THE SEGAL CONJECTURE FOR TOPOLOGICAL HOCHSCHILD HOMOLOGY OF COMPLEX COBORDISM

SVERRE LUNØE–NIELSEN AND JOHN ROGNES

Abstract. We study the C_{p}-equivariant Tate construction on the topological Hochschild homology THH(B) of a symmetric ring spectrum B by relating it to a topological version R_{+}(B) of the Singer construction, extended by a natural circle action. This enables us to prove that the fixed and homotopy fixed point spectra of THH(B) are p-adically equivalent for B = MU and BP. This generalizes the classical C_{p}-equivariant Segal conjecture, which corresponds to the case B = S.

Contents

1. Introduction 1 2. Strategy of proof 2 3. Equivariant approximations 4 4. Tate representatives 8 5. Cyclotomic structure 13 6. The comparison map 15 7. The Segal conjecture 19 References 23

1. Introduction

Let p be a prime, let 𝕋 be the circle group, and let C_{p} ⊂ 𝕋 be the cyclic subgroup of order p. Let B be any symmetric ring spectrum, in the sense of [19]. Its topological Hochschild homology THH(B), as constructed in [18], is then a genuinely 𝕋-equivariant spectrum in the sense of [20]. Let E𝕋 be a free, contractible 𝕋-CW complex and let c: E𝕋_{+} → S^{0} be the usual collapse map. Forming the induced map THH(B) = F (S^{0}, THH(B)) → F (E𝕋_{+}, THH(B)) and passing to C_{p}-fixed points, we obtain the canonical map Γ: THH(B)^{C_{p}} → THH(B)^{hC_{p}} comparing fixed and homotopy fixed points.

When B = S is the sphere spectrum, THH(S) and the 𝕋-equivariant sphere spectrum are C_{p}- equivariantly equivalent, so the Segal conjecture for C_{p} is the assertion that Γ: THH(S)^{C_{p}} → THH(S)^{hC_{p}} is a p-adic equivalence, which was proven in [21] and [2]. Our main theorem generalizes this result to the cases when B = MU is the complex cobordism spectrum or B = BP is the p-local Brown–Peterson spectrum.

Theorem 1.1. The maps

Γ: THH(MU)^{C_{p}} ⟶ THH(MU)^{hC_{p}}

and

Γ: THH(BP)^{C_{p}} ⟶ THH(BP)^{hC_{p}}

are p-adic equivalences.

The corresponding assertion is not true as stated for general symmetric ring spectra B, but there are special cases for which it is approximately true. For example, when B = H𝔽_{p}, Hℤ (the Eilenberg– Mac Lane spectra) or ℓ (the Adams summand of connective topological K-theory ku) the map Γ induces an isomorphism of homotopy groups in sufficiently high degrees, with suitable coefficients, as proved in [18], [11] and [4], respectively. In particular, in those cases the homotopy fixed point spectrum is not connective, even if the fixed point spectrum is. We believe that the explanation for this phenomenon is related to the conjectured “red-shift” property of topological cyclic homology and algebraic K-theory [5], since in the examples mentioned above the difference between the fixed and homotopy fixed points stems from the first chromatic truncation present in B, while S, MU and BP are not truncated at any finite chromatic complexity. The p-adic equivalences discussed above are thus rather special properties of the symmetric ring spectra S, MU and BP. This is reflected in our proof of Theorem 1.1, which depends on calculational facts particular to these cases.

Date: October 20th 2010.

Let C_{p^{n}} ⊂ 𝕋 be the cyclic subgroup of order p^{n}. The original Segal conjecture for C_{p^{n}} is known to follow from the one for C_{p}, without further explicit computations [26], much like the Segal conjecture for general p-groups follows from the elementary abelian case [15]. Similarly, combining [12, 1.8] with Theorem 1.1 implies the following.

Corollary 1.2. The maps

Γ_{n}: THH(MU)^{C_{p^{n}}} → THH(MU)^{hC_{p^{n}}}

and

Γ_{n}: THH(BP)^{C_{p^{n}}} → THH(BP)^{hC_{p^{n}}}

are p-adic equivalences, for all n ≥ 1.

2. Strategy of proof

Let Ẽ𝕋 be the mapping cone of the collapse map c: E𝕋_{+} → S^{0}. There is a homotopy cartesian square of 𝕋-equivariant spectra

\xymatrix{THH(B) \ar[r] \ar[d] & Ẽ𝕋 ∧ THH(B) \ar[d] \\ F (E𝕋_{+}, THH(B)) \ar[r] & Ẽ𝕋 ∧ F (E𝕋_{+}, THH(B)) ,}

see [17]. Passing to C_{p}-fixed points, we get a homotopy cartesian square of 𝕋/C_{p}-equivariant spectra

(2.1)  \xymatrix{THH(B)^{C_{p}} \ar[r]^{R} \ar[d]_{Γ} & [Ẽ𝕋 ∧ THH(B)]^{C_{p}} \ar[d]^{Γ̂} \\ THH(B)^{hC_{p}} \ar[r]^{R^{h}} & THH(B)^{tC_{p}} .}

Here THH(B)^{tC_{p}} denotes the C_{p}-Tate construction on THH(B). In order to prove Theorem 1.1 we use the homotopy cartesian square (2.1) to translate the problem into a question about the map Γ̂: [Ẽ𝕋 ∧ THH(B)]^{C_{p}} → THH(B)^{tC_{p}}.

Suppose hereafter that B is connective, meaning that π_{∗}(B) = 0 for ∗ < 0. By the cyclotomic structure of THH(B), see [18], there is a natural equivalence [Ẽ𝕋 ∧ THH(B)]^{C_{p}} ⟶^{≃} THH(B), and we are faced with the problem of showing that the composite stable map

γ: THH(B) ⟶ THH(B)^{tC_{p}}

is a p-adic equivalence for B = MU and BP. We do this using the methods of [16] and [22], by realizing the Tate construction THH(B)^{tC_{p}} as the homotopy limit

THH(B)^{tC_{p}} ≃ holim_{n} THH(B)^{tC_{p}}[n]

of a Tate tower {THH(B)^{tC_{p}}[n]}_{n} (see (4.5)) of bounded below spectra of finite type mod p, and comparing the Adams spectral sequence

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

to the inverse limit of Adams spectral sequences

(2.3)  E_{2}^{s,t} = Ext_{𝒜}^{s,t}(H_{c}^{∗}(THH(B)^{tC_{p}}), 𝔽_{p}) ⟹ π_{t−s}(THH(B)^{tC_{p}})_{p}

associated to that tower. We always use 𝔽_{p}-coefficients for homology and cohomology, 𝒜 denotes the Steenrod algebra, and

H_{c}^{∗}(THH(B)^{tC_{p}}) = colim_{n} H^{∗}(THH(B)^{tC_{p}}[n])

is the continuous cohomology of the Tate tower. The key computational input is now that, in the cases B = S, MU and BP, the induced 𝒜 -module homomorphism

γ^{∗}: H_{c}^{∗}(THH(B)^{tC_{p}}) ⟶ H^{∗}(THH(B))

is an Ext-equivalence, in the sense that it induces an isomorphism between the E_{2}-terms of the spectral sequences (2.2) and (2.3). Both spectral sequences are strongly convergent for connective B of finite type mod p, so this implies that γ, Γ̂ and Γ are p-adic equivalences. More precisely, we can prove the following theorem. Recall from [2], [22] that the Singer construction on an 𝒜 -module M is an 𝒜 -module R_{+}(M), which comes equipped with a natural Ext-equivalence ϵ: R_{+}(M) → M.

Theorem 2.1. When B = S, MU or BP there is an 𝒜 -module isomorphism

Φ_{B}^{∗}: H_{c}^{∗}(THH(B)^{tC_{p}}) ⟶^{≅} R_{+}(H^{∗}(THH(B)))

such that γ^{∗} = ϵ ∘ Φ_{B}^{∗}.

As summarized above, this implies Theorem 1.1. When B = S, the 0-simplex inclusion η_{p}: S^{∧p} → sd_{p} THH(S) ≅ THH(S) (see Definition 3.6) is a C_{p}-equivariant equivalence, so the theorem is already covered by [22, 5.13], or the original proofs of the Segal conjecture. On the other hand, the cases B = MU and BP involve new ideas. For one thing, we will make use of the ring spectrum structure on THH(B)^{tC_{p}}, which means that it is more convenient to work with the continuous homology

H_{∗}^{c}(THH(B)^{tC_{p}}) = lim_{n} H_{∗}(THH(B)^{tC_{p}}[n])

of the Tate tower, and its induced algebra structure, than with the continuous cohomology. The continuous homology must be viewed as a topological graded 𝔽_{p}-vector space, with the linear topology generated by the neighborhood basis of the origin given by the kernels

F_{n}H_{∗}^{c}(THH(B)^{tC_{p}}) = ker(H_{∗}^{c}(THH(B)^{tC_{p}}) → H_{∗}(THH(B)^{tC_{p}}[n])) .

We may also view H_{∗}^{c}(THH(B)^{tC_{p}}) as the limit of its quotients

F^{n}H_{∗}^{c}(THH(B)^{tC_{p}}) = im(H_{∗}^{c}(THH(B)^{tC_{p}}) → H_{∗}(THH(B)^{tC_{p}}[n])) .

Let 𝒜_{∗} denote the dual of the Steenrod algebra. The 𝒜_{∗}-comodule structure on H_{∗}(THH(B)^{tC_{p}}[n]) for each integer n makes the limit H_{∗}^{c}(THH(B)^{tC_{p}}) a complete 𝒜_{∗}-comodule in the sense of [22, 2.7]. Similarly, for each 𝒜_{∗}-comodule M_{∗} that is bounded below and of finite type, the homological Singer construction R_{+}(M_{∗}) from [22, 3.7] is a complete 𝒜_{∗}-comodule, equipped with a natural continuous homomorphism ϵ_{∗}: M_{∗} → R_{+}(M_{∗}) of complete 𝒜_{∗}-comodules. The linear topologies on H_{∗}^{c}(THH(B)^{tC_{p}}) and R_{+}(M_{∗}) allow us to recover the continuous cohomology and the (cohomological) Singer construction, respectively, as their continuous 𝔽_{p}-linear duals

H_{c}^{∗}(THH(B)^{tC_{p}}) ≅ Hom^{c}(H_{∗}^{c}(THH(B)^{tC_{p}}), 𝔽_{p})
R_{+}(M) ≅ Hom^{c}(R_{+}(M_{∗}), 𝔽_{p})

where M = H^{∗}(THH(B)) and M_{∗} = H_{∗}(THH(B)), see [22, 2.6, 2.9]. The corresponding assertions for the full linear duals would be false, since neither H_{c}^{∗}(THH(B)^{tC_{p}}) nor R_{+}(M) will be of finite type. The 𝒜 -module homomorphism γ^{∗} is the continuous dual of the complete 𝒜_{∗}-comodule homomorphism

γ_{∗}: H_{∗}(THH(B)) ⟶ H_{∗}^{c}(THH(B)^{tC_{p}}) ,

where H_{∗}(THH(B)) has the discrete topology, and similarly ϵ is the continuous dual of ϵ_{∗}. Theorem 2.1 now follows immediately from its homological analogue, stated below, by letting Φ_{B}^{∗} be the continuous dual of Φ_{B}.

Theorem 2.2. When B = S, MU or BP there is a continuous isomorphism of complete 𝒜_{∗}-comodules

Φ_{B}: R_{+}(H_{∗}(THH(B))) ⟶^{≅} H_{∗}^{c}(THH(B)^{tC_{p}}) ,

with continuous inverse, such that γ_{∗} = Φ_{B} ∘ ϵ_{∗}.

This is then the main technical result that we will need to prove. The argument relies on computations, and is not of a formal nature. To effect these computations, we establish in Theorem 5.3 that for connective symmetric ring spectra B there is a natural commutative square

(2.4)  \xymatrix{𝕋 ⋉ B \ar[r]^{ω} \ar[d]_{ρ⋉ϵ_{B}} & THH(B) \ar[d]^{γ} \\ 𝕋/C_{p} ⋉ R_{+}(B) \ar[r]^{ω^{t}} & THH(B)^{tC_{p}}}

in the stable homotopy category, where γ is as above, ρ: 𝕋 → 𝕋/C_{p} is the p-th root isomorphism of groups, R_{+}(B) = (B^{∧p})^{tC_{p}} is the topological Singer construction from [22, 5.8] realizing the homological Singer construction in continuous homology, and ϵ_{B}: B → R_{+}(B) is a natural map inducing the continuous homomorphism ϵ_{∗}: H_{∗}(B) → R_{+}(H_{∗}(B)) ≅ H_{∗}^{c}(R_{+}(B)) of complete 𝒜_{∗}-comodules. The map ω extends the inclusion of 0-simplices η: B → THH(B) using the 𝕋-action on the target, and the construction of ω^{t} is similar, but more elaborate.

To compute the continuous homology of THH(B)^{tC_{p}}, we use the homological Tate spectral sequence

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

recalled in Proposition 4.1. In the cases we are interested in, H_{∗}(THH(B)) is generated as an algebra by the image of ω_{∗}, so we can use the commutative square (2.4) above, known formulas [22, §3.2.1] for ϵ_{∗}, and new explicit formulas established in Theorem 4.10 for ω_{∗}^{t}, in order to understand the completed 𝒜_{∗}- comodule homomorphism γ_{∗}. These calculations are carried out for B = MU and BP in Propositions 6.3 and 6.8, and in Theorems 6.4 and 6.9. The proof of Theorem 2.2 is completed in Section 7, the main steps being Propositions 7.2 and 7.3.

3. Equivariant approximations

We assume some familiarity with equivariant spectra [20], symmetric ring spectra [19], edgewise subdivision of cyclic objects [10, §1], and the topological Hochschild homology spectrum [18, §2.4].

Recall Connes’ extension Λ of the category Δ. A cyclic space X_{•} is a contravariant functor from Λ to spaces, and its geometric realization |X_{•}| has a natural 𝕋-action. The 𝕋-action on a 0-simplex x in X_{0} traces out the closed loop in |X_{•}| given by the 1-simplex t_{1}s_{0}(x) in X_{1}.

Let B be a symmetric ring 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.

Definition 3.1. For each k ≥ 0 and each finite-dimensional 𝕋-representation V let

thh(B; S^{V})_{k} = hocolim_{(n_{0},...,n_{k})∈I^{k+1}} Map(S^{n_{0}} ∧ ⋯ ∧ S^{n_{k}}, B_{n_{0}} ∧ ⋯ ∧ B_{n_{k}} ∧ S^{V}) .

These spaces combine to a cyclic space thh(B; S^{V})_{•}, with Hochschild-style structure maps d_{i}, s_{j} and t_{k}. Its geometric realization thh(B; S^{V}) = | thh(B; S^{V})_{•}| has a natural 𝕋-action, given as the diagonal of the 𝕋-action coming from the cyclic structure and the 𝕋-action on S^{V}. These 𝕋-spaces assemble to a 𝕋-equivariant prespectrum thh(B), with V -th space thh(B)(V) = thh(B; S^{V}). We let

THH(B) = L(thh(B)^{τ})

be the genuine 𝕋-spectrum obtained by spectrification from a functorial good thickening of this prespectrum [18, §A.1].

Definition 3.2. Let B_{pre}^{∧1} be the prespectrum with V -th space

B_{pre}^{∧1}(V) = thh(B; S^{V})_{0} = hocolim_{n∈I} Map(S^{n}, B_{n} ∧ S^{V}) ,

and let B^{∧1} = L((B_{pre}^{∧1})^{τ}). The inclusion of 0-simplices defines a natural map

η: B^{∧1} → THH(B)

of spectra. Using the 𝕋-action on the target, we can uniquely extend η to a map

ω: 𝕋 ⋉ B^{∧1} → THH(B)

of 𝕋-equivariant spectra, as in [20, II.4.1].

We suppose hereafter that B is connective. Without loss of generality we may also assume that B is flat and convergent, as defined in [22, §5.2]. Then the natural maps

B_{pre}^{∧1}(V) ⟵^{≃} B_{pre}^{∧1}(V)^{τ} ⟶^{≃} B^{∧1}(V)

and

thh(B)(V) ⟵^{≃} thh(B)(V)^{τ} ⟶^{≃} THH(B)(V)

are C-equivariant equivalences for each finite subgroup C ⊂ 𝕋, by [18, Prop. 2.4]. Furthermore, there is a natural chain of weak equivalences relating B^{∧1} and the Lewis–May spectrum associated to B, see [22, 5.5]. Hence we shall simply write B for B^{∧1}, especially in the context of the maps η: B → THH(B) and ω: 𝕋 ⋉ B → THH(B).

For any simplicial space X_{•}, its p-fold edgewise subdivision sd_{p} X_{•} is obtained by precomposing the contravariant functor X_{•} with the functor

sd_{p}: Δ → Δ

that takes the object [k] = {0 < 1 < ⋯ < k} to its p-fold concatenation

sd_{p}([k]) = [k] ⊔ ⋯ ⊔ [k] = [p(k+1)−1] ,

for each k ≥ 0, and likewise for morphisms: sd_{p}(f) = f ⊔ ⋯ ⊔ f. Hence (sd_{p} X)_{k} = X_{p(k+1)−1}. There is a natural homeomorphism [10, 1.1]

D: | sd_{p} X_{•}| ⟶^{≅} |X_{•}| ,

induced by the diagonal embeddings Δ^{k} → Δ^{p(k+1)−1} given in barycentric coordinates as

u ↦ (u/p, …, u/p)

for u = (u_{0}, …, u_{k}) ∈ Δ^{k}, with ∑_{i} u_{i} = 1, u_{i} ≥ 0. In more detail, this is the map

D: ∐_{k≥0}(sd_{p} X)_{k} × Δ^{k}/∼ ⟶ ∐_{ℓ≥0} X_{ℓ} × Δ^{ℓ}/∼

that takes the k-th summand to the ℓ-th one, with ℓ = p(k + 1) − 1, via the identity (sd_{p} X)_{k} = X_{ℓ} and the given embedding Δ^{k} → Δ^{ℓ}.

Definition 3.3. There is a natural simplicial map e_{•}: sd_{p} X_{•} → X_{•}, induced by the natural transformation id → sd_{p} of functors Δ → Δ, with components [k] → sd_{p}([k]) given by inclusion on the p-th (= last) copy of [k], for k ≥ 0. This is the order-preserving function that omits the (p − 1)(k + 1) first elements in the target, so in simplicial degree k,

(3.1)  e_{k} = d_{0}^{(p−1)(k+1)} : (sd_{p} X)_{k} = X_{p(k+1)−1} → X_{k}

is equal to the (p − 1)(k + 1)-fold iterate of the 0-th face map. After geometric realization,

e = |e_{•}|: | sd_{p} X_{•}| → |X_{•}|

is induced by the corresponding iterated face maps δ_{0}^{(p−1)(k+1)} : Δ^{k} → Δ^{p(k+1)−1}, given in barycentric coordinates as

u ↦ (0, …, 0, u)

for u ∈ Δ^{k}, as above.

Lemma 3.4. There is a natural homotopy D ≃ e = |e_{•}| of maps | sd_{p} X_{•}| → |X_{•}|.

Proof. The homotopy is given by the convex linear combination

(t, u) ↦ (1 − t)(u/p, …, u/p) + t(0, …, 0, u)

for t ∈ [0, 1], u ∈ Δ^{k}, of the two maps Δ^{k} → Δ^{p(k+1)−1}. □

Recall the “p-fold cover” Λ_{p} → Λ [10, 1.5]. A Λ_{p}-space is a contravariant functor from Λ_{p} to spaces. When X_{•} is a cyclic space, with cyclic operators t_{k}: X_{k} → X_{k} satisfying t_{k}^{k+1} = id for k ≥ 0, the p-fold edgewise subdivision sd_{p} X_{•} is a Λ_{p}-space, with operators t_{k}: (sd_{p} X)_{k} → (sd_{p} X)_{k} satisfying t_{k}^{p(k+1)} = id, defined to be equal to the operators t_{p(k+1)−1} : X_{p(k+1)−1} → X_{p(k+1)−1}.

Lemma 3.5. Let X_{•} be a cyclic space. There is a natural 𝕋-action on | sd_{p} X_{•}| such that D: | sd_{p} X_{•}| → |X_{•}| is a 𝕋-equivariant homeomorphism. The action of the subgroup C_{p} ⊂ 𝕋 is simplicial, in the sense that it comes from a C_{p}-action on sd_{p} X_{•}, such that a generator of C_{p} acts by

t_{k}^{k+1} : (sd_{p} X)_{k} → (sd_{p} X)_{k}

for k ≥ 0. In particular, the inclusion of 0-simplices (sd_{p} X)_{0} ⊂ | sd_{p} X_{•}| is C_{p}-equivariant.

The 𝕋-action on a 0-simplex y in (sd_{p} X)_{0} = X_{p−1} traces out the closed loop in | sd_{p} X_{•}| given by the chain of p 1-simplices

t_{1}s_{0}(y) , t_{1}^{3}s_{0}(y) , …, t_{1}^{2p−1}s_{0}(y)

in (sd_{p} X)_{1} = X_{2p−1}. These meet at the p 0-simplices

y , t_{1}^{2}(y) , …, t_{1}^{2p−2}(y)

that constitute the C_{p}-orbit of y.

Proof. See [10, 1.6] and the surrounding discussion. □

Definition 3.6. Let sd_{p} thh(B) = | sd_{p} thh(B)_{•}| be the 𝕋-equivariant prespectrum with V -th space sd_{p} thh(B)(V) = | sd_{p} thh(B; S^{V})_{•}|, and let sd_{p} THH(B) = L(sd_{p} thh(B)^{τ}) be the spectrification of its good thickening, as in Definition 3.1. In simplicial degree 0,

sd_{p} thh(B; S^{V})_{0} = thh(B; S^{V})_{p−1} = B_{pre}^{∧p}(V)

in the notation of [22, 5.3]. Hence the inclusion of 0-simplices defines a natural map

η_{p}: B^{∧p} → sd_{p} THH(B)

of C_{p}-equivariant spectra, where B^{∧p} = L((B_{pre}^{∧p})^{τ}). Using the full 𝕋-action on the target, we can uniquely extend η_{p} to a map

ω_{p}: 𝕋 ⋉_{C_{p}} B^{∧p} → sd_{p} THH(B)

of 𝕋-equivariant spectra, again as in [20, II.4.1].

Combining these constructions, we have a 𝕋-equivariant homeomorphism

D: sd_{p} THH(B) ⟶^{≅} THH(B) .

Lemma 3.7. The composite

D ∘ η_{p}: B^{∧p} → THH(B)

is homotopic to the composite

e ∘ η_{p} = η ∘ d_{0}^{p−1} : B^{∧p} → THH(B)

where d_{0}^{p−1} : B^{∧p} → B^{∧1} ≃ B is the p-fold multiplication map.

Proof. The homotopy is that of Lemma 3.4. The identity e ∘ η_{p} = η ∘ d_{0}^{p−1} amounts to the formula (3.1) for e_{k}, in the special case k = 0. □

To analyze the extension ω_{p} of η_{p}, we pass to homology.

Definition 3.8. The circle 𝕋, with the subgroup action of C_{p}, is a free C_{p}-CW complex. Its mod p cellular complex C_{∗}(𝕋) = 𝔽_{p}[C_{p}]{e_{0}, e_{1}} has boundary operator d(e_{1}) = (T − 1)e_{0}, where T ∈ C_{p} is the generator mapping to exp(2πi/p) ∈ 𝕋, e_{1} is the 1-cell covering the arc from 1 to T, and e_{0} is the 0-cell 1.

Let C_{∗}(B) be the cellular complex of a CW-approximation to B, and choose a quasi-isomorphism H_{∗}(B) ≃ C_{∗}(B), taking each homology class to a representing cycle in its class.

Lemma 3.9. There are quasi-isomorphisms

C_{∗}(𝕋 ⋉_{C_{p}} B^{∧p}) ≃ C_{∗}(𝕋) ⊗_{C_{p}} C_{∗}(B)^{⊗p} ≃ C_{∗}(𝕋) ⊗_{C_{p}} H_{∗}(B)^{⊗p}

and natural isomorphisms

H_{∗}(𝕋 ⋉_{C_{p}} B^{∧p}) ≅ H_{∗}(C_{∗}(𝕋) ⊗_{C_{p}} H_{∗}(B)^{⊗p})
≅ cok(T − 1){e_{0}} ⊕ ker(T − 1){e_{1}}

where

T − 1: H_{∗}(B)^{⊗p} → H_{∗}(B)^{⊗p}

is the difference between the cyclic permutation, induced by the action of T ∈ C_{p} on B^{∧p}, and the identity.

Proof. The first quasi-isomorphism follows from [20, VIII.1.6] and [20, VIII.2.9], combined with cellular approximation. We use [22, 5.5] to compare B^{∧p} with the p-th external power B^{(p)}. The second quasiisomorphism is then clear, since C_{∗}(𝕋) is C_{p}-free. This implies the first isomorphism. The second isomorphism is simply the computation of the homology of the complex

d: H_{∗}(B)^{⊗p}{e_{1}} → H_{∗}(B)^{⊗p}{e_{0}} ,

where d maps e_{1} ⊗ (α_{1} ⊗ ⋯ ⊗ α_{p}) to e_{0} ⊗ (T − 1)(α_{1} ⊗ ⋯ ⊗ α_{p}). □

Definition 3.10. For α ∈ H_{q}(B) let e_{1}⊗α^{⊗p} ∈ H_{1+pq}(𝕋⋉_{C_{p}} B^{∧p}) denote the homology class of e_{1}⊗α^{⊗p} in C_{∗}(𝕋) ⊗_{C_{p}} H_{∗}(B)^{⊗p}, and let

α^{p−1} ∧ α = (D ∘ ω_{p})_{∗}(e_{1} ⊗ α^{⊗p}) ∈ H_{1+pq}(THH(B))

denote its image under D ∘ ω_{p}: 𝕋 ⋉_{C_{p}} B^{∧p} → THH(B).

The following spectral sequence was used by Bökstedt [9] to compute THH(ℤ/p) and THH(ℤ). See [24, 3.1], [3, §4] for more background.

Lemma 3.11. In the Bökstedt spectral sequence

E_{∗,∗}^{2} = HH_{∗}(H_{∗}(B)) ⟹ H_{∗}(THH(B))

the class α^{p−1} ∧ α is represented modulo filtration 0 by the homology class of the Hochschild 1-cycle

α^{p−1} ⊗ α ∈ H_{∗}(B) ⊗ H_{∗}(B) .

Proof. By Lemma 3.4 we may compute α^{p−1} ∧ α as the image of e_{1} ⊗ α^{⊗p} under the cellular map e ∘ ω_{p}. By the second part of Lemma 3.5, and the formula (3.1) for e = |e_{•}| in simplicial degree k = 1, the cycle e_{1} ⊗ α^{⊗p} maps under e ∘ ω_{p} to the cycle

d_{0}^{2p−2}t_{1}s_{0}(α^{⊗p}) = d_{0}^{2p−2}(1 ⊗ α ⊗ ⋯ ⊗ 1 ⊗ α) = α^{p−1} ⊗ α

in the Bökstedt spectral sequence E^{1}-term. □

We can be more specific about α^{p−1} ∧ α under additional commutativity hypotheses. The homotopy cofiber sequence

B ⟶^{i} 𝕋 ⋉ B ⟶^{j} S^{1} ∧ B

is stably split by the collapse map c: 𝕋⋉B → B, and its homotopy fiber map, a section s: S^{1}∧B → 𝕋⋉B. We let σ: S^{1} ∧ B → THH(B) be the composite map

S^{1} ∧ B ⟶^{s} 𝕋 ⋉ B ⟶^{ω} THH(B) .

For α ∈ H_{q}(B), the homology class σα = σ_{∗}(α) ∈ H_{1+q}(THH(B)) is represented by the Hochschild 1-cycle 1 ⊗ α ∈ H_{∗}(B) ⊗ H_{∗}(B) in the Bökstedt spectral sequence above, see [24, 3.2].

When B is an E_{2} symmetric ring spectrum, it follows from [13] that THH(B) is an E_{1} = A_{∞} ring spectrum, hence equivalent to an (associative) S-algebra. Furthermore, the Bökstedt spectral sequence is an algebra spectral sequence, with product induced by the shuffle product on the Hochschild complex.

Lemma 3.12. Suppose that B is an E_{2} symmetric ring spectrum. Then for α ∈ H_{q}(B) we have

α^{p−1} ∧ α ≡ α^{p−1} · σα

in H_{∗}(THH(B)), modulo the image of η_{∗}: H_{∗}(B) → H_{∗}(THH(B)).

Proof. The shuffle product of the 0-cycle α^{p−1} and the 1-cycle 1 ⊗ α is the 1-cycle α^{p−1} ⊗ α. Hence the product α^{p−1} · σα and α^{p−1} ∧ α have the same representative in Hochschild filtration 1, and must be equal modulo the image of Hochschild filtration 0. □

The following corollary applies to B = S, MU and BP. (According to work in preparation [7], [8] by Basterra and Mandell, BP is an E_{4} ring spectrum.)

Corollary 3.13. Suppose that B is an E_{2} symmetric ring spectrum with H_{∗}(B) concentrated in even degrees. Then for α ∈ H_{q}(B) we have

α^{p−1} ∧ α = α^{p−1} · σα .

Proof. There is only something to prove when q is even, in which case H_{1+pq}(B) = 0 by hypothesis, so the indeterminacy is 0. □

When B is a commutative symmetric ring spectrum, it follows similarly that THH(B) is an E_{∞} ring spectrum, hence equivalent to a commutative S-algebra. The multiplication maps THH(B)_{k} ≃ B^{∧(k+1)} → B then combine to an augmentation map ϵ: THH(B) → B (not to be confused with the stable map ϵ_{B}), such that ϵη = id_{B} and ϵσ ≃ ∗. See e.g. [3, §3]. We can thus use ϵ to determine the error term in Lemma 3.12.

The commutative product on B makes the associated Lewis–May spectrum an E_{∞} ring spectrum, with structure maps

ξ_{p}: EΣ_{p} ⋉_{Σ_{p}} B^{(p)} ⟶ B

among others. Here EΣ_{p} is a free, contractible Σ_{p}-space. Up to homotopy, there is a unique map χ: 𝕋 → EΣ_{p} that is equivariant with respect to the inclusion C_{p} ⊂ Σ_{p}. Furthermore, the composite map

𝕋 ⋉_{C_{p}} B^{(p)} ⟶^{χ⋉id} EΣ_{p} ⋉_{Σ_{p}} B^{(p)} ⟶^{ξ_{p}} B

exhibits the commuting homotopy μ^{p} ≃ μ^{p}T that is preferred by the E_{∞} structure, where μ^{p}: B^{(p)} → B is the multiplication map and T: B^{(p)} → B^{(p)} is the cyclic permutation. Using [22, 5.5], we can identify the source here with the source 𝕋 ⋉_{C_{p}} B^{∧p} of ω_{p}.

Lemma 3.14. Let B be a commutative symmetric ring spectrum. The composite map

𝕋 ⋉_{C_{p}} B^{∧p} ⟶^{ω_{p}} sd_{p} THH(B) ⟶^{D} THH(B) ⟶^{ϵ} B

is homotopic to the composite map

𝕋 ⋉_{C_{p}} B^{∧p} ≃ 𝕋 ⋉_{C_{p}} B^{(p)} ⟶^{χ⋉id} EΣ_{p} ⋉_{Σ_{p}} B^{(p)} ⟶^{ξ_{p}} B .

Proof. Using Lemma 3.4 we may replace D by the simplicial map e = |e_{•}|. When restricted to 1 ∈ C_{p} ⊂ 𝕋, both maps then agree with the p-fold multiplication map d_{0}^{p−1} : B^{∧p} → B, taking α_{1} ⊗ ⋯ ⊗ α_{p} to α_{1} ⋯ α_{p} (in homology). Similarly, when restricted to T ∈ C_{p} ⊂ 𝕋, both maps agree with the cyclically permuted multiplication map d_{0}^{p−1}t_{p−1} : B^{∧p} → B, taking α_{1}⊗⋯⊗α_{p} to α_{p}α_{1} ⋯ α_{p−1}. When restricted to the 1-cell e_{1} ⊂ 𝕋, connecting 1 to T, both maps exhibit the commuting homotopy between these two maps that is preferred by the given E_{∞} structure. (This is how the E_{∞} structure enters into the definition of ϵ.) In particular the homotopies agree, up to homotopy relative to the endpoints. □

The following corollary applies to B = H𝔽_{p}, Hℤ, ku and ℓ.

Corollary 3.15. Let B be a commutative symmetric ring spectrum. For α ∈ H_{q}(B) we have

α^{p−1} ∧ α = α^{p−1} · σα + η_{∗}(Q_{1}(α))

in H_{∗}(THH(B)), where Q_{1}(α) = ξ_{p∗}(e_{1} ⊗ α^{⊗p}) is the image of α under the Dyer–Lashof operation Q_{1} : H_{q}(B) → H_{1+pq}(B).

Proof. We know that α^{p−1}∧α = (Dω_{p})_{∗}(e_{1}⊗α^{⊗p}) equals α^{p−1}·σα+η_{∗}(x) for some x ∈ H_{∗}(B). Applying ϵ_{∗}, we get that ϵ_{∗}(α^{p−1} ∧α) = (ϵDω_{p})_{∗}(e_{1} ⊗α^{⊗p}) = ξ_{p∗}(e_{1} ⊗α^{⊗p}) = Q_{1}(α) equals ϵ_{∗}(α^{p−1} ·σα+η_{∗}(x)) = α^{p−1} · 0 + x = x. □

4. Tate representatives

To prove Theorem 2.2, we use the following homological Tate spectral sequence to study the continuous homology of Tate constructions. We only need to consider the C_{p}-equivariant case.

Proposition 4.1. Let X be a C_{p}-equivariant spectrum. Assume that X is bounded below with H_{∗}(X) of finite type. Then the homological Tate spectral sequence

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

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

If, furthermore, X is a C_{p}-equivariant ring spectrum, then this is an 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. See Propositions 4.15 and 4.17 in [22]. □

When B is a connective symmetric ring spectrum, with H_{∗}(B) of finite type, then both X = B^{∧p} and X = THH(B) are C_{p}-spectra that are bounded below with H_{∗}(X) of finite type. This is clear from the Künneth formula and the Bökstedt spectral sequence in Lemma 3.11.

Remark 4.2. In the case when X = S^{∧p} ≅ THH(S) is the C_{p}-equivariant sphere spectrum, H_{∗}(X) = 𝔽_{p} is concentrated in degree 0, so

Ê_{∗,∗}^{2}(S^{∧p}) = Ĥ^{−∗}(C_{p}; 𝔽_{p})

is concentrated on the horizontal axis. Here Ĥ^{−∗}(C_{p}; 𝔽_{p}) = P (u, u^{−1}) for p = 2 and Ĥ^{−∗}(C_{p}; 𝔽_{p}) = E(u) ⊗ P (t, t^{−1}) for p odd, with u ∈ Ĥ^{1} and t ∈ Ĥ^{2}. There cannot be any differentials in this spectral sequence, for bidegree reasons, so each class u^{i}t^{r} is an infinite cycle. (This formula applies when p is odd—the reader should always replace u^{i}t^{r} by u^{i+2r} when p = 2.)

By naturality with respect to the unit maps S^{∧p} → B^{∧p} and THH(S) → THH(B) it follows that the classes u^{i}t^{r} are infinite cycles, also in the homological Tate spectral sequences for X = B^{∧p} and X = THH(B). Hence all of these spectral sequences exhibit a horizontal periodicity, with each Ê^{r}-term being determined by the part on the vertical axis via an isomorphism

Ê_{∗,∗}^{r}(X) ≅ Ĥ^{−∗}(C_{p}; 𝔽_{p}) ⊗ Ê_{0,∗}^{r}(X) .

Remark 4.3. In the case X = B^{∧p} the C_{p}-action on H_{∗}(X) = H_{∗}(B)^{⊗p} is given by cyclic permutation of the tensor factors, so

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

where α ranges over an 𝔽_{p}-basis of H_{∗}(B). Also in this case the spectral sequence collapses at the Ê^{2}-term, and converges to

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

see [22, 5.14]. Here R_{+}(B) = (B^{∧p})^{tC_{p}} is the topological Singer construction on B, and R_{+}(H_{∗}(B)) is the homological Singer construction on H_{∗}(B), discussed in Definitions 5.8 and 3.7 of [22], respectively.

The right hand isomorphism is given in Theorem 5.9 of that paper. Implicit in this isomorphism is the fact that for each u^{i}t^{r} ∈ Ĥ^{−∗}(C_{p}; 𝔽_{p}) and α ∈ H_{∗}(B) there is a preferred class in H_{∗}^{c}((B^{∧p})^{tC_{p}}) that is represented in the Tate spectral sequence by u^{i}t^{r} ⊗ α^{⊗p} in Ê_{∗,∗}^{2}(B^{∧p}). It is obtained by representing α by a map f: S^{q} → H ∧ B, and taking u^{i}t^{r} ⊗ α^{⊗p} to be in the image of an induced homomorphism

H_{∗}^{c}((f^{p})^{tC_{p}}): H_{∗}^{c}((S^{pq})^{tC_{p}}) → H_{∗}^{c}((B^{∧p})^{tC_{p}}) .

See the proof of [22, 5.14] for details. Hence u^{i}t^{r} ⊗ α^{⊗p} is a well-defined class in H_{∗}^{c}((B^{∧p})^{tC_{p}}), not just defined modulo the Tate filtration.

Remark 4.4. In the case X = THH(B) the C_{p}-action on H_{∗}(X) = H_{∗}(THH(B)) is obtained by restriction from a 𝕋-action, hence is algebraically trivial. Thus

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

In this case the spectral sequence does not generally collapse at the Ê^{2}-term. For example, the d^{2}- differential satisfies

d^{2}(u^{i}t^{r} ⊗ α) = u^{i}t^{r+1} ⊗ σα

for p odd, and similarly for p = 2, see [28, 3.3]. When B is an E_{∞} ring spectrum, so that THH(B) is equivalent to a 𝕋-equivariant commutative S-algebra, there are many cases where this spectral sequence collapses at the Ê^{3}-term, see [14, 1.2]. For example, this is the case for B = MU, as we shall prove directly in Proposition 6.3 below.

By naturality with respect to the C_{p}-equivariant map D ∘ η_{p}: B^{∧p} → THH(B), we get a map of homological Tate spectral sequences

Ê_{∗,∗}^{2}(B^{∧p}) ⟶ Ê_{∗,∗}^{2}(THH(B))

taking u^{i}t^{r} ⊗ α^{⊗p} to u^{i}t^{r} ⊗ α^{p} (recall Lemma 3.7), which converges to a complete 𝒜_{∗}-comodule homomorphism

η_{∗}^{t}: H_{∗}^{c}((B^{∧p})^{tC_{p}}) ⟶ H_{∗}^{c}(THH(B)^{tC_{p}}) .

This is useful for determining differentials and 𝒜_{∗}-comodule structure in the target, but there is an even more useful extension of this homomorphism,

ω_{∗}^{t}: H_{∗}(𝕋/C_{p}) ⊗ H_{∗}^{c}((B^{∧p})^{tC_{p}}) ⟶ H_{∗}^{c}(THH(B)^{tC_{p}}) ,

which is constructed by taking the full 𝕋-action on THH(B) into account. We now explain this construction.

Consider the diagram

(4.1)  \xymatrix{B^{∧p} \ar[r]^{i_{p}} \ar[rr]^{η_{p}} & 𝕋 ⋉_{C_{p}} B^{∧p} \ar[r]^{ω_{p}} & sd_{p} THH(B) \ar[d]^{D}_{≅} \\  &  & THH(B)}

of C_{p}-equivariant spectra, where i_{p} is induced by the inclusion C_{p} ⊂ 𝕋. Note that ω_{p} and D are 𝕋-equivariant maps. Applying the C_{p}-Tate construction

X^{tC_{p}} = [Ẽ𝕋 ∧ F (E𝕋_{+}, X)]^{C_{p}}

to (4.1) we can form a commutative diagram:

(4.2)  [figure: 𝕋/C_{p} ⋉ (B^{∧p})^{tC_{p}}; κ; ≃; ω^{t}; (B^{∧p})^{tC_{p}}; i_{p}^{tCp}; η^{t}; i′; (𝕋 ⋉_{C_{p}} B^{∧p})^{tC_{p}}; ω_{p}^{tCp}; sd_{p} THH(B)^{tC_{p}}; D^{tCp}; ≅; THH(B)^{tC_{p}}]

Definition 4.5. Let

η^{t}: (B^{∧p})^{tC_{p}} → THH(B)^{tC_{p}}

be the map η^{t} = (D ∘ η_{p})^{tC_{p}}, let

κ: 𝕋/C_{p} ⋉ (B^{∧p})^{tC_{p}} → (𝕋 ⋉_{C_{p}} B^{∧p})^{tC_{p}}

be the unique 𝕋/C_{p}-equivariant map that extends i_{p}^{tC_{p}}, and let

ω^{t}: 𝕋/C_{p} ⋉ (B^{∧p})^{tC_{p}} → THH(B)^{tC_{p}}

be the 𝕋/C_{p}-equivariant composite ω^{t} = (D ∘ ω_{p})^{tC_{p}} ∘ κ. Lemma 4.6. The map

κ: 𝕋/C_{p} ⋉ (B^{∧p})^{tC_{p}} ⟶^{≃} (𝕋 ⋉_{C_{p}} B^{∧p})^{tC_{p}}

is an equivalence. Proof. The C_{p}-equivariant cofiber sequence

(4.3)  \xymatrix{C_{p+} \ar[r]^{i_{p}} & 𝕋_{+} \ar[r]^{j_{p}} & S^{1} ∧ C_{p+}}

induces a map of cofiber sequences

\xymatrix{(B^{∧p})^{tC_{p}} \ar[r]^{i′} \ar[d]_{=} & 𝕋/C_{p} ⋉ (B^{∧p})^{tC_{p}} \ar[r] \ar[d]^{κ} & S^{1} ∧ (B^{∧p})^{tC_{p}} \ar[d]^{≃} \\ (B^{∧p})^{tC_{p}} \ar[r]^{i_{p}^{tCp}} & (𝕋 ⋉_{C_{p}} B^{∧p})^{tC_{p}} \ar[r] & (S^{1} ∧ B^{∧p})^{tC_{p}} ,}

which implies the result. □ Remark 4.7. In the intermediate case of the homological Tate spectral sequence for X = 𝕋 ⋉_{C_{p}} B^{∧p}, the homology H_{∗}(X) = H_{∗}(𝕋 ⋉_{C_{p}} B^{∧p}) was given in Lemma 3.9. The C_{p}-action extends to a 𝕋-action, hence is algebraically trivial. Thus in this case

Ê_{∗,∗}^{2}(𝕋 ⋉_{C_{p}} B^{∧p}) = Ĥ^{−∗}(C_{p}; H_{∗}(𝕋 ⋉_{C_{p}} B^{∧p})) ≅ Ĥ^{−∗}(C_{p}; 𝔽_{p}) ⊗ H_{∗}(𝕋 ⋉_{C_{p}} B^{∧p})
⟹ H_{∗}^{c}((𝕋 ⋉_{C_{p}} B^{∧p})^{tC_{p}}) .

It is not difficult to determine the d^{2}-differentials, and to prove that this spectral sequence collapses at the Ê^{3}-term, but we shall not need this result.

Recall the C_{p}-CW structure on 𝕋 from Definition 3.8, which descends to a CW structure on the orbit space (= quotient group) 𝕋/C_{p}, with mod p cellular complex C_{∗}(𝕋/C_{p}) = 𝔽_{p}{e_{0}, e_{1}} having boundary operator d(e_{1}) = 0. Hence H_{∗}(𝕋/C_{p}) = 𝔽_{p}{e_{0}, e_{1}}, with deg(e_{j}) = j for j ∈ {0, 1}. Proposition 4.8. The map κ induces the complete 𝒜_{∗}-comodule isomorphism

κ_{∗}: H_{∗}(𝕋/C_{p}) ⊗ H_{∗}^{c}((B^{∧p})^{tC_{p}}) ⟶^{≅} H_{∗}^{c}((𝕋 ⋉_{C_{p}} B^{∧p})^{tC_{p}})

given, modulo Tate filtration in the target, by

e_{j} ⊗ u^{i}t^{r} ⊗ α^{⊗p} ⟼ u^{i}t^{r} ⊗ e_{j} ⊗ α^{⊗p}

for (i, j) = (0, 0), (1, 0) or (0, 1), while

(e_{1} ⊗ ut^{r} − e_{0} ⊗ t^{r}) ⊗ α^{⊗p} ⟼ ut^{r} ⊗ e_{1} ⊗ α^{⊗p}

in the case (i, j) = (1, 1).

Remark 4.9. Note that the 𝕋/C_{p}-action has the effect of increasing by +1 the Tate filtration of the classes represented in odd filtration, since the image of e_{1} ⊗ ut^{r} ⊗ α^{⊗p} is represented by t^{r} ⊗ e_{0} ⊗ α^{⊗p}, modulo a term of lower Tate filtration.

Proof. For definiteness, we assume that our models for E𝕋 and Ẽ𝕋 are the unit sphere S(∞ℂ) and the one-point compactification S^{∞ℂ} of ∞ℂ = ℂ^{∞} with the diagonal 𝕋-action, respectively. We fix a based 𝕋-CW structure on Ẽ𝕋 with S^{nℂ} as 𝕋-equivariant (2n−1)- and 2n-skeleton, so that

S^{nℂ}/S^{(n−1)ℂ} ≅ Σ^{2n−1}𝕋_{+} .

Restricting the action to C_{p} ⊂ 𝕋, there is a based C_{p}-CW structure on Ẽ𝕋 with S^{nℂ} as C_{p}-equivariant 2n-skeleton, so that the subquotient C_{p}-CW structure on S^{nℂ}/S^{(n−1)ℂ} is the (2n−1)-th suspension of the C_{p}-CW structure on 𝕋 from Definition 3.8, based at a disjoint base point. For n ≥ 0 let Ẽ_{n} be the n-skeleton of this based C_{p}-CW structure on Ẽ𝕋, viewed as a C_{p}-CW spectrum, and recall that the Greenlees filtration

⋯ → Ẽ_{n−1} → Ẽ_{n} → ⋯ → Ẽ𝕋

of C_{p}-spectra extends this notation to all integers n, so that Ẽ_{n}/Ẽ_{n−1} ≅ Σ^{n}C_{p+} and Ẽ_{2n} = S^{nℂ} for all n ∈ ℤ. The cofiber sequence

(4.4)  Ẽ_{2n−1}/Ẽ_{2n−2} → Ẽ_{2n}/Ẽ_{2n−2} → Ẽ_{2n}/Ẽ_{2n−1}

equals the (2n−1)-th suspension of (4.3). If we only consider the even-indexed spectra, we get a coarser filtration

⋯ → Ẽ_{2n−2} → Ẽ_{2n} → ⋯ → Ẽ𝕋

of 𝕋-spectra, so that Ẽ_{2n}/Ẽ_{2n−2} ≅ Σ^{2n−1}𝕋_{+} for all n.

The homological Tate spectral sequence in Proposition 4.1 is obtained by expressing the C_{p}-Tate construction as a homotopy limit

X^{tC_{p}} ≃ holim_{n→−∞} X^{tC_{p}}[n]

of a tower of spectra, where

(4.5)  X^{tC_{p}}[n] = [Ẽ𝕋/Ẽ_{n−1} ∧ F (E𝕋_{+}, X)]^{C_{p}} .

There are homotopy (co-)fiber sequences

[Ẽ_{n}/Ẽ_{n−1} ∧ F (E𝕋_{+}, X)]^{C_{p}} → X^{tC_{p}}[n] → X^{tC_{p}}[n+1]

and equivalences

[Ẽ_{n}/Ẽ_{n−1} ∧ F (E𝕋_{+}, X)]^{C_{p}} ≃ [Ẽ_{n}/Ẽ_{n−1} ∧ X]^{C_{p}}
≃ (Ẽ_{n}/Ẽ_{n−1} ∧ X)/C_{p} ≅ Σ^{n}X

for all n, since Ẽ_{n}/Ẽ_{n−1} is C_{p}-free. The spectral sequence in question is associated to the exact couple with A_{n,∗} = H_{n+∗}(X^{tC_{p}}[n]) and

Ê_{n,∗}^{1} = H_{n+∗}((Ẽ_{n}/Ẽ_{n−1} ∧ X)/C_{p}) ≅ H_{∗}(X) .

In our case of interest, X = 𝕋 ⋉_{C_{p}} B^{∧p} is a 𝕋-equivariant spectrum, and we seek to understand the residual 𝕋/C_{p}-action on X^{tC_{p}}. The group 𝕋/C_{p} acts (only) on the even-indexed terms in the Greenlees filtration, hence (only) on the odd-indexed terms in the homotopy limit above. Restricting to this coarser tower of spectra, we can express the C_{p}-Tate construction as the homotopy limit

X^{tC_{p}} ≃ holim_{n→−∞} X^{tC_{p}}[2n−1]

of 𝕋/C_{p}-equivariant spectra. There are 𝕋/C_{p}-equivariant homotopy (co-)fiber sequences

(4.6)  [Ẽ_{2n}/Ẽ_{2n−2} ∧ F (E𝕋_{+}, X)]^{C_{p}} → X^{tC_{p}}[2n−1] → X^{tC_{p}}[2n+1]

and equivalences

[Ẽ_{2n}/Ẽ_{2n−2} ∧ F (E𝕋_{+}, X)]^{C_{p}} ≃ [Ẽ_{2n}/Ẽ_{2n−2} ∧ X]^{C_{p}}
≃ (Ẽ_{2n}/Ẽ_{2n−2} ∧ X)/C_{p} ≅ Σ^{2n−1}𝕋_{+} ∧_{C_{p}} X

for all n, since Ẽ_{2n}/Ẽ_{2n−2} is C_{p}-free.

The map κ is then realized as the homotopy limit of a tower of maps

κ[2n−1]: 𝕋/C_{p} ⋉ (B^{∧p})^{tC_{p}}[2n−1] → (𝕋 ⋉_{C_{p}} B^{∧p})^{tC_{p}}[2n−1]

induced by the C_{p}-equivariant inclusion i_{p}: B^{∧p} → 𝕋 ⋉_{C_{p}} B^{∧p} and the 𝕋/C_{p}-action on the displayed target. Passing to homotopy fibers as in (4.6), we see that κ in Tate filtrations (2n−1) and 2n is represented by the map

κ̄: 𝕋/C_{p} ⋉ (Σ^{2n−1}𝕋_{+} ∧_{C_{p}} B^{∧p}) ⟶ Σ^{2n−1}𝕋_{+} ∧_{C_{p}} (𝕋 ⋉_{C_{p}} B^{∧p})

induced by id ∧_{C_{p}}i_{p} and the 𝕋/C_{p}-action. More explicitly, this is the map

κ̄ = Σ^{2n−1}(ξ ⋉_{C_{p}} id)

induced by the C_{p}-equivariant homeomorphism

ξ: 𝕋/C_{p} × 𝕋 ⟶^{≅} 𝕋 ×_{C_{p}} 𝕋

that takes ([z], w) to [zw, z]. Here z, w ∈ 𝕋 and square brackets indicate C_{p}-orbits.

A cellular approximation to ξ induces the C_{p}-equivariant chain isomorphism

ξ_{∗}: C_{∗}(𝕋/C_{p}) ⊗ C_{∗}(𝕋) ⟶^{≅} C_{∗}(𝕋) ⊗_{C_{p}} C_{∗}(𝕋)

given by

e_{j} ⊗ e_{k} ⟼ e_{k} ⊗ e_{j}

for (j, k) = (0, 0), (0, 1) or (1, 1), while

e_{1} ⊗ e_{0} ⟼ e_{0} ⊗ e_{1} + e_{1} ⊗ T e_{0}

in the case (j, k) = (1, 0). This follows by combining the cellular model e_{0} ↦ e_{0}, e_{1} ↦ e_{0} ⊗ e_{1} + e_{1} ⊗ T e_{0} for the diagonal map z ↦ (z, z) with the cellular model e_{j} ⊗ e_{k} ↦ e_{j+k} for j + k ≤ 1, e_{1} ⊗ e_{1} ↦ 0 for the multiplication map (z, w) ↦ zw.

Recall also that we made a choice of a chain equivalence H_{∗}(B) ≃ C_{∗}(B), inducing a chain equivalence H_{∗}(B)^{⊗p} ≃ C_{∗}(B)^{⊗p} ≃ C_{∗}(B^{∧p}) that is compatible with the C_{p}-actions. Combining these facts, we see that κ̄ has a chain level model

κ̄_{∗}: C_{∗}(𝕋/C_{p}) ⊗ Σ^{2n−1}C_{∗}(𝕋) ⊗_{C_{p}} H_{∗}(B)^{⊗p} ⟶^{≅} Σ^{2n−1}C_{∗}(𝕋) ⊗_{C_{p}} C_{∗}(𝕋) ⊗_{C_{p}} H_{∗}(B)^{⊗p}

given by

(4.7)  e_{j} ⊗ Σ^{2n−1}e_{k} ⊗ x ⟼ Σ^{2n−1}e_{k} ⊗ e_{j} ⊗ x

for (j, k) = (0, 0), (0, 1) or (1, 1), and

(4.8)  e_{1} ⊗ Σ^{2n−1}e_{0} ⊗ x ⟼ (Σ^{2n−1}e_{0} ⊗ e_{1} + Σ^{2n−1}e_{1} ⊗ T e_{0}) ⊗ x

in the case (j, k) = (1, 0), where x ∈ H_{∗}(B)^{⊗p}. Note that, as a consequence of linearity,

e_{1} ⊗ Σ^{2n−1}e_{0} ⊗ x − e_{0} ⊗ Σ^{2n−1}e_{1} ⊗ T x ⟼ Σ^{2n−1}e_{0} ⊗ e_{1} ⊗ x .

Now consider a class u^{i}t^{r} ⊗ α^{⊗p} in H_{∗}^{c}((B^{∧p})^{tC_{p}}). Let n = −r and k = 1 − i. Then u^{i}t^{r} ⊗ α^{⊗p} lies in Tate filtration −(i + 2r) = 2n − 1 + k, and is represented by the homology class of

Σ^{2n−1}e_{k} ⊗ α^{⊗p}

in H_{∗}(Σ^{2n−1}𝕋_{+} ∧_{C_{p}} B^{∧p}). Hence e_{j} ⊗ u^{i}t^{r} ⊗ α^{⊗p} is represented by the class of

e_{j} ⊗ Σ^{2n−1}e_{k} ⊗ α^{⊗p}

in H_{∗}(𝕋/C_{p} ⋉ (Σ^{2n−1}𝕋_{+} ∧_{C_{p}} B^{∧p})).

Its image under κ̄_{∗} is given by the formulas (4.7) and (4.8) above, with x = α^{⊗p}, hence equals the class of

Σ^{2n−1}e_{k} ⊗ e_{j} ⊗ α^{⊗p}

in H_{∗}(Σ^{2n−1}𝕋_{+} ∧_{C_{p}} (𝕋 ⋉_{C_{p}} B^{∧p})) for (j, k) = (0, 0), (0, 1) or (1, 1), and the class of

(Σ^{2n−1}e_{0} ⊗ e_{1} + Σ^{2n−1}e_{1} ⊗ e_{0}) ⊗ α^{⊗p}

for (j, k) = (1, 0), since T (α^{⊗p}) = α^{⊗p}. It also follows that

(e_{1} ⊗ ut^{r} − e_{0} ⊗ t^{r}) ⊗ α^{⊗p}

is represented by

(e_{1} ⊗ Σ^{2n−1}e_{0} − e_{0} ⊗ Σ^{2n−1}e_{1}) ⊗ α^{⊗p} ,

which maps under κ̄_{∗} to Σ^{2n−1}e_{0} ⊗ e_{1} ⊗ α^{⊗p}.

To find the Tate representative of κ_{∗}(e_{j} ⊗ u^{i}t^{r} ⊗ α^{⊗p}) we now use the cofiber sequence (4.4) and the associated cofiber sequence

Σ^{2n−1}(𝕋 ⋉_{C_{p}} B^{∧p}) → Σ^{2n−1}𝕋_{+} ∧_{C_{p}} (𝕋 ⋉_{C_{p}} B^{∧p}) → Σ^{2n}(𝕋 ⋉_{C_{p}} B^{∧p})

coming from the Tate filtration of the middle term. It shows that Σ^{2n−1}e_{k} ⊗ e_{j} ⊗ α^{⊗p} in

H_{∗}(Σ^{2n−1}𝕋_{+} ∧_{C_{p}} (𝕋 ⋉_{C_{p}} B^{∧p}))

has Tate representative u^{i}t^{r} ⊗ e_{j} ⊗ α^{⊗p}, where i = 1 − k and r = −n. Noting that the case (i, j) = (1, 1) corresponds to the case (j, k) = (1, 0), and chasing the formulas, we get the asserted result. □

Theorem 4.10. The map

ω^{t}: 𝕋/C_{p} ⋉ (B^{∧p})^{tC_{p}} ⟶ THH(B)^{tC_{p}}

induces the complete 𝒜_{∗}-comodule homomorphism

ω_{∗}^{t}: H_{∗}(𝕋/C_{p}) ⊗ H_{∗}^{c}((B^{∧p})^{tC_{p}}) ⟶ H_{∗}^{c}(THH(B)^{tC_{p}})

given, modulo Tate filtration in the target, by

e_{0} ⊗ u^{i}t^{r} ⊗ α^{⊗p} ⟼ u^{i}t^{r} ⊗ α^{p}
e_{1} ⊗ t^{r} ⊗ α^{⊗p} ⟼ t^{r} ⊗ α^{p−1} ∧ α
(e_{1} ⊗ ut^{r} − e_{0} ⊗ t^{r}) ⊗ α^{⊗p} ⟼ ut^{r} ⊗ α^{p−1} ∧ α

for i ∈ {0, 1}, r ∈ ℤ.

Proof. This is clear from Proposition 4.8, Definition 3.10, and naturality of the homological Tate spectral sequence with respect to the map D ∘ ω_{p}: 𝕋 ⋉_{C_{p}} B^{∧p} → THH(B). □

5. Cyclotomic structure

Recall from diagram (2.1) that we are interested in the natural map

Γ̂: [Ẽ𝕋 ∧ X]^{C_{p}} → [Ẽ𝕋 ∧ F (E𝕋_{+}, X)]^{C_{p}} = X^{tC_{p}}

for X ≅ THH(B). We now study the source of this map.

Definition 5.1. Let G ⊆ 𝕋 be a closed subgroup containing C_{p}. For each good G-equivariant prespectrum Y, with spectrification X = LY, the geometric fixed point prespectrum Φ^{C_{p}}(Y) is the G/C_{p}- equivariant prespectrum with

Φ^{C_{p}}(Y)(V^{C_{p}}) = Y (V)^{C_{p}} ,

and the geometric fixed point spectrum Φ^{C_{p}}(X) is its spectrification Φ^{C_{p}}(X) = LΦ^{C_{p}}(Y). See [20, II.9.7] and [18, §2.1].

There is a natural equivalence of G/C_{p}-spectra

s̄: [Ẽ𝕋 ∧ X]^{C_{p}} ⟶^{≃} Φ^{C_{p}}(X) ,

see [20, II.9.8] and the proof of [18, Lem. 2.1]. We are concerned with the cases G = C_{p}, Y = (B_{pre}^{∧p})^{τ} and G = 𝕋, Y = sd_{p} thh(B)^{τ}, corresponding to X = B^{∧p} and X = sd_{p} THH(B), respectively.

With notation as in Definition 3.1, there are natural isomorphisms and maps

sd_{p} thh(B; S^{V})_{k}^{C_{p}} = thh(B; S^{V})_{p(k+1)−1}^{C_{p}}

= ( hocolim_{n⃗∈I^{p(k+1)}} Map(S^{n_{0}} ∧ ⋯ ∧ S^{n_{p(k+1)−1}}, B_{n_{0}} ∧ ⋯ ∧ B_{n_{p(k+1)−1}} ∧ S^{V}))^{C_{p}}

≅ hocolim_{n⃗∈I^{k+1}} Map((S^{n_{0}} ∧ ⋯ ∧ S^{n_{k}})^{∧p}, (B_{n_{0}} ∧ ⋯ ∧ B_{n_{k}})^{∧p} ∧ S^{V})^{C_{p}}

→ hocolim_{n⃗∈I^{k+1}} Map(S^{n_{0}} ∧ ⋯ ∧ S^{n_{k}}, B_{n_{0}} ∧ ⋯ ∧ B_{n_{k}} ∧ S^{V^{Cp}}) = thh(B; S^{V^{Cp}})_{k}

induced from the identifications S^{n_{0}} ∧ ⋯ ∧ S^{n_{k}} ≅ ((S^{n_{0}} ∧ ⋯ ∧ S^{n_{k}})^{∧p})^{C_{p}} and B_{n_{0}} ∧ ⋯ ∧ B_{n_{k}} ≅ ((B_{n_{0}} ∧ ⋯ ∧ B_{n_{k}})^{∧p})^{C_{p}}. These define a natural map of prespectra

r′_{k}: Φ^{C_{p}}(sd_{p} thh(B)_{k}) → thh(B)_{k}

for each k ≥ 0, and likewise after the natural good thickening.

Lemma 5.2. Suppose that B is connective. The spectrum maps

r′_{0} : Φ^{C_{p}}(B^{∧p}) ⟶^{≃} B^{∧1}

(corresponding to the case k = 0) and

r′: Φ^{C_{p}}(sd_{p} THH(B)) ⟶^{≃} THH(B)

(obtained from the cases k ≥ 0 by geometric realization) are natural equivalences.

Proof. The second case is proved in [18, Prop. 2.5], and the first case is part of their proof. In more detail, they prove that the connectivity of the map

Ω^{V^{Cp} −W} r′_{0}(V^{C_{p}}): Ω^{V^{Cp} −W} thh(B; S^{V})_{p−1}^{C_{p}} ⟶ Ω^{V^{Cp} −W} thh(B; S^{V^{Cp}})_{0}

grows to infinity with V, for each fixed W. (The connectivity hypothesis enters at the bottom of page 42 in the cited paper.) □

In the case G = 𝕋, let ρ: 𝕋 → 𝕋/C_{p} be the p-th root isomorphism of groups, with inverse ρ^{−1} : 𝕋/C_{p} → 𝕋 taking [z] to z^{p}. The cyclic structures on sd_{p} thh(B)_{•} and thh(B)_{•} induce a 𝕋/C_{p}-equivariant structure on Φ^{C_{p}}(sd_{p} THH(B)) and a 𝕋-equivariant structure on THH(B). These are compatible, in the sense that the cyclotomic structure equivalence r′ is ρ^{−1}-equivariant. See [18, Def. 2.2].

The C_{p}-equivariant map η_{p}: B^{∧p} → sd_{p} THH(B) is induced from the inclusion of 0-simplices B_{pre}^{∧p} → sd_{p} thh(B)_{•}, hence is compatible with s̄ by naturality, and with the maps r′_{0} and r′ by the construction of the latter via geometric realization. Hence the outer part of the following diagram commutes.

(5.1)  \xymatrix{B^{∧1} \ar[r]^{i} & 𝕋 ⋉ B^{∧1} \ar[r]^{ω} & THH(B) \\ Φ^{C_{p}}(B^{∧p}) \ar[r] \ar[u]^{≃}_{r′_{0}} & 𝕋/C_{p} ⋉ Φ^{C_{p}}(B^{∧p}) \ar[r] \ar[u]^{≃}_{ρ^{−1}⋉r′_{0}} & Φ^{C_{p}}(sd_{p} THH(B)) \ar[u]^{≃}_{r′} \\ [Ẽ𝕋 ∧ B^{∧p}]^{C_{p}} \ar[r] \ar[u]^{≃}_{s̄} & 𝕋/C_{p} ⋉ [Ẽ𝕋 ∧ B^{∧p}]^{C_{p}} \ar[r] \ar[u]^{≃}_{id ⋉s̄} & [Ẽ𝕋 ∧ sd_{p} THH(B)]^{C_{p}} \ar[u]^{≃}_{s̄}}

The right hand horizontal maps are the unique equivariant extensions, given by the 𝕋/C_{p}-action on [Ẽ𝕋∧ sd_{p} THH(B)]^{C_{p}} and Φ^{C_{p}}(sd_{p} THH(B)), and the 𝕋-action on THH(B), respectively. These equivariant extensions are compatible, via the identity id: 𝕋/C_{p} → 𝕋/C_{p} and the isomorphism ρ^{−1} : 𝕋/C_{p} → 𝕋, respectively, in view of the equivariance properties of s̄ and r′ discussed above. Hence the whole diagram commutes.

Theorem 5.3. Let B be a connective symmetric ring spectrum. There is a natural commutative diagram in the stable homotopy category

\xymatrix{B \ar[r]^{i} \ar[d]_{ϵ_{B}} \ar[rr]^{η} & 𝕋 ⋉ B \ar[r]^{ω} \ar[d]^{ρ⋉ϵ_{B}} & THH(B) \ar[d]^{γ} \\ (B^{∧p})^{tC_{p}} \ar[r]^{i′} \ar[rr]_{η^{t}} & 𝕋/C_{p} ⋉ (B^{∧p})^{tC_{p}} \ar[r]^{ω^{t}} & THH(B)^{tC_{p}}}

where ϵ_{B} is the composite

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

and γ is the composite

THH(B) ⟵^{≃} [Ẽ𝕋 ∧ sd_{p} THH(B)]^{C_{p}} ⟶^{Γ̂} (sd_{p} THH(B))^{tC_{p}} ≅ THH(B)^{tC_{p}} .

Proof. By naturality of the map Γ̂ with respect to the inclusion of 0-simplices η_{p}: B^{∧p} → sd_{p} THH(B) we get that the outer part of the following diagram commutes.

(5.2)  \xymatrix{[Ẽ𝕋 ∧ B^{∧p}]^{C_{p}} \ar[r] \ar[rr]^{[id ∧η_{p}]^{Cp}} \ar[d]_{Γ̂} & 𝕋/C_{p} ⋉ [Ẽ𝕋 ∧ B^{∧p}]^{C_{p}} \ar[r] \ar[d]^{id ⋉Γ̂} & [Ẽ𝕋 ∧ sd_{p} THH(B)]^{C_{p}} \ar[d]^{Γ̂} \\ (B^{∧p})^{tC_{p}} \ar[r]^{i′} \ar[rr]_{η_{p}^{tCp}} & 𝕋/C_{p} ⋉ (B^{∧p})^{tC_{p}} \ar[r]^{ω_{p}^{tCp} ∘κ} & (sd_{p} THH(B))^{tC_{p}}}

The right hand column is 𝕋/C_{p}-equivariant, and the right hand square is constructed to be the unique 𝕋/C_{p}-equivariant extension of the outer part. Hence the whole diagram commutes.

By combining the right hand parts of diagrams (5.1), (5.2) and (4.2), we get the central part of the commutative diagram below:

[figure: 𝕋 ⋉ B^{∧1}; ω; ρ⋉ϵ_{B}; THH(B); γ; 𝕋/C_{p} ⋉ [Ẽ𝕋 ∧ B^{∧p}]^{C_{p}}; ≃; ρ^{−1}⋉r′_{0}s̄; id ⋉Γ̂; [Ẽ𝕋 ∧ sd_{p} THH(B)]^{C_{p}}; ≃; r′s̄; Γ̂; 𝕋/C_{p} ⋉ (B^{∧p})^{tC_{p}}; ω^{t}; sd_{p} THH(B)^{tC_{p}}; ≅; D^{tCp}; THH(B)^{tC_{p}}]

This proves the theorem. □

6. The comparison map

In this section we compute the effect of the comparison map γ: THH(B) → THH(B)^{tC_{p}} for B = MU and BP. Let H = H𝔽_{p} be the mod p Eilenberg–Mac Lane spectrum, realized as a commutative symmetric ring spectrum. Recall [25] the structure

𝒜_{∗} = H_{∗}(H) = P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ 0)

of the dual Steenrod algebra, where |ξ̄_{k}| = 2p^{k} − 2 and |τ̄_{k}| = 2p^{k} − 1. (This assumes p is odd—we leave the details for p = 2 to the reader.) The Hopf algebra coproduct is given by the formulas

ψ(ξ̄_{k}) = ∑_{i+j=k} ξ̄_{i} ⊗ ξ̄_{j}^{p^{i}}
ψ(τ̄_{k}) = 1 ⊗ τ̄_{k} + ∑_{i+j=k} τ̄_{i} ⊗ ξ̄_{j}^{p^{i}} ,

where ξ̄_{0} = 1.

6.1. The case of complex cobordism. Let MU be the complex cobordism spectrum, realized as a commutative symmetric ring spectrum [23, IV.2]. Recall [1, pp. 75–77] the 𝒜_{∗}-comodule algebra isomorphism

H_{∗}(MU) ≅ P (ξ̄_{k} | k ≥ 1) ⊗ P (m_{ℓ} | ℓ ≠ p^{k} − 1) ,

where m_{ℓ} is an 𝒜_{∗}-comodule primitive of degree 2ℓ, for each ℓ ≥ 1 not of the form p^{k} − 1. Note that H_{∗}(MU) is concentrated in even degrees.

Definition 6.1. For ℓ = p^{k} − 1, let m_{ℓ} = ξ̄_{k}, so that H_{∗}(MU) ≅ P (m_{ℓ} | ℓ ≥ 1).

Lemma 6.2. There is an 𝒜_{∗}-comodule algebra isomorphism

H_{∗}(THH(MU)) ≅ H_{∗}(MU) ⊗ E(σm_{ℓ} | ℓ ≥ 1) .

The classes σm_{ℓ} are 𝒜_{∗}-comodule primitive, for all ℓ ≥ 1.

Proof. We use the (first quadrant) Bökstedt spectral sequence

E_{∗,∗}^{2} = HH_{∗}(H_{∗}(B)) ⟹ H_{∗}(THH(B))

arising from the skeleton filtration on THH(B). See [3, §5] for the tools used in this computation, including the facts that σ is a derivation, and that it commutes with the Dyer–Lashof operations Q^{i}.

The E^{2}-term for B = MU is

E_{∗,∗}^{2} = H_{∗}(MU) ⊗ E(σm_{ℓ} | ℓ ≥ 1) .

We have E_{∗,∗}^{2} = E_{∗,∗}^{∞}, since all algebra generators lie in filtrations ≤ 1. There are no algebra extensions, since σm_{ℓ} is in an odd degree, hence is an exterior class for p odd by graded commutativity, and for p = 2 by the Dyer–Lashof calculation (σm_{ℓ})^{2} = Q^{2ℓ+1}(σm_{ℓ}) = σQ^{2ℓ+1}(m_{ℓ}) = 0, since Q^{2ℓ+1}(m_{ℓ}) ∈ H_{4ℓ+1}(MU) = 0. The 𝒜_{∗}-coaction ν: H_{∗}(X) → 𝒜_{∗} ⊗ H_{∗}(X) for X = THH(MU) is given by ν(σm_{ℓ}) = (1 ⊗ σ)ν(m_{ℓ}) = (1 ⊗ σ)(1 ⊗ m_{ℓ}) = 1 ⊗ σm_{ℓ} for ℓ ≠ p^{k} − 1, while

ν(σξ̄_{k}) = (1 ⊗ σ)( ∑_{i+j=k} ξ̄_{i} ⊗ ξ̄_{j}^{p^{i}}) = ∑_{i+j=k} ξ̄_{i} ⊗ σ(ξ̄_{j}^{p^{i}}) = 1 ⊗ σξ̄_{k} ,

since σ(ξ̄_{j}^{p^{i}}) = 0 for i ≥ 1. □

The following was proved by a different method in [14, 6.4].

Proposition 6.3. The homological Tate spectral sequence

Ê_{∗,∗}^{2}(THH(MU)) = Ĥ^{−∗}(C_{p}; H_{∗}(THH(MU))) ⟹ H_{∗}^{c}(THH(MU)^{tC_{p}})

collapses at the Ê^{3} = Ê^{∞}-term, with

Ê_{∗,∗}^{∞} = Ĥ^{−∗}(C_{p}; 𝔽_{p}) ⊗ P (m_{ℓ}^{p} | ℓ ≥ 1) ⊗ E(m_{ℓ}^{p−1}σm_{ℓ} | ℓ ≥ 1) .

Proof. The Ê^{2}-term is

Ê_{∗,∗}^{2} = Ĥ^{−∗}(C_{p}; 𝔽_{p}) ⊗ P (m_{ℓ} | ℓ ≥ 1) ⊗ E(σm_{ℓ} | ℓ ≥ 1)

where Ĥ^{−∗}(C_{p}; 𝔽_{p}) = E(u) ⊗ P (t, t^{−1}) for p odd. The classes u^{i}t^{r} are all infinite cycles, as recalled in Remark 4.2. The d^{2}-differentials are given by the formula

d^{2}(u^{i}t^{r} ⊗ α) = u^{i}t^{r+1} ⊗ σα .

The homology of P (m_{ℓ})⊗E(σm_{ℓ}) with respect to σ is P (m_{ℓ}^{p})⊗E(m_{ℓ}^{p−1}σm_{ℓ}), so by the Künneth formula

Ê_{∗,∗}^{3} = Ĥ^{−∗}(C_{p}; 𝔽_{p}) ⊗ P (m_{ℓ}^{p} | ℓ ≥ 1) ⊗ E(m_{ℓ}^{p−1}σm_{ℓ} | ℓ ≥ 1) .

By Theorem 4.10 and Corollary 3.13 the map

ω^{t}: 𝕋/C_{p} ⋉ (B^{∧p})^{tC_{p}} → THH(B)^{tC_{p}}

for B = MU takes the classes e_{0} ⊗ 1 ⊗ m_{ℓ}^{⊗p} and e_{1} ⊗ 1 ⊗ m_{ℓ}^{⊗p} in H_{∗}(𝕋/C_{p}) ⊗ H_{∗}^{c}((B^{∧p})^{tC_{p}}) to classes represented by 1 ⊗ m_{ℓ}^{p} and 1 ⊗ m_{ℓ}^{p−1}σm_{ℓ} in the Tate spectral sequence, respectively. Hence the Ê^{3}-term is generated by infinite cycles, and there cannot be any further differentials. □

Theorem 6.4. The map

γ: THH(MU) → THH(MU)^{tC_{p}}

induces a complete 𝒜_{∗}-comodule algebra homomorphism

γ_{∗}: H_{∗}(THH(MU)) ⟶ H_{∗}^{c}(THH(MU)^{tC_{p}})

mapping

σm_{ℓ} ⟼ (−1)^{ℓ}t^{(p−1)ℓ} ⊗ m_{ℓ}^{p−1}σm_{ℓ}

modulo Tate filtration in the target, for each ℓ ≥ 1.

Proof. We use the commutative diagram

\xymatrix{𝕋 ⋉ MU \ar[r]^{ω} \ar[d]_{ρ⋉ϵ_{MU}} & THH(MU) \ar[d]^{γ} \\ 𝕋/C_{p} ⋉ (MU^{∧p})^{tC_{p}} \ar[r]^{ω^{t}} & THH(MU)^{tC_{p}}}

from Theorem 5.3 for B = MU. Let [𝕋] ∈ H_{1}(𝕋) be the fundamental class, which maps by ρ_{∗} to the fundamental class e_{1} ∈ H_{1}(𝕋/C_{p}). Then ω_{∗}([𝕋] ⊗ α) = σα for all α ∈ H_{∗}(MU), so we can compute γ_{∗}(σα) as the image under ω_{∗}^{t} of e_{1} ⊗ (ϵ_{MU})_{∗}(α). We make separate calculations for the cases α = ξ̄_{k} (with ℓ = p^{k} − 1) and α = m_{ℓ} (with ℓ ≠ p^{k} − 1).

Recall from [22, 3.2.1] that the homomorphism

ϵ_{∗}: H_{∗}(MU) → R_{+}(H_{∗}(MU))

is given by the formula

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

in the homological Singer construction R_{+}(H_{∗}(MU)), where P_{∗}^{r} is the homology operation dual to the r-th Steenrod power. (The terms involving (β P^{r})_{∗}(α) vanish in this case, since H_{∗}(MU) is concentrated in even degrees.)

By Lemma 6.5 below, we obtain

ϵ_{∗}(ξ̄_{k}) = ∑_{i=0}^{k} t^{−(p^{i}−1)} ⊗ ξ̄_{k−i}^{p^{i}}
= 1 ⊗ ξ̄_{k} + t^{−(p−1)} ⊗ ξ̄_{k−1}^{p} + ⋯ + t^{−(p^{k}−1)} ⊗ 1

in R_{+}(H_{∗}(MU)), for each k ≥ 1. To control the terms with 1 ≤ i ≤ k, it will be convenient to compare with the ϵ_{∗}-image of ξ̄_{k−1}^{p}. By Lemma 6.6 below, we obtain

ϵ_{∗}(ξ̄_{k−1}^{p}) = ∑_{i=0}^{k−1} t^{−p(p^{i}−1)} ⊗ ξ̄_{k−1−i}^{p^{i+1}} = ∑_{i=1}^{k} t^{−(p^{i}−p)} ⊗ ξ̄_{k−i}^{p^{i}} .

Hence

(6.1)  ϵ_{∗}(ξ̄_{k}) = 1 ⊗ ξ̄_{k} + t^{−(p−1)} · ϵ_{∗}(ξ̄_{k−1}^{p})

in R_{+}(H_{∗}(MU)), for each k ≥ 1. Here the multiplication by t^{−(p−1)} refers to the R_{+}(H_{∗}(S)) = Ĥ^{−∗}(C_{p}; 𝔽_{p})-module structure on R_{+}(H_{∗}(MU)), coming from the S-module structure on MU.

Next recall the isomorphism R_{+}(H_{∗}(MU)) ≅ H_{∗}^{c}((MU^{∧p})^{tC_{p}}) of [22, 5.14], taking t^{r} ⊗α to a preferred class represented by

(−1)^{ℓ}t^{r+(p−1)ℓ} ⊗ α^{⊗p}

in the Tate spectral sequence, where |α| = 2ℓ is assumed to be even. (The coefficient is more complicated when |α| is odd.) By [22, 5.12], the map ϵ_{MU}: MU → (MU^{∧p})^{tC_{p}} induces the composite of ϵ_{∗}: H_{∗}(MU) → R_{+}(H_{∗}(MU)) and this isomorphism. Hence, the identity (6.1) tells us that the difference

(ϵ_{MU})_{∗}(ξ̄_{k}) − t^{−(p−1)} · (ϵ_{MU})_{∗}(ξ̄_{k−1}^{p})

is the preferred class represented by

t^{(p−1)(p^{k}−1)} ⊗ ξ̄_{k}^{⊗p}

in the Tate spectral sequence converging to H_{∗}^{c}((MU^{∧p})^{tC_{p}}).

We now chase the class [𝕋] ⊗ ξ̄_{k−1}^{p} in H_{∗}(𝕋 ⋉ MU) around the commutative square above. Going to the right,

ω_{∗}([𝕋] ⊗ ξ̄_{k−1}^{p}) = σ(ξ̄_{k−1}^{p}) = 0

in H_{∗}(THH(MU)), since σ is a derivation. Hence the image under γ_{∗} is also 0, which implies that

(ω^{t})_{∗}(e_{1} ⊗ (ϵ_{MU})_{∗}(ξ̄_{k−1}^{p})) = 0 .

It follows by the Ĥ^{−∗}(C_{p}; 𝔽_{p})-module structure that

(ω^{t})_{∗}(e_{1} ⊗ t^{−(p−1)} · (ϵ_{MU})_{∗}(ξ̄_{k−1}^{p})) = 0 .

Finally we chase the class [𝕋] ⊗ ξ̄_{k} around the square, to see that γ_{∗}(σξ̄_{k}) equals

(ω^{t})_{∗}(e_{1} ⊗ (ϵ_{MU})_{∗}(ξ̄_{k})) = (ω^{t})_{∗}(e_{1} ⊗ t^{(p−1)(p^{k}−1)} ⊗ ξ̄_{k}^{⊗p}) ,

which by Theorem 4.10 and Corollary 3.13 is represented by

t^{(p−1)(p^{k}−1)} ⊗ ξ̄_{k}^{p−1}σξ̄_{k}

in the Tate spectral sequence converging to H_{∗}^{c}(THH(MU)^{tC_{p}}). This proves the theorem for m_{ℓ} = ξ̄_{k}, with ℓ = p^{k} − 1, since (−1)^{ℓ} ≡ +1 mod p in these cases.

The remaining cases, of m_{ℓ} with ℓ ≠ p^{k} − 1, are simpler. We have

ϵ_{∗}(m_{ℓ}) = 1 ⊗ m_{ℓ}

in the homological Singer construction, since these m_{ℓ}’s are 𝒜_{∗}-comodule primitives. Hence (ϵ_{MU})_{∗}(m_{ℓ}) is the preferred class represented by

(−1)^{ℓ}t^{(p−1)ℓ} ⊗ m_{ℓ}^{⊗p}

in the Tate spectral sequence converging to H_{∗}^{c}((MU^{∧p})^{tC_{p}}). Chasing [𝕋] ⊗ m_{ℓ} around the commutative diagram, we find that γ_{∗}(σm_{ℓ}) equals

ω_{∗}^{t}(e_{1} ⊗ (−1)^{ℓ}t^{(p−1)ℓ} ⊗ m_{ℓ}^{⊗p}) = (−1)^{ℓ}t^{(p−1)ℓ} ⊗ m_{ℓ}^{p−1}σm_{ℓ} .
□

Lemma 6.5.

(−1)^{r} P_{∗}^{r}(ξ̄_{k}) = \begin{cases}ξ̄_{k−i}^{p^{i}} & if r = (p^{i} − 1)/(p − 1), \\ 0 & otherwise. \end{cases}

Proof. The Steenrod power P^{r} is dual to ξ_{1}^{r} in the Milnor basis (ξ^{I}τ^{J}) for 𝒜_{∗}, so each term of the form ξ_{1}^{r}⊗α′′ in the coaction ν(α) contributes a term α′′ in P_{∗}^{r}(α). From the recursive relation 0 = ∑_{i+j=k} ξ_{i}^{p^{j}} ·ξ̄_{j}

for k ≥ 1, we see that ξ̄_{i} ≡ (−1)^{i}ξ_{1}^{(p^{i}−1)/(p−1)} for i ≥ 0, modulo the ideal J(0) ⊂ 𝒜_{∗} generated by the ξ_{k} with k ≥ 2 and the τ_{k} with k ≥ 1. Hence ν(ξ̄_{k}) = ∑_{i+j=k} ξ̄_{i} ⊗ ξ̄_{j}^{p^{i}} is congruent to

∑_{i=0}^{k}(−1)^{i}ξ_{1}^{(p^{i}−1)/(p−1)} ⊗ ξ̄_{k−i}^{p^{i}} ,

and contributes (−1)^{i}ξ̄_{k−i}^{p^{i}} to P_{∗}^{r}(ξ̄_{k}) precisely if r = (p^{i} − 1)/(p − 1). In this case (−1)^{i} ≡ (−1)^{r} mod p. □

Lemma 6.6.

(−1)^{r} P_{∗}^{r}(ξ̄_{k−1}^{p}) = \begin{cases}ξ̄_{k−1−i}^{p^{i+1}} & if r = p(p^{i} − 1)/(p − 1), \\ 0 & otherwise. \end{cases}

Proof. The coaction ν(ξ̄_{k−1}^{p}) = ∑_{i+j=k−1} ξ̄_{i}^{p} ⊗ ξ̄_{j}^{p^{i+1}} is congruent to

∑_{i=0}^{k−1}(−1)^{i}ξ_{1}^{p(p^{i}−1)/(p−1)} ⊗ ξ̄_{k−1−i}^{p^{i+1}} ,

and contributes (−1)^{i}ξ̄_{k−1−i}^{p^{i+1}} to P_{∗}^{r}(ξ̄_{k−1}^{p}) precisely if r = p(p^{i} − 1)/(p − 1). □

6.2. The Brown–Peterson case. Let BP be the p-local Brown–Peterson spectrum, realized as an E_{4} symmetric ring spectrum [7], [8]. We could avoid using the E_{4} structure on BP by appealing to the symmetric ring spectrum map MU → BP of [6] and naturality, as in [14, 6.4], but this would make some arguments longer.

Recall [27, 4.1.12] the 𝒜_{∗}-comodule algebra isomorphism

H_{∗}(BP) ≅ P (ξ̄_{k} | k ≥ 1) .

Note that H_{∗}(BP) is concentrated in even degrees.

Lemma 6.7. There is an 𝒜_{∗}-comodule algebra isomorphism

H_{∗}(THH(BP)) ≅ H_{∗}(BP) ⊗ E(σξ̄_{k} | k ≥ 1) .

The classes σξ̄_{k} are 𝒜_{∗}-comodule primitive, for all k ≥ 1.

Proof. This is similar to the MU-case, see [3, 5.12]. □

Proposition 6.8. The homological Tate spectral sequence

Ê_{∗,∗}^{2}(THH(BP)) = Ĥ^{−∗}(C_{p}; H_{∗}(THH(BP))) ⟹ H_{∗}^{c}(THH(BP)^{tC_{p}})

collapses at the Ê^{3} = Ê^{∞}-term, with

Ê_{∗,∗}^{∞} = Ĥ^{−∗}(C_{p}; 𝔽_{p}) ⊗ P (ξ̄_{k}^{p} | k ≥ 1) ⊗ E(ξ̄_{k}^{p−1}σξ̄_{k} | k ≥ 1) .

Proof. This is similar to the MU-case. See also [14, 6.4]. □

Theorem 6.9. The map

γ: THH(BP) → THH(BP)^{tC_{p}}

induces a complete 𝒜_{∗}-comodule algebra homomorphism

γ_{∗}: H_{∗}(THH(BP)) ⟶ H_{∗}^{c}(THH(BP)^{tC_{p}})

mapping

σξ̄_{k} ⟼ t^{(p−1)(p^{k}−1)} ⊗ ξ̄_{k}^{p−1}σξ̄_{k}

modulo Tate filtration in the target, for each k ≥ 1.

Proof. This is similar to the MU-case, using the commutative diagram

\xymatrix{𝕋 ⋉ BP \ar[r]^{ω} \ar[d]_{ρ⋉ϵ_{BP}} & THH(BP) \ar[d]^{γ} \\ 𝕋/C_{p} ⋉ (BP^{∧p})^{tC_{p}} \ar[r]^{ω^{t}} & THH(BP)^{tC_{p}}}

from Theorem 5.3, or naturality with respect to the symmetric ring spectrum map MU → BP. □

7. The Segal conjecture

In this section we prove Theorem 2.2.

Remark 7.1. The basic idea, in the case B = MU, is that the homological Tate spectral sequences

′Ê_{∗,∗}^{2} = Ĥ^{−∗}(C_{p}; H_{∗}(THH(MU))) ⟹ H_{∗}^{c}(THH(MU)^{tC_{p}})

and

′′Ê_{∗,∗}^{2} = Ĥ^{−∗}(C_{p}; H_{∗}(THH(MU)^{∧p})) ⟹ R_{+}(H_{∗}(THH(MU)))

have Ê^{∞}-terms

′Ê_{∗,∗}^{∞} = E(u) ⊗ P (t, t^{−1}) ⊗ P (m_{ℓ}^{p} | ℓ ≥ 1) ⊗ E(m_{ℓ}^{p−1}σm_{ℓ} | ℓ ≥ 1)

and

′′Ê_{∗,∗}^{∞} = E(u) ⊗ P (t, t^{−1}) ⊗ P (m_{ℓ}^{⊗p} | ℓ ≥ 1) ⊗ E(σm_{ℓ}^{⊗p} | ℓ ≥ 1) ,

that are isomorphic by way of a filtration-shifting isomorphism given by

m_{ℓ}^{p} ↦ m_{ℓ}^{⊗p}  and  m_{ℓ}^{p−1}σm_{ℓ} ↦ t^{m} ⊗ σm_{ℓ}^{⊗p} ,

where m = (p − 1)/2. The difficulty is to promote this isomorphism to an 𝒜_{∗}-comodule isomorphism of the abutments, respecting the linear topologies. One approach is to compare the two Tate towers via a third tower, given by base changing the Tate tower for MU^{∧p} along η: MU → THH(MU), as we now explain.

Consider any connective symmetric ring spectrum B. We have a commutative diagram

(7.1)  \xymatrix{THH(B) \ar[d]_{ϵ_{THH(B)}} & B \ar[r]^{η} \ar[l]_{η} \ar[d]_{ϵ_{B}} & THH(B) \ar[d]^{γ} \\ R_{+}(THH(B)) \ar[d] & R_{+}(B) \ar[r]^{η^{t}} \ar[l]_{R_{+}(η)} \ar[d] & THH(B)^{tC_{p}} \ar[d] \\ (THH(B)^{∧p})^{tC_{p}}[n] & (B^{∧p})^{tC_{p}}[n] \ar[l] \ar[r] & THH(B)^{tC_{p}}[n]}

by Theorem 5.3 and naturality of ϵ. We recall that R_{+}(B) = (B^{∧p})^{tC_{p}}, while (B^{∧p})^{tC_{p}}[n] denotes the nth term in the Tate tower (4.5), and similarly with THH(B) in place of B. We are principally concerned with the limit behavior as n → −∞.

Passing to continuous homology, we get a commutative diagram

(7.2)  \xymatrix{H_{∗}(THH(B)) \ar[d]_{ϵ_{∗}} & H_{∗}(B) \ar[r]^{η_{∗}} \ar[l]_{η_{∗}} \ar[d]_{ϵ_{∗}} & H_{∗}(THH(B)) \ar[d]^{γ_{∗}} \\ R_{+}(H_{∗}(THH(B))) \ar@{->>}[d] & R_{+}(H_{∗}(B)) \ar[r]^{η_{∗}^{t}} \ar[l]_{R_{+}(η_{∗})} \ar@{->>}[d] & H_{∗}^{c}(THH(B)^{tC_{p}}) \ar@{->>}[d] \\ F^{n}R_{+}(H_{∗}(THH(B))) & F^{n}R_{+}(H_{∗}(B)) \ar[l] \ar[r] & F^{n}H_{∗}^{c}(THH(B)^{tC_{p}})}

of complete 𝒜_{∗}-comodules. Here

F^{n}R_{+}(H_{∗}(B)) = im(H_{∗}^{c}(R_{+}(B)) → H_{∗}((B^{∧p})^{tC_{p}}[n]))

so that R_{+}(H_{∗}(B)) = lim_{n} F^{n}R_{+}(H_{∗}(B)), and similarly with THH(B) in place of B. We also use the notation

F^{n}H_{∗}^{c}(THH(B)^{tC_{p}}) = im(H_{∗}^{c}(THH(B)^{tC_{p}}) → H_{∗}(THH(B)^{tC_{p}}[n]))

so that H_{∗}^{c}(THH(B)^{tC_{p}}) = lim_{n} F^{n}H_{∗}^{c}(THH(B)^{tC_{p}}).

Suppose now that B is an E_{2} symmetric ring spectrum, so that THH(B) is a ring spectrum and the upper two rows of (7.1) form a diagram of ring spectra. Then the upper two rows of (7.2) form a diagram of complete 𝒜_{∗}-comodule algebras. Using the algebra structures, we get a commutative diagram given by the solid arrows below:

(7.3)  \xymatrix{ & H_{∗}(THH(B)) \ar[ddl]_{ϵ_{∗}} \ar[d] \ar[ddr]^{γ_{∗}} \\  & R_{+}(H_{∗}(B)) ⊗_{H_{∗}(B)} H_{∗}(THH(B)) \ar[dl]^{f} \ar@{-->}[dr]_{g} \\ R_{+}(H_{∗}(THH(B)) \ar@{-->}[rr]_{Φ_{B}} &  & H_{∗}^{c}(THH(B)^{tC_{p}})}

Here f(α ⊗ β) = R_{+}(η_{∗})(α) · ϵ_{∗}(β) while g(α ⊗ β) = η_{∗}^{t}(α) · γ_{∗}(β). We wish to construct a suitably structured isomorphism Φ_{B}: R_{+}(H_{∗}(THH(B))) → H_{∗}^{c}(THH(B)^{tC_{p}}) making the whole diagram commute.

7.1. The case B = MU. In this case the central term in diagram (7.3) is

R_{+}(H_{∗}(MU)) ⊗_{H_{∗}(MU)} H_{∗}(THH(MU)) ≅ R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1) ,

since H_{∗}(THH(MU)) ≅ H_{∗}(MU) ⊗ E(σm_{ℓ} | ℓ ≥ 1).

By Lemma 6.2 each σm_{ℓ} is 𝒜_{∗}-comodule primitive, so by [22, 3.2.1] we have ϵ_{∗}(σm_{ℓ}) = 1 ⊗ σm_{ℓ} in R_{+}(H_{∗}(THH(MU)) on the left hand side. It has Tate representative

t^{m(2ℓ+1)} ⊗ σm_{ℓ}^{⊗p}

(up to a unit in 𝔽_{p}) in Tate filtration −(p − 1)(2ℓ + 1), by [22, 5.14], where m = (p − 1)/2. On the right hand side, we showed in Theorem 6.4 that γ_{∗}(σm_{ℓ}) has Tate representative

t^{(p−1)ℓ} ⊗ m_{ℓ}^{p−1}σm_{ℓ}

(up to a sign) in Tate filtration −2(p − 1)ℓ.

Since both ϵ_{∗}(σm_{ℓ}) and γ_{∗}(σm_{ℓ}) are in negative Tate filtration, we find that f maps α ⊗ β, with α in Tate filtration < n and β ∈ E(σm_{ℓ} | ℓ ≥ 1), to a class in Tate filtration < n, and likewise for g. Hence we get a commutative diagram

\xymatrix{R_{+}(H_{∗}(THH(MU))) \ar@{->>}[d] & R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1) \ar[l]_{f} \ar[r]^{g} \ar@{->>}[d] & H_{∗}^{c}(THH(MU)^{tC_{p}}) \ar@{->>}[d] \\ F^{n}R_{+}(H_{∗}(THH(MU))) & F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1) \ar[l]_{f_{n}} \ar[r]^{g_{n}} & F^{n}H_{∗}^{c}(THH(MU)^{tC_{p}}) .}

Each f_{n} and g_{n} is an 𝒜_{∗}-comodule homomorphism. Theorem 2.2 for MU now follows from the proposition below by letting

Φ_{MU} = ĝ ∘ f̂^{−1} = (lim_{n} g_{n}) ∘ (lim_{n} f_{n})^{−1} .

Recall the completed tensor product ˆ⊗ from [22, §2.5].

Proposition 7.2. The homomorphisms

f_{n}: F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1) ⟶ F^{n}R_{+}(H_{∗}(THH(MU)))

and

g_{n}: F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1) ⟶ F^{n}H_{∗}^{c}(THH(MU)^{tC_{p}})

define strict maps {f_{n}}_{n} and {g_{n}}_{n} of inverse systems of bounded below 𝒜_{∗}-comodules of finite type. These are pro-isomorphisms, in each homological degree. Hence each of the limiting homomorphisms

f̂ = lim_{n} f_{n}: R_{+}(H_{∗}(MU)) ˆ⊗ E(σm_{ℓ} | ℓ ≥ 1) ⟶ R_{+}(H_{∗}(THH(MU)))

and

ĝ = lim_{n} g_{n}: R_{+}(H_{∗}(MU)) ˆ⊗ E(σm_{ℓ} | ℓ ≥ 1) ⟶ H_{∗}^{c}(THH(MU)^{tC_{p}})

is a continuous isomorphism of complete 𝒜_{∗}-comodules, with continuous inverse.

Proof. We begin with the proof for f̂. The part of the Tate spectral sequence Ê^{∞}-term

Ê_{∗,∗}^{∞} = E(u) ⊗ P (t, t^{−1}) ⊗ P (m_{ℓ}^{⊗p} | ℓ ≥ 1)
⟹ R_{+}(H_{∗}(MU))

in bidegrees (s, t) with Tate filtration s ≥ n is the associated graded of the preferred filtration of F^{n}R_{+}(H_{∗}(MU)). Hence the part of

E(u) ⊗ P (t, t^{−1}) ⊗ P (m_{ℓ}^{⊗p} | ℓ ≥ 1) ⊗ E(σm_{ℓ} | ℓ ≥ 1)

in Tate filtration s ≥ n is the associated graded of the filtration of F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1). Similarly the part of the Tate spectral sequence Ê^{∞}-term

Ê_{∗,∗}^{∞} = E(u) ⊗ P (t, t^{−1}) ⊗ P (m_{ℓ}^{⊗p} | ℓ ≥ 1) ⊗ E(σm_{ℓ}^{⊗p} | ℓ ≥ 1)
⟹ R_{+}(H_{∗}(THH(MU)))

in Tate filtration s ≥ n is the associated graded of the filtration of F^{n}R_{+}(H_{∗}(THH(MU))).

The 𝒜_{∗}-comodule homomorphism f_{n} is the identity on F^{n}R_{+}(H_{∗}(MU)), and takes σm_{ℓ} to ϵ_{∗}(σm_{ℓ}) represented by a unit times t^{m(2ℓ+1)} ⊗ σm_{ℓ}^{⊗p} modulo filtration. Notice that t^{m(2ℓ+1)} ⊗ σm_{ℓ}^{⊗p} lies in bidegree (−(p−1)(2ℓ+1), p(2ℓ+1)), on the line of slope −p/(p−1) through the origin in the (s, t)-plane.

Now restrict attention to one total degree d, indicated by subscripts. We thus have compatible 𝔽_{p}- linear homomorphisms

f_{n,d}: [F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d} ⟶ F^{n}R_{+}(H_{∗}(THH(MU)))_{d}

for all integers n. To provide a pro-inverse, we shall define compatible 𝔽_{p}-linear homomorphisms

ϕ_{n,d}: F^{N}R_{+}(H_{∗}(THH(MU)))_{d} ⟶ [F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d}

with N = N(n, d) = p(n − d) + d, for all integers n. Write the source of the inclusion

[R_{+}(H_{∗}(MU)) ⊗ E(ϵ_{∗}(σm_{ℓ}) | ℓ ≥ 1)]_{d} ⟶ R_{+}(H_{∗}(THH(MU)))_{d}

as a direct sum

⨁_{L} [R_{+}(H_{∗}(MU)) ⊗ 𝔽_{p}{ϵ_{L}}]_{d}

with L ranging over the strictly increasing sequences L = (ℓ_{1} < ⋯ < ℓ_{r}) of natural numbers, of length r ≥ 0. Here

ϵ_{L} = ϵ_{∗}(σm_{ℓ_{1}}) · . . . · ϵ_{∗}(σm_{ℓ_{r}})

has bidegree (s_{L}, t_{L}) = (−(p − 1)(2|L| + r), p(2|L| + r)), where |L| = ℓ_{1} + ⋯ + ℓ_{r}. The inclusion descends to an isomorphism

⨁_{L} [F^{N−s_{L}}R_{+}(H_{∗}(MU)) ⊗ 𝔽_{p}{ϵ_{L}}]_{d} ⟶^{≅} F^{N}R_{+}(H_{∗}(THH(MU)))_{d} ,

as can be seen from the Ê^{∞}-terms. Since F^{N−s_{L}}R_{+}(H_{∗}(MU)) is concentrated in total degrees ≥ N −s_{L}, only the summands with (N − s_{L}) + (s_{L} + t_{L}) ≤ d are nonzero, and this is equivalent to 2|L| + r ≤ d − n. This inequality, in turn, implies that N − s_{L} ≤ n. Hence we have homomorphisms

[F^{N−s_{L}}R_{+}(H_{∗}(MU)) ⊗ 𝔽_{p}{ϵ_{L}}]_{d} ↣ [F^{N−s_{L}}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d}
↠ [F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d}

taking ϵ_{L} to σm_{ℓ_{1}} · . . . · σm_{ℓ_{r}}. Taking the direct sum over L, and factoring through the isomorphism displayed above, we get the desired homomorphism ϕ_{n,d}.

For varying n, the collection {ϕ_{n,d}}_{n} defines a pro-map, such that f_{n,d} ∘ ϕ_{n,d} is equal to the structural surjection

F^{N}R_{+}(H_{∗}(THH(MU)))_{d} ↠ F^{n}R_{+}(H_{∗}(THH(MU)))_{d} ,

and ϕ_{n,d} ∘ f_{N,d} is equal to the structural surjection

[F^{N}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d} ↠ [F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d} .

Hence {f_{n,d}}_{n} is a pro-isomorphism, with pro-inverse {ϕ_{n,d}}_{n}, in each total degree d.

The proof for ĝ relies on similar filtration shift estimates. The associated graded of the filtration of F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1) was discussed in the first part of the proof. The part of the Tate spectral sequence Ê^{∞}-term

Ê_{∗,∗}^{∞} = E(u) ⊗ P (t, t^{−1}) ⊗ P (m_{ℓ}^{p} | ℓ ≥ 1) ⊗ E(m_{ℓ}^{p−1}σm_{ℓ} | ℓ ≥ 1)
⟹ H_{∗}^{c}(THH(MU)^{tC_{p}})

in Tate filtration s ≥ n is the associated graded of the filtration of F^{n}H_{∗}^{c}(THH(MU)^{tC_{p}}).

The 𝒜_{∗}-comodule homomorphism g_{n} identifies F^{n}R_{+}(H_{∗}(MU)) with the Tate filtration s ≥ n part of E(u) ⊗ P (t, t^{−1}) ⊗ P (m_{ℓ}^{p} | ℓ ≥ 1), taking m_{ℓ}^{⊗p} to m_{ℓ}^{p}. Furthermore, it takes σm_{ℓ} to γ_{∗}(σm_{ℓ}), represented by a sign times t^{(p−1)ℓ} ⊗m_{ℓ}^{p−1}σm_{ℓ} modulo filtration. Here t^{(p−1)ℓ} ⊗m_{ℓ}^{p−1}σm_{ℓ} lies in bidegree (−2(p − 1)ℓ, 2pℓ + 1), on the line of slope −p/(p − 1) through the point (s, t) = (0, 1).

In total degree d we have the strict pro-map {g_{n,d}}_{n}, with components

g_{n,d}: [F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d} ⟶ F^{n}H_{∗}^{c}(THH(MU)^{tC_{p}})_{d} .

We define an 𝔽_{p}-linear pro-inverse {ψ_{n,d}}_{n}, with components

ψ_{n,d}: F^{N}H_{∗}^{c}(THH(MU)^{tC_{p}})_{d} ⟶ [F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d} .

Here N = N(n, d) = p(n − d) + d, as in the f̂-case.

Write the source of the inclusion

[R_{+}(H_{∗}(MU)) ⊗ E(γ_{∗}(σm_{ℓ}) | ℓ ≥ 1)]_{d} ⟶ H_{∗}^{c}(THH(MU)^{tC_{p}})_{d}

as a direct sum

⨁_{L} [R_{+}(H_{∗}(MU)) ⊗ 𝔽_{p}{γ_{L}}]_{d}

with L = (ℓ_{1} < ⋯ < ℓ_{r}) as above and

γ_{L} = γ_{∗}(σm_{ℓ_{1}}) · . . . · γ_{∗}(σm_{ℓ_{r}})

in bidegree (s′_{L}, t′_{L}) = (−2(p − 1)|L|, 2p|L| + r). The inclusion descends to an isomorphism

⨁_{L} [F^{N−s′_{L}}R_{+}(H_{∗}(MU)) ⊗ 𝔽_{p}{γ_{L}}]_{d} ⟶^{≅} F^{N}H_{∗}^{c}(THH(MU)^{tC_{p}})_{d} .

Only the summands with (N −s′_{L})+(s′_{L} +t′_{L}) ≤ d are nonzero, and this implies 2|L| ≤ d−n since r ≥ 0. This, in turn, implies N − s′_{L} ≤ n. Hence we have homomorphisms

[F^{N−s′_{L}}R_{+}(H_{∗}(MU)) ⊗ 𝔽_{p}{γ_{L}}]_{d} ↣ [F^{N−s′_{L}}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d}
↠ [F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d}

taking γ_{L} to σm_{ℓ_{1}} · . . . · σm_{ℓ_{r}}. Summing over L, and using the isomorphism above, we get the required homomorphism ψ_{n,d}.

The collection {ψ_{n,d}}_{n} defines a pro-map, such that g_{n,d} ∘ ψ_{n,d} is equal to the structural surjection

F^{N}H_{∗}^{c}(THH(MU)^{tC_{p}})_{d} ↠ F^{n}H_{∗}^{c}(THH(MU)^{tC_{p}})_{d} ,

and ψ_{n,d} ∘ g_{N,d} is equal to the structural surjection

[F^{N}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d} ↠ [F^{n}R_{+}(H_{∗}(MU)) ⊗ E(σm_{ℓ} | ℓ ≥ 1)]_{d} .

Hence {g_{n,d}}_{n} is a pro-isomorphism. □

7.2. The case B = BP. In this case the central term in diagram (7.3) is

R_{+}(H_{∗}(BP)) ⊗_{H_{∗}(BP)} H_{∗}(THH(BP)) ≅ R_{+}(H_{∗}(BP)) ⊗ E(σξ̄_{k} | k ≥ 1) .

By Lemma 6.7 each σξ̄_{k} is 𝒜_{∗}-comodule primitive, so by [22, 3.2.1] we have ϵ_{∗}(σξ̄_{k}) = 1 ⊗ σξ̄_{k} in R_{+}(H_{∗}(THH(BP)). It has Tate representative

t^{m(2p^{k}−1)} ⊗ σξ̄_{k}^{⊗p}

in Tate filtration −(p − 1)(2p^{k} − 1), by [22, 5.14]. We showed in Theorem 6.9 that γ_{∗}(σξ̄_{k}) has Tate representative

t^{(p−1)(p^{k}−1)} ⊗ ξ̄_{k}^{p−1}σξ̄_{k}

in Tate filtration −2(p−1)(p^{k}−1). Since both images are in negative Tate filtration, we get a commutative diagram

\xymatrix{R_{+}(H_{∗}(THH(BP))) \ar@{->>}[d] & R_{+}(H_{∗}(BP)) ⊗ E(σξ̄_{k} | k ≥ 1) \ar[l]_{f} \ar[r]^{g} \ar@{->>}[d] & H_{∗}^{c}(THH(BP)^{tC_{p}}) \ar@{->>}[d] \\ F^{n}R_{+}(H_{∗}(THH(BP))) & F^{n}R_{+}(H_{∗}(BP)) ⊗ E(σξ̄_{k} | k ≥ 1) \ar[l]_{f_{n}} \ar[r]^{g_{n}} & F^{n}H_{∗}^{c}(THH(BP)^{tC_{p}}) .}

Theorem 2.2 for BP now follows from the proposition below by letting

Φ_{BP} = ĝ ∘ f̂^{−1} = (lim_{n} g_{n}) ∘ (lim_{n} f_{n})^{−1} .

Proposition 7.3. The homomorphisms

f_{n}: F^{n}R_{+}(H_{∗}(BP)) ⊗ E(σξ̄_{k} | k ≥ 1) ⟶ F^{n}R_{+}(H_{∗}(THH(BP)))

and

g_{n}: F^{n}R_{+}(H_{∗}(BP)) ⊗ E(σξ̄_{k} | k ≥ 1) ⟶ F^{n}H_{∗}^{c}(THH(BP)^{tC_{p}})

define strict maps {f_{n}}_{n} and {g_{n}}_{n} of inverse systems of bounded below 𝒜_{∗}-comodules of finite type. These are pro-isomorphisms, in each homological degree. Hence each of the limiting homomorphisms

f̂ = lim_{n} f_{n}: R_{+}(H_{∗}(BP)) ˆ⊗ E(σξ̄_{k} | k ≥ 1) ⟶ R_{+}(H_{∗}(THH(BP)))

and

ĝ = lim_{n} g_{n}: R_{+}(H_{∗}(BP)) ˆ⊗ E(σξ̄_{k} | k ≥ 1) ⟶ H_{∗}^{c}(THH(BP)^{tC_{p}})

is a continuous isomorphism of complete 𝒜_{∗}-comodules, with continuous inverse.

Proof. This is similar to the MU-case. □

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Department of Mathematics, University of Oslo, Norway

   E-mail address: sverreln@math.uio.no
   URL: http://folk.uio.no/sverreln

Department of Mathematics, University of Oslo, Norway

   E-mail address: rognes@math.uio.no
   URL: http://folk.uio.no/rognes