5(ii)]), to

∑_{j} \binom{r + mq − 1 − (p − 1)j}{s − pj}\frac{(−1)^{q}}{ν(q − n)} · xy^{r+(s−j)(p−1)+mq−1} ⊗ P^{j}(a) .

These expressions are obviously equal. □

5.2. The relationship between the Tate and Singer constructions. We now show that the inverse system (5.1) for G = C_{p} and B a bounded below spectrum can be realized, up to a single suspension, as the Tate tower (4.7) for a C_{p}-spectrum X = B^{∧p}. As a naive C_{p}-spectrum, B^{∧p} is equivalent to the p-fold smash product B ∧ ⋯ ∧ B, with the C_{p}-action given by cyclic permutation of the factors. The genuinely equivariant definition of B^{∧p} is obtained by specialization from Bökstedt’s definition in [6], [4] of the topological Hochschild homology THH(B) of a symmetric ring spectrum B, in the sense of [17]. Namely, B^{∧p} = sd_{p} THH(B)_{0} = THH(B)_{p−1} equals the 0-simplices of the p-fold edgewise subdivision of THH(B), which in turn equals the (p − 1)-simplices of THH(B). The ring structure on B is only relevant for the simplicial structure on THH(B), and is not needed for the formation of its (p − 1)-simplices. However, it is necessary to assume that the spectrum B is realized as a symmetric spectrum. We now make a review of definitions, in order to compare the Bökstedt-style smash powers of symmetric spectra with the external powers of Lewis–May spectra. From now on, let 𝒰 be the complete C_{p}-universe

𝒰 = ℝ^{∞} ⊕ ⋯ ⊕ ℝ^{∞} = (ℝ^{∞})^{p}

with C_{p}-action given by cyclic permutation of summands. The inclusion i: ℝ^{∞} → 𝒰 is the diagonal embedding Δ: ℝ^{∞} → (ℝ^{∞})^{p}. Let 𝔄 = {ℝ^{n} | n ≥ 0} be the sequential indexing set in ℝ^{∞}, and let 𝔄^{p} = {ℝ^{n} ⊕ ⋯ ⊕ ℝ^{n} = (ℝ^{n})^{p} | n ≥ 0} be the associated diagonal indexing set [18, §VI.5] in 𝒰 . Recall that an equivariant prespectrum D is Σ-cofibrant in the sense of [18, I.8.7], hence good in the sense of Hesselholt and Madsen [15, Def. A.1], if each structure map Σ^{W −V} D(V) → D(W) is an equivariant cofibration for V ⊆ W in the indexing set. There is a functorial thickening D^{τ} of equivariant prespectra, which produces Σ-cofibrant, hence good, prespectra, and there is a natural spacewise equivariant equivalence D^{τ} → D, see [15, Lem. A.1]. Let B be a symmetric spectrum of topological spaces, with n-th space B_{n} for each n ≥ 0. Recall that B is S-cofibrant in the sense of [17, 5.3.6] if the natural map ν_{n}: L_{n}B → B is a cofibration for each n, where the latching space L_{n}B is the n-th space in the spectrum B ∧ S̄. Here S̄ is the symmetric spectrum with 0-th space ∗ and n-th space S^{n}, for n > 0. We prefer to follow the terminology in [24, III.1.2] and refer to the S-cofibrant symmetric spectra as being flat. Every symmetric spectrum is level equivalent, hence stably equivalent, to a flat symmetric spectrum, so any spectrum may be modeled by a flat symmetric spectrum. Each symmetric spectrum B has an underlying sequential prespectrum indexed on 𝔄, with B(ℝ^{n}) = B_{n} equal to the n-th space of B. The struc-

ture map σ: B(ℝ^{n−1}) ∧ S^{1} → B(ℝ^{n}) factors as B_{n−1} ∧ S^{1} ⟶^{ι_{n}} L_{n}B ⟶^{ν_{n}} B_{n},

where ι_{n} is always a cofibration. Hence the underlying prespectrum of a flat symmetric spectrum is Σ-cofibrant.

Definition 5.3. Let B be a symmetric spectrum, with n-th space B_{n} for each n ≥ 0, and let I be Bökstedt’s category of finite sets n = {1, 2, …, n} for n ≥ 0 and injective functions. Let B_{pre}^{∧p} be the C_{p}-equivariant prespectrum with V -th space

B_{pre}^{∧p}(V) = hocolim_{(n_{1},...,n_{p})∈I^{p}} Map(S^{n_{1}} ∧ ⋯ ∧ S^{n_{p}}, B_{n_{1}} ∧ ⋯ ∧ B_{n_{p}} ∧ S^{V})

for each finite dimensional V ⊂ 𝒰 . Here C_{p} acts by cyclically permuting the n_{i}, the S^{n_{i}} and the B_{n_{i}}, as well as acting on S^{V}. Let

B^{∧p} = L((B_{pre}^{∧p})^{τ})

be the genuine C_{p}-spectrum in C_{p}𝒮 𝒰 obtained by spectrification from the functorial good thickening (B_{pre}^{∧p})^{τ} of this prespectrum. The natural maps

B_{pre}^{∧p}(V) ⟵^{≃} (B_{pre}^{∧p})^{τ} (V) ⟶^{≃} B^{∧p}(V)

are C_{p}-equivariant equivalences by the proof of [15, Prop. 2.4].

Definition 5.4. Let B be a prespectrum indexed on 𝔄. The p-fold external smash product B^{(p)} is the spectrification in C_{p}𝒮 𝒰 of the C_{p}-equivariant prespectrum

B_{pre}^{(p)}((ℝ^{n})^{p}) = B(ℝ^{n}) ∧ ⋯ ∧ B(ℝ^{n}) = B(ℝ^{n})^{∧p}

indexed on 𝔄^{p}. When B is Σ-cofibrant, so is B_{pre}^{(p)}, hence the V -th space of the spectrification is given by the colimit

B^{(p)}(V) = colim_{V ⊆(ℝ^{n})^{p}} Map(S^{(ℝ^{n})^{p}−V} , B(ℝ^{n})^{∧p}) .

Here the colimit runs over the n ∈ ℕ_{0} such that V ⊆ (ℝ^{n})^{p}, and (ℝ^{n})^{p} − V denotes the orthogonal complement of V in (ℝ^{n})^{p}. Suspension by V induces an isomorphism

B^{(p)}(V) ≅ colim_{n∈ℕ_{0}} Map((S^{n})^{∧p}, B(ℝ^{n})^{∧p} ∧ S^{V}) ,

with inverse given by suspension by (ℝ^{m})^{p} − V, followed by p instances of the stabilization B(ℝ^{n}) ∧ S^{m} → B(ℝ^{n+m}), for m sufficiently large.

We say that a symmetric spectrum B is convergent if there are integers λ(n) that grow to infinity with n, such that B_{n} is ((n/2) + λ(n))-connected and the structure map ΣB_{n} → B_{n+1} is (n+λ(n))-connected, for all sufficiently large n. These hypotheses suffice for the use of Bökstedt’s approximation lemma in the following proof. Every symmetric spectrum is stably equivalent to a convergent one.

Lemma 5.5. Let B be a flat, convergent symmetric spectrum. There is a natural chain of weak equivalences of naive C_{p}-spectra

i^{∗}B^{(p)} ≃ i^{∗}B^{∧p} .

Proof. For every finite dimensional V ⊂ ℝ^{∞} we have a natural chain of C_{p}- equivariant maps

\xymatrix{colim_{n∈ℕ_{0}} Map((S^{n})^{∧p}, B(ℝ^{n})^{∧p} ∧ S^{V}) \\ hocolim_{n∈ℕ_{0}} Map((S^{n})^{∧p}, B(ℝ^{n})^{∧p} ∧ S^{V}) \ar[u]_{≃} \ar[d]^{≃} \\ hocolim_{(n_{1},...,n_{p})∈ℕ_{0}^{p}} Map(S^{n_{1}} ∧ ⋯ ∧ S^{n_{p}}, B_{n_{1}} ∧ ⋯ ∧ B_{n_{p}} ∧ S^{V}) \ar[d]^{≃} \\ hocolim_{(n_{1},...,n_{p})∈I^{p}} Map(S^{n_{1}} ∧ ⋯ ∧ S^{n_{p}}, B_{n_{1}} ∧ ⋯ ∧ B_{n_{p}} ∧ S^{V})}

connecting B^{(p)}(V) to B_{pre}^{∧p}(V). The upper map is a weak equivalence because B is flat, hence Σ-cofibrant. The middle map is a weak homotopy equivalence because the diagonal ℕ_{0} → ℕ_{0}^{p} is (co-)final. The lower map is a weak equivalence by convergence, the fact that ℕ_{0}^{p} is filtering, and Bökstedt’s approximation lemma [6, 1.5], see [8, 2.5.1] for a published proof. Applying the thickening construction and spectrifying, we get the desired chain of naively C_{p}-equivariant weak equivalences. □

Definition 5.6. For p = 2, let ℝ(1) be the sign representation of C_{2}. For p odd, let ℂ(1) be the standard rank 1 representation of C_{p} ⊂ S^{1}, and let ℂ(i) be its ith tensor power. For all primes p, let W ⊂ ℝ^{p} be the orthogonal complement of the diagonal copy of ℝ. Then W ≅ ℝ(1) for p = 2, and W ≅ ℂ(1) ⊕ ⋯ ⊕ ℂ(m) for p odd, where m = (p − 1)/2. Let EC_{p} = S(∞W), ẼC_{p} = S^{∞W}, and give EC_{p} a C_{p}-CW structure so that

ẼC_{p}^{((p−1)n)} = S^{nW}

for all n ≥ 0. Then Ẽ_{(p−1)n} = S^{nW} for all integers n.

Proposition 5.7. Let B be a flat and convergent symmetric spectrum, and give EC_{p} a free C_{p}-CW structure as in the definition above. There is a natural weak equivalence

(B^{∧p})_{hC_{p}} = (EC_{p+} ∧ i^{∗}B^{∧p})/C_{p} ≃ EC_{p} ⋉_{C_{p}} B^{(p)} = D_{C_{p}}(B) .

More generally, there are weak equivalences

(B^{∧p})^{tC_{p}}[1−(p−1)n] ≃ Σ^{1+n}D_{C_{p}}(Σ^{−n}B)

for all n ≥ 0, which are compatible with the (p − 1)-fold composites of maps in the Tate tower (4.7) for X = B^{∧p}

… ⟶ (B^{∧p})^{tC_{p}}[1−(p−1)(n+1)] ⟶ (B^{∧p})^{tC_{p}}[1−(p−1)n] ⟶ …,

and the suspension of the inverse system (5.1) for G = C_{p}

… ⟶ Σ^{1+n+1}D_{C_{p}}(Σ^{−n−1}B) ⟶ Σ^{1+n}D_{C_{p}}(Σ^{−n}B) ⟶ ….

Proof. By the untwisting theorem [18, VI.1.17] and Lemma 5.5 there are C_{p}- equivariant equivalences

EC_{p} ⋉ B^{(p)} ≃ EC_{p+} ∧ i^{∗}B^{(p)} ≃ EC_{p+} ∧ i^{∗}B^{∧p} ,

since EC_{p} is C_{p}-free. The first claim follows by passage to C_{p}-orbit spectra. More generally, there are C_{p}-equivariant equivalences

Σ^{1+n}EC_{p} ⋉ (Σ^{−n}B)^{(p)} ≃ ΣF (S^{nW} , EC_{p} ⋉ B^{(p)}) ≃ ΣF (S^{nW} , EC_{p+} ∧ i^{∗}B^{∧p})

by [18, VI.1.5], since S^{n} ∧ S^{nW} ≅ (S^{n})^{∧p}. Passing to C_{p}-orbits, and using the Adams transfer equivalence (4.2), we get the equivalences

Σ^{1+n}D_{C_{p}}(Σ^{−n}B) ≃ ΣF (S^{nW} , EC_{p+} ∧ B^{∧p})^{C_{p}} .

The right hand side is a model for (B^{∧p})^{tC_{p}}[1−(p−1)n], by Lemma 4.10, since Ẽ_{(p−1)n} = S^{nW}. The stabilization of the left hand side given by Δ: S^{1} → S^{p} is compatible under all of these equivalences with the stabilization of the right hand side given by the inclusion S^{0} → S^{W}, again by [18, VI.1.5]. □

Definition 5.8. For each symmetric spectrum B let the topological Singer construction on B be the spectrum

R_{+}(B) = (B^{∧p})^{tC_{p}} .

The topological Singer construction realizes the algebraic Singer constructions, in the following sense.

Theorem 5.9. Let B be a symmetric spectrum that is bounded below and of finite type over 𝔽_{p}. There are natural isomorphisms

ω: H_{c}^{∗}(R_{+}(B)) ⟶^{≅} R_{+}(H^{∗}(B))

and

ω_{∗}: R_{+}(H_{∗}(B)) ⟶^{≅} H_{∗}^{c}(R_{+}(B))

of 𝒜 -modules and complete 𝒜_{∗}-comodules, respectively.

Proof. We may replace B by a stably equivalent flat and convergent symmetric spectrum, without changing the (co-)homology of B^{∧p} and R_{+}(B). By Proposition 5.7 there are 𝒜 -module isomorphisms

H^{∗}((B^{∧p})^{tC_{p}}[1−(p−1)n]) ≅ Σ^{1+n}H^{∗}(D_{C_{p}}(Σ^{−n}B))

for each n, which by Theorem 5.2 induce 𝒜 -module isomorphisms

H_{c}^{∗}((B^{∧p})^{tC_{p}}) ≅ R_{+}(H^{∗}(B))

after passage to colimits. The dual 𝒜_{∗}-comodule isomorphisms then induce complete 𝒜_{∗}-comodule isomorphisms

H_{∗}^{c}((B^{∧p})^{tC_{p}}) ≅ Hom(R_{+}(H^{∗}(B)), 𝔽_{p}) = R_{+}(H_{∗}(B))

after passage to limits. □

5.3. The topological Singer ϵ-map. We now turn to the construction of a stable map ϵ_{B}: B → R_{+}(B) that realizes the Singer homomorphism ϵ on passage to cohomology. Having done so will enable us to prove Theorem 5.13. Let B be a symmetric spectrum that is bounded below and of finite type over 𝔽_{p}. The C_{p}-spectrum X = B^{∧p} introduced in Definition 5.3 is then also bounded below and of finite type over 𝔽_{p}. We shall make use of parts of the Hesselholt–Madsen proof that THH(B) is a cyclotomic spectrum. By the first half of the proof of [15, Prop. 2.1] there is a natural equivalence

s̄_{C_{p}}: [ẼC_{p} ∧ B^{∧p}]^{C_{p}} ⟶^{≃} Φ^{C_{p}}(B^{∧p}) ,

where Φ^{C_{p}}(X) denotes the geometric fixed point spectrum of X. Furthermore, by the simplicial degree k = p − 1 part of the proof of [15, Prop. 2.5] there is a natural equivalence

r′_{C_{p}}: Φ^{C_{p}}(B^{∧p}) ⟶^{≃} B .

If B is a ring spectrum, both of these equivalences are ring spectrum maps. We abuse notation and write R also for the composite map

(B^{∧p})^{C_{p}} ⟶^{R} [ẼC_{p} ∧ B^{∧p}]^{C_{p}} ⟶^{≃} Φ^{C_{p}}(B^{∧p}) ⟶^{≃} B ,

which corresponds (at the level of 0-simplices) to the restriction map R: sd_{p} THH(B)^{C_{p}} → THH(B) of [4] and [15].

Definition 5.10. Let ϵ_{B}: B → R_{+}(B) be the natural stable map given by the composite

B ⟵^{≃} Φ^{C_{p}}(B^{∧p}) ⟵^{≃} [ẼC_{p} ∧ B^{∧p}]^{C_{p}} ⟶^{Γ̂} (B^{∧p})^{tC_{p}} = R_{+}(B) .

If B is a ring spectrum then ϵ_{B} is a ring spectrum map.

With this notation we can rewrite the homotopy cartesian square in (4.3) for X = B^{∧p} as follows:

(5.4)  \xymatrix{(B^{∧p})^{C_{p}} \ar[r]^{R} \ar[d]_{Γ} & B \ar[d]^{ϵ_{B}} \\ (B^{∧p})^{hC_{p}} \ar[r]^{R^{h}} & R_{+}(B)}

We thank M. Bökstedt for a helpful discussion on the following two results.

Lemma 5.11. The stable map ϵ_{B}: B → R_{+}(B) commutes with suspension, in the sense that ϵ_{ΣB} = ΔΣϵ_{B}.

Proof. Consider the commutative diagram:

\xymatrix{ΣB \ar[d]_{=} & Σ[ẼC_{p} ∧ B^{∧p}]^{C_{p}} \ar[l]_{≃} \ar[r]^{ΣΓ̂} \ar[d]^{Δ} & ΣR_{+}(B) \ar[d]^{Δ} \\ ΣB & [ẼC_{p} ∧ (ΣB)^{∧p}]^{C_{p}} \ar[l]_{≃} \ar[r]^{Γ̂} & R_{+}(ΣB)}

The vertical maps labeled Δ are induced by the diagonal inclusion S^{1} → S^{p}, which on the right hand side is the same map as was used in the interpretation (Proposition 5.7) of R_{+}(B) as the inverse system of suspended extended power constructions. Hence these maps are weak equivalences. □

Proposition 5.12. Let B be a bounded below spectrum of finite type over 𝔽_{p}. Then the homomorphism

(ϵ_{B})^{∗}: H_{c}^{∗}(R_{+}(B)) ⟶ H^{∗}(B)

induced on continuous cohomology by the spectrum map ϵ_{B}: B → R_{+}(B) is equal to Singer’s homomorphism

ϵ_{H^{∗}(B)} : R_{+}(H^{∗}(B)) ⟶ H^{∗}(B)

associated to the 𝒜 -module H^{∗}(B), via the identification ω : H_{c}^{∗}(R_{+}(B)) ≅ R_{+}(H^{∗}(B)) of Theorem 5.9.

Proof. By Corollary 3.5 there is a unique 𝒜 -module homomorphism g_{B}: H^{∗}(B) → H^{∗}(B) that makes the square

\xymatrix{R_{+}(H^{∗}(B)) \ar[d]_{ϵ_{H^{∗}(B)}} & H_{c}^{∗}(R_{+}(B)) \ar[d]^{(ϵ_{B})^{∗}} \ar[l]_{ω}^{≅} \\ H^{∗}(B) \ar[r]^{g_{B}} & H^{∗}(B)}

commute. We must show that g_{B} equals the identity. First consider the case B = H. The homological Tate spectral sequence

Ê_{∗,∗}^{2}(H) = Ĥ^{−∗}(C_{p}; H_{∗}(H)^{⊗p}) ⟹ H_{∗}^{c}(R_{+}(H)) is an algebra spectral sequence (Proposition 4.17), and ϵ_{H}: H → R_{+}(H) is a ring spectrum map, so the image of the unit 1 ∈ H_{0}(H) under (ϵ_{H})_{∗} is represented by the unit 1 ⊗ 1^{⊗p} in Ĥ^{0}(C_{p}; H_{0}(H)^{⊗p}). Hence (ϵ_{H})^{∗} maps the dual class represented by 1 ⊗ 1^{⊗p} in Ĥ_{0}(C_{p}; H^{0}(H)^{⊗p}) to 1 ∈ H^{0}(H). Now ω(1 ⊗ 1^{⊗p}) = Σxy^{−1} ⊗ 1 ∈ R_{+}(𝒜) and ϵ_{𝒜} (Σxy^{−1} ⊗ 1) = 1 ∈ 𝒜, so g_{H}(1) = 1. (We replace xy^{−1} by x^{−1} for p = 2.) Since g_{H} is an 𝒜 -module homomorphism, it must be equal to the identity morphism of 𝒜 = H^{∗}(H). The case B = Σ^{n}H then follows by Lemma 5.11. In the general case any element in H^{n}(B) is represented by a map f: B → Σ^{n}H, which induces an 𝒜 -module homomorphism f^{∗}: Σ^{n}𝒜 → H^{∗}(B). By naturality of the isomorphism ω, the Singer homomorphism ϵ, and the spectrum map ϵ, we get a diagram

\xymatrix{R_{+}(Σ^{n}𝒜) \ar[ddd]_{ϵ_{Σ^{n}𝒜}} \ar[dr]_{R_{+}(f^{∗})} &  &  & H_{c}^{∗}(R_{+}(Σ^{n}H)) \ar[ddd]^{(ϵ_{Σ^{n}H})^{∗}} \ar[lll]_{ω}^{≅} \ar[dl]^{R_{+}(f)^{∗}} \\  & R_{+}(H^{∗}(B)) \ar[d]_{ϵ_{H^{∗}(B)}} & H_{c}^{∗}(R_{+}(B)) \ar[d]^{(ϵ_{B})^{∗}} \ar[l]_{ω}^{≅} \\  & H^{∗}(B) \ar[r]_{g_{B}} & H^{∗}(B) \\ Σ^{n}𝒜 \ar[rrr]^{=} \ar[ur]^{f^{∗}} &  &  & H^{∗}(Σ^{n}H) \ar[ul]_{f^{∗}}}

where the left hand, upper and right hand trapezoids all commute. The inner square commutes by construction, and the outer square commutes by the case B = Σ^{n}H. Since ϵ_{Σ^{n}𝒜} is surjective, it follows that the lower trapezoid also commutes. Hence g_{B} equals the identity on the class f^{∗}(Σ^{n}1) ∈ H^{n}(B). Since n and f were arbitrary, this proves that g_{B} equals the identity on all of H^{∗}(B).

□

The following theorem generalizes the Segal conjecture for C_{p}.

Theorem 5.13. Let B be a bounded below spectrum of finite type over 𝔽_{p}. Then the natural maps

ϵ_{B}: B ⟶ R_{+}(B) = (B^{∧p})^{tC_{p}}

and

Γ: (B^{∧p})^{C_{p}} ⟶ (B^{∧p})^{hC_{p}}

are p-adic equivalences of spectra.

Proof. The map ϵ_{B} induces a map of spectral sequences

E_{2}^{∗,∗}(B) = Ext_{𝒜}^{∗,∗}(H^{∗}(B), 𝔽_{p}) ⟶ Ext_{𝒜}^{∗,∗}(H_{c}^{∗}(R_{+}(B)), 𝔽_{p}) = E_{2}^{∗,∗}(R_{+}(B))

where the first is the Adams spectral sequence of B, and the second is the inverse limit of Adams spectral sequences associated to the Tate tower {(B^{∧p})^{tC_{p}}[n]}_{n}, as in Proposition 2.2. The map converges strongly to the homomorphism

π_{∗}(ϵ_{B})_{p}: π_{∗}(B_{p}) ⟶ π_{∗}(R_{+}(B)_{p}) .

By Propositions 5.12 and 3.3 and Theorem 3.4, the map of E_{2}-terms is an isomorphism, hence so is the map of E_{∞}-terms and of the abutments. In other words, ϵ_{B} is a p-adic equivalence. The corresponding assertion for Γ follows immediately, since diagram (5.4) is homotopy cartesian. □

5.4. The Tate spectral sequence for the topological Singer construction. We conclude by relating the homological Tate spectral sequence for B^{∧p} to the Tate filtration on the homological Singer construction for H_{∗}(B), and likewise in cohomology.

Proposition 5.14. Let B be a bounded below spectrum of finite type over 𝔽_{p}. The homological Tate spectral sequence

Ê_{∗,∗}^{2} = Ĥ^{−∗}(C_{p}; H_{∗}(B)^{⊗p}) ⟹ H_{∗}^{c}((B^{∧p})^{tC_{p}})

converging to H_{∗}^{c}(R_{+}(B)) ≅ R_{+}(H_{∗}(B)) collapses at the Ê^{2}-term. Hence the Ê^{2} = Ê^{∞}-term is given by

Ê_{∗,∗}^{∞} = P (u, u^{−1}) ⊗ 𝔽_{2}{α^{⊗2}}

for p = 2, and by

Ê_{∗,∗}^{∞} = E(u) ⊗ P (t, t^{−1}) ⊗ 𝔽_{p}{α^{⊗p}}

for p odd. In each case α runs through an 𝔽_{p}-basis for H_{∗}(B). For p = 2 and any r ∈ ℤ, α ∈ H_{q}(B), the element u^{r} ⊗ α ∈ R_{+}(H_{∗}(B)) is represented in the Tate spectral sequence by

u^{r+q} ⊗ α^{⊗2} ∈ Ê_{−r−q,2q}^{∞} .

For p odd and any i ∈ {0, 1}, r ∈ ℤ and α ∈ H_{q}(B), the element u^{i}t^{r} ⊗ α ∈ R_{+}(H_{∗}(B)) is represented in the Tate spectral sequence by

\frac{(−1)^{q}}{ν(q)} · u^{i}t^{r+mq} ⊗ α^{⊗p} ∈ Ê_{−i−2r−(p−1)q,pq}^{∞} ,

where m = (p − 1)/2 and ν(2j + ϵ) = (−1)^{j}(m!)^{ϵ} for ϵ ∈ {0, 1}.

Proof. Consider first the case B = S^{q}. Then the result is trivial for dimensional reasons. The Ê^{2}-term is concentrated in bidegrees (∗, pq), so there is no room for differentials, and there are no extension problems to be solved. The formula for the homological Tate spectral sequence representative for u^{i}t^{r} ⊗α will follow by dualization from the cohomological case, given below. Let H be a model for the mod p Eilenberg–Mac Lane spectrum as a commutative symmetric ring spectrum, and form the spectrum H^{∧p} in C_{p}𝒮 𝒰 as in Definition 5.3. The iterated multiplication on H then induces a naively C_{p}- equivariant map H^{∧p} → H. Let f: S^{q} → H ∧ B represent a class α ∈ H_{q}(B), and consider the naively C_{p}-equivariant composite map

f^{p}: H ∧S^{pq} ⟶ H ∧(H ∧B)^{∧p} ≃ H ∧H^{∧p}∧B^{∧p} ⟶ H ∧H ∧B^{∧p} ⟶ H ∧B^{∧p} .

On homotopy groups it induces the homomorphism H_{∗}(S^{q})^{⊗p} → H_{∗}(B)^{⊗p} that takes ι_{q}^{⊗p} to α^{⊗p}, where ι_{q} = Σ^{q}1 is the fundamental class in H_{q}(S^{q}). By applying the Tate construction to this map, we get a map of spectra

(f^{p})^{tC_{p}}: (H ∧ S^{pq})^{tC_{p}} ⟶ (H ∧ B^{∧p})^{tC_{p}}

and an associated map of homotopical Tate spectral sequences, converging strongly to a homomorphism π_{∗}(H ∧ S^{pq})^{tC_{p}} → π_{∗}(H ∧ B^{∧p})^{tC_{p}}. We can rewrite the latter as an 𝔽_{p}-linear map

(5.5)  H_{∗}^{c}(R_{+}(S^{q})) ⟶ H_{∗}^{c}(R_{+}(B))

by Proposition 4.16. For p odd, this map is given at the level of Ê^{2}-terms as sending u^{i}t^{r} ⊗ ι_{q}^{⊗p} to u^{i}t^{r} ⊗ α^{⊗p}, and similarly for p = 2. The statement of the proposition then follows by naturality. □

Note that the 𝔽_{p}-linear map (5.5) is not a homomorphism of 𝒜_{∗}-comodules in general, because f^{p} is not of the form 1_{H} ∧ (−).

Proposition 5.15. Let B be a bounded below spectrum of finite type over 𝔽_{p}. The cohomological Tate spectral sequence

Ê_{2}^{∗,∗} = Ĥ_{−∗}(C_{p}; H^{∗}(B)^{⊗p}) ⟹ H_{c}^{∗}((B^{∧p})^{tC_{p}})

converging to H_{c}^{∗}(R_{+}(B)) ≅ R_{+}(H^{∗}(B)) collapses at the Ê_{2}-term, so that

Ê_{∞}^{∗,∗} = ΣP (x, x^{−1}) ⊗ 𝔽_{2}{a^{⊗2}}

for p = 2, and

Ê_{∞}^{∗,∗} = ΣE(x) ⊗ P (y, y^{−1}) ⊗ 𝔽_{p}{a^{⊗p}}

for p odd. In each case a runs through an 𝔽_{p}-basis for H^{∗}(B). For p = 2 and any r ∈ ℤ, a ∈ H^{q}(B), the element Σx^{r} ⊗ α ∈ R_{+}(H^{∗}(B)) is represented in the Tate spectral sequence by

Σx^{r−q} ⊗ a^{⊗2} ∈ Ê_{∞}^{1+r−q,2q} .

For p odd and any i ∈ {0, 1}, r ∈ ℤ and a ∈ H^{q}(B), the element Σx^{i}y^{r} ⊗ a ∈ R_{+}(H^{∗}(B)) is represented in the Tate spectral sequence by

(−1)^{q}ν(q) · Σx^{i}y^{r−mq} ⊗ a^{⊗p} ∈ Ê_{∞}^{1+i+2r−(p−1)q,pq} .

Proof. This follows by dualization from the homological case. In the special case B = S^{q}, Σx^{i}y^{r} ⊗ a^{⊗p} ∈ H_{c}^{∗}(R_{+}(B)) ≅ Ĥ_{−∗}(C_{p}; H^{∗}(S^{q})^{⊗p}) is represented by (−1)^{q}ν(q)^{−1} · Σx^{i}y^{r+mq} ⊗ a ∈ R_{+}(H^{∗}(S^{q})), by the explicit isomorphism given in the proof of Theorem 5.2. The formula for the cohomological Tate spectral sequence representative follows. □

Corollary 5.16. The Tate filtration

{F^{n}R_{+}(H^{∗}(B))}_{n}

of the Singer construction R_{+}(H^{∗}(B)) corresponds, under the isomorphism R_{+}(H^{∗}(B)) ≅ H_{c}^{∗}(R_{+}(B)), to the Boardman filtration

{F^{n}H_{c}^{∗}(R_{+}(B))}_{n}

of H_{c}^{∗}(R_{+}(B)).

Proof. For each integer n, the Boardman filtration F^{n}H_{c}^{∗}(R_{+}(B)) equals the image of H^{∗}((B^{∧p})^{tC_{p}}[n]) in H^{∗}((B^{∧p})^{tC_{p}}), which is the part of H^{∗}((B^{∧p})^{tC_{p}}) represented in filtrations ≥ n at the Ê_{∞}-term. This corresponds to the part of the Singer construction H^{∗}(R_{+}(B)) spanned by the monomials Σx^{r} ⊗ a with 1+r−q ≥ n for p = 2, and by the monomials Σx^{i}y^{r}⊗a with 1+i+2r−(p−1)q ≥ n for p odd, which precisely equals the n-th term F^{n}R_{+}(H^{∗}(B)) of the Tate filtration, as defined in §3.2. □

References

[1] J. F. Adams, J. H. Gunawardena, and H. Miller, The Segal conjecture for elementary abelian p-groups, Topology 24 (1985), no. 4, 435–460.

[2] Ch. Ausoni and J. Rognes, Algebraic K-theory of topological K-theory, Acta Math. 188 (2002), no. 1, 1–39.

[3] J. M. Boardman, Conditionally convergent spectral sequences, Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 49–84.

[4] M. Bökstedt, W. C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), no. 3, 465–539.

[5] M. Bökstedt and I. Madsen, Topological cyclic homology of the integers, Astérisque 226 (1994), 7–8, 57–143. K-theory (Strasbourg, 1992).

[6] M. Bökstedt, Topological Hochschild homology (ca. 1986). Bielefeld preprint.

[7] M. Bökstedt, R. R. Bruner, S. Lunøe-Nielsen, and J. Rognes, On cyclic fixed points of spectra (2007). arXiv:0712.3476 preprint.

[8] M. Brun, Topological Hochschild homology of Z/p^{n}, J. Pure Appl. Algebra 148 (2000), no. 1, 29–76.

[9] R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, H_{∞} ring spectra and their applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag, Berlin, 1986.

[10] H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956.

[11] J. Caruso, J. P. May, and S. B. Priddy, The Segal conjecture for elementary abelian p-groups. II. p-adic completion in equivariant cohomology, Topology 26 (1987), no. 4, 413–433.

[12] J. P. C. Greenlees, Representing Tate cohomology of G-spaces, Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 3, 435–443.

[13] , Tate cohomology in commutative algebra, J. Pure Appl. Algebra 94 (1994), no. 1, 59–83.

[14] J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178.

[15] L. Hesselholt and I. Madsen, On the K-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), no. 1, 29–101.

[16] , On the K-theory of local fields, Ann. of Math. (2) 158 (2003), no. 1, 1–113.

[17] M. Hovey, B. Shipley, and J. Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208.

[18] L. G. Lewis Jr., J. P. May, and M. Steinberger, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure.

[19] H. H. Li and W. M. Singer, Resolutions of modules over the Steenrod algebra and the classical theory of invariants, Math. Z. 181 (1982), no. 2, 269–286.

[20] W. H. Lin, D. M. Davis, M. E. Mahowald, and J. F. Adams, Calculation of Lin’s Ext groups, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 3, 459–469.

[21] S. Lunøe-Nielsen, The Segal conjecture for topological Hochschild homology of commutative S-algebras, PhD dissertation, University of Oslo, Norway, 2005.

[22] S. Lunøe-Nielsen and J. Rognes, The Segal conjecture for topological Hochschild homology of complex cobordism, Journal of Topology 4 (2011), no. 3, 591-622.

[23] J. Rognes, Topological cyclic homology of the integers at two, J. Pure Appl. Algebra 134 (1999), no. 3, 219–286.

[24] S. Schwede, An untitled book project about symmetric spectra, 2007. http://www.math.uni-bonn.de/people/schwede/SymSpec.pdf (in preparation).

[25] W. M. Singer, On the localization of modules over the Steenrod algebra, J. Pure Appl. Algebra 16 (1980), no. 1, 75–84.

[26] , A new chain complex for the homology of the Steenrod algebra, Math. Proc. Cambridge Philos. Soc. 90 (1981), no. 2, 279–292.

[27] C. Wilkerson, Classifying spaces, Steenrod operations and algebraic closure, Topology 16 (1977), no. 3, 227–237.

Sverre Lunøe–Nielsen

Dept. of Mathematics

University of Oslo

Norway

sverreln@math.uio.no

http://folk.uio.no/sverreln John Rognes Dept. of Mathematics University of Oslo Norway rognes@math.uio.no http://folk.uio.no/rognes 910

Documenta Mathematica 17 (2012)