Algebraic K-theory of the first Morava K-theory

Christian Ausoni  John Rognes

3. February 2011

Abstract

For a prime p ≥ 5, we compute the algebraic K-theory modulo p and v_{1} of the mod p Adams summand, using topological cyclic homology. On the way, we evaluate its modulo p and v_{1} topological Hochschild homology. Using a localization sequence, we also compute the K-theory modulo p and v_{1} of the first Morava Ktheory.

Keywords. Algebraic K-theory, Morava K-theory, topological cyclic homology, topological Hochschild homology

1 Introduction

In this paper we continue the investigation from [AR02] and [Aus10] of the algebraic Ktheory of topological K-theory and related S-algebras. Let ℓ_{p} be the p-complete Adams summand of connective complex K-theory, and let ℓ/p = k(1) be the first connective Morava K-theory. It has a unique S-algebra structure [Ang, Th. A], and we show in Section 2 that ℓ/p is an ℓ_{p}-algebra (in uncountably many ways), so that K(ℓ/p) is a K(ℓ_{p})- module spectrum.

Let V (1) = S/(p, v_{1}) be the type 2 Smith–Toda complex (see Section 4 below for a definition). It is a homotopy commutative ring spectrum for p ≥ 5, with a preferred periodic class v_{2} ∈ V (1)_{∗} of degree 2p^{2} − 2. We write V (1)_{∗}(X) = π_{∗}(V (1) ∧ X) for the V (1)-homotopy of a spectrum X. Multiplication by v_{2} makes V (1)_{∗}(X) a P (v_{2})- module, where P (v_{2}) denotes the polynomial algebra over 𝔽_{p} generated by v_{2}. We denote by 𝔽_{p}{x_{1}, …, x_{n}} the 𝔽_{p}-vector space generated by x_{1}, …, x_{n}, and by E(x_{1}, …, x_{n}) the exterior algebra over 𝔽_{p} generated by x_{1}, …, x_{n}.

We computed the V (1)-homotopy of K(ℓ_{p}) in [AR02, Th. 9.1], showing that it is essentially a free P (v_{2})-module on (4p + 4) generators. The following is our main result, corresponding to Theorem 7.7 in the body of the paper.

Ch. Ausoni: Mathematical Institute, University of Münster, DE–48149 Münster, Germany; e-mail:

ausoni@uni-muenster.de

J. Rognes: Department of Mathematics, University of Oslo, NO–0316 Oslo, Norway; e-mail:

rognes@math.uio.no

Mathematics Subject Classification (2010): 19D55, 55N15

Theorem 1.1. Let p ≥ 5 be a prime and let ℓ/p = k(1) be the first connective Morava K-theory spectrum. There is an isomorphism of P (v_{2})-modules

V (1)_{∗}K(ℓ/p) ≅ P (v_{2}) ⊗ E(ϵ̄_{1}) ⊗ 𝔽_{p}{1, ∂λ_{2}, λ_{2}, ∂v_{2}}
⊕ P (v_{2}) ⊗ E(dlog v_{1}) ⊗ 𝔽_{p}{t^{d}v_{2} | 0 < d < p^{2} − p, p ∤ d}
⊕ P (v_{2}) ⊗ E(ϵ̄_{1}) ⊗ 𝔽_{p}{t^{dp}λ_{2} | 0 < d < p} .

Here |λ_{1}| = |ϵ̄_{1}| = 2p − 1, |λ_{2}| = 2p^{2} − 1, |v_{2}| = 2p^{2} − 2, | dlog v_{1}| = 1, |∂| = −1 and |t| = −2. This is a free P (v_{2})-module of rank (2p^{2} − 2p + 8) and of zero Euler characteristic.

We prove this theorem by means of the cyclotomic trace map [BHM93] to topological cyclic homology T C(ℓ/p; p). Along the way we evaluate V (1)_{∗}THH(ℓ/p), where THH denotes topological Hochschild homology, as well as V (1)_{∗}T C(ℓ/p; p), see Proposition 4.2 and Theorem 7.6.

Let L_{p} be the p-complete Adams summand of periodic complex K-theory, and let L/p = K(1) be the first periodic Morava K-theory. The localization cofiber sequence K(ℤ_{p}) → K(ℓ_{p}) → K(L_{p}) → ΣK(ℤ_{p}) of Blumberg and Mandell [BM08, p. 157] has the mod p Adams analogue

K(ℤ/p) → K(ℓ/p) → K(L/p) → ΣK(ℤ/p) ,

see Proposition 2.2 below. Using Quillen’s computation [Qui72, Th. 7] of K(ℤ/p), we obtain the following consequence:

Corollary 1.2. Let p ≥ 5 be a prime and let L/p = K(1) be the first Morava K-theory spectrum. There is an isomorphism of P (v_{2}^{±1})-modules

V (1)_{∗}K(L/p)[v_{2}^{−1}] ≅ V (1)_{∗}K(ℓ/p)[v_{2}^{−1}] .

If there is a class dlog v_{1} ∈ V (1)_{1}K(L/p) with λ_{2} = v_{2} · dlog v_{1}, then there is an isomorphism of P (v_{2})-modules

V (1)_{∗}K(L/p) ≅ P (v_{2}) ⊗ E(ϵ̄_{1}) ⊗ 𝔽_{p}{1, ∂λ_{2}, dlog v_{1}, ∂v_{2}}
⊕ P (v_{2}) ⊗ E(dlog v_{1}) ⊗ 𝔽_{p}{t^{d}v_{2} | 0 < d < p^{2} − p, p ∤ d}
⊕ P (v_{2}) ⊗ E(ϵ̄_{1}) ⊗ 𝔽_{p}{t^{dp}v_{2} dlog v_{1} | 0 < d < p} ,

where the degrees of the generators are as in Theorem 1.1. This is a free P (v_{2})-module of rank (2p^{2} − 2p + 8) and of zero Euler characteristic.

Our far-reaching aim, which partially motivated the computations presented here, is to conceptually understand the algebraic K-theory of ℓ_{p} and other commutative S-algebras in terms of localization and Galois descent, in the same way as we understand the algebraic K-theory of rings of integers in (local) number fields or more general regular rings. The first task is to relate K(ℓ_{p}) to the algebraic K-theory of its “residue fields” and “fraction field”, for which we expect a description in terms of Galois cohomology to exist, starting with the Galois theory for commutative S-algebras developed by the second author [Rog08]. The residue rings of ℓ_{p} appear to be ℓ/p, Hℤ_{p} and Hℤ/p, while the fraction field ff(ℓ_{p}) is more mysterious. For our purposes, its algebraic K-theory K(ff(ℓ_{p})) should fit in a natural localization cofibre sequence of spectra

K(L/p) → K(L_{p}) → K(ff(ℓ_{p})) → ΣK(L/p) .

An obvious candidate for ff(ℓ_{p}) is provided by the algebraic localization L_{p}[p^{−1}] = Lℚ_{p}, having as coefficients the graded field ℚ_{p}[v_{1}^{±1}]. However, by the following corollary, this is too naive.

Corollary 1.3. The spectra K(L/p), K(L_{p}) and K(Lℚ_{p}) cannot possibly fit in a cofibre sequence

K(L/p) → K(L_{p}) → K(Lℚ_{p}) → ΣK(L/p) .

Indeed, the above computation implies that V (1)_{∗}K(L/p)[v_{2}^{−1}] and V (1)_{∗}K(L_{p})[v_{2}^{−1}] are not abstractly isomorphic, while V (1)_{∗}K(Lℚ_{p})[v_{2}^{−1}] is zero since it is an algebra over V (1)_{∗}K(ℚ_{p})[v_{2}^{−1}] = 0. The later equality follows from the computation of the p-primary homotopy type of K(ℚ_{p}) [HM03, Th. D], which shows that V (1)_{∗}K(ℚ_{p}) is v_{2}-torsion.

In conclusion, the conjectural fraction field ff(ℓ_{p}) appears to be a localization of L_{p} away from L/p less drastic than the algebraic localization L_{p}[p^{−1}] = Lℚ_{p}. We elaborate more on this issue in [AR].

The paper is organized as follows. In Section 2 we fix our notations, show that ℓ/p admits the structure of an associative ℓ_{p}-algebra, and give a similar discussion for ku/p and the periodic versions L/p and KU/p. Section 3 contains the computation of the mod p homology of THH(ℓ/p), and in Section 4 we evaluate its V (1)-homotopy. In Section 5 we show that the C_{p^{n}}-fixed points and C_{p^{n}}-homotopy fixed points of THH(ℓ/p) are closely related, and use this to inductively determine their V (1)-homotopy in Section 6. Finally, in Section 7 we achieve the computation of T C(ℓ/p; p) and K(ℓ/p) in V (1)-homotopy.

Notations and conventions. Let p be fixed prime. We write E(x) = 𝔽_{p}[x]/(x^{2}) for the exterior algebra, P (x) = 𝔽_{p}[x] for the polynomial algebra and P (x^{±1}) = 𝔽_{p}[x, x^{−1}] for the Laurent polynomial algebra on one generator x, and similarly for a list of generators. We will also write Γ(x) = 𝔽_{p}{γ_{i}(x) | i ≥ 0} for the divided power algebra, with γ_{i}(x) · γ_{j}(x) = (i, j)γ_{i+j}(x), where (i, j) = (i + j)!/i!j! is the binomial coefficient. We use the obvious abbreviations γ_{0}(x) = 1 and γ_{1}(x) = x. Finally, we write P_{h}(x) = 𝔽_{p}[x]/(x^{h}) for the truncated polynomial algebra of height h, and recall the isomorphism Γ(x) ≅ P_{p}(γ_{p^{e}}(x) | e ≥ 0) in characteristic p. We write X_{(p)} and X_{p} for the p-localization and the p-completion, respectively, of any spectrum or abelian group X. In the spectral sequences (of 𝔽_{p}-modules) discussed below, we often determine differentials only up to multiplication by a unit. We use the notation d(x) ≐ y to indicate that the equation d(x) = αy holds for some unit α ∈ 𝔽_{p}.

2 Base change squares of S-algebras

Let p be a prime, even or odd for now. Let ku and KU be the connective and the periodic complex K-theory spectra, with homotopy rings ku_{∗} = ℤ[u] and KU_{∗} = ℤ[u^{±1}], where |u| = 2. Let ℓ = BP ⟨1⟩ and L = E(1) be the p-local Adams summands, with ℓ_{∗} = ℤ_{(p)}[v_{1}] and L_{∗} = ℤ_{(p)}[v_{1}^{±1}], where |v_{1}| = 2p − 2. The inclusion ℓ → ku_{(p)} maps v_{1} to u^{p−1}. Alternate notations in the p-complete cases are KU_{p} = E_{1} and L_{p} = { \atop E(1)}. These ring spectra are all commutative S-algebras, in the sense that each admits a unique E_{∞} ring spectrum structure. See [BR05, p. 692] for proofs of uniqueness in the periodic cases.

Let ku/p and KU/p be the connective and periodic mod p complex K-theory spectra, with coefficients (ku/p)_{∗} = ℤ/p[u] and (KU/p)_{∗} = ℤ/p[u^{±1}]. These are 2-periodic versions of the first Morava K-theory spectra ℓ/p = k(1) and L/p = K(1), with (ℓ/p)_{∗} = ℤ/p[v_{1}] and (L/p)_{∗} = ℤ/p[v_{1}^{±1}]. Each of these can be constructed as the cofiber of the multiplication by p map, as a module over the corresponding commutative S-algebra. For example, there is a cofiber sequence of ku-modules ku ⟶^{p} ku ⟶^{i} ku/p → Σku.

Let HR be the Eilenberg–Mac Lane spectrum of a ring R. When R is associative, HR admits a unique associative S-algebra structure, and when R is commutative, HR admits a unique commutative S-algebra structure. The zeroth Postnikov section defines unique maps of commutative S-algebras π: ku → Hℤ and π: ℓ → Hℤ_{(p)}, which can be followed by unique commutative S-algebra maps to Hℤ/p.

The ku-module spectrum ku/p does not admit the structure of a commutative kualgebra. It cannot even be an E_{2} or H_{2} ring spectrum, since the homomorphism induced in mod p homology by the resulting map π: ku/p → Hℤ/p of H_{2} ring spectra would not commute with the homology operation Q^{1}(τ̄_{0}) = τ̄_{1} in the target H_{∗}(Hℤ/p; 𝔽_{p}) [BMMS86, III.2.3]. Similar remarks apply for KU/p, ℓ/p and L/p. Associative algebra structures, or A_{∞} ring spectrum structures, are easier to come by. The following result is a direct application of the methods of [Laz01, §§9–11]. We adapt the notation of [BJ02, §3] to provide some details in our case.

Proposition 2.1. The ku-module spectrum ku/p admits the structure of an associative ku-algebra, but the structure is not unique. Similar statements hold for KU/p as a KUalgebra, ℓ/p as an ℓ-algebra and L/p as an L-algebra.

Proof. We construct ku/p as the (homotopy) limit of its Postnikov tower of associative ku-algebras P^{2m−2} = ku/(p, u^{m}), with coefficient rings ku/(p, u^{m})_{∗} = ku_{∗}/(p, u^{m}) for m ≥ 1. To start the induction, P^{0} = Hℤ/p is a ku-algebra via i∘π: ku → Hℤ → Hℤ/p. Assume inductively for m ≥ 1 that P = P^{2m−2} has been constructed. We will define P^{2m} by a (homotopy) pullback diagram

\xymatrix{P^{2m} \ar[r] \ar[d] & P \ar[d]^{in_{1}} \\ P \ar[r]^{d} & P ∨ Σ^{2m+1}Hℤ/p}

in the category of associative ku-algebras. Here

d ∈ ADer_{ku}^{2m+1}(P, Hℤ/p) ≅ THH_{ku}^{2m+2}(P, Hℤ/p)

is an associative ku-algebra derivation of P with values in Σ^{2m+1}Hℤ/p, and the group of such can be identified with the indicated topological Hochschild cohomology group of P over ku. We recall that these are the homotopy groups (cohomologically graded) of the function spectrum F_{P ∧_{ku}P^{op}}(P, Hℤ/p). The composite map pr_{2} ∘ d: P → Σ^{2m+1}Hℤ/p of ku-modules, where pr_{2} projects onto the second wedge summand, is restricted to equal the ku-module Postnikov k-invariant of ku/p in

H_{ku}^{2m+1}(P; ℤ/p) = π_{0}F_{ku}(P, Σ^{2m+1}Hℤ/p) .

We compute that π_{∗}(P ∧_{ku} P^{op}) = ku_{∗}/(p, u^{m}) ⊗ E(τ_{0}, τ_{1,m}), where |τ_{0}| = 1, |τ_{1,m}| = 2m+1 and E(−) denotes the exterior algebra on the given generators. (For p = 2, the use of the opposite product is essential here [Ang08, §3].) The function spectrum description of topological Hochschild cohomology leads to the spectral sequence

E_{2}^{∗,∗} = Ext_{π_{∗}(P ∧_{ku}P^{op})}^{∗,∗}(π_{∗}(P), ℤ/p)
≅ ℤ/p[y_{0}, y_{1,m}]
⟹ THH_{ku}^{∗}(P, Hℤ/p) ,

where y_{0} and y_{1,m} have cohomological bidegrees (1, 1) and (1, 2m + 1), respectively. The spectral sequence collapses at E_{2} = E_{∞}, since it is concentrated in even total degrees. In particular,

ADer_{ku}^{2m+1}(P, Hℤ/p) ≅ 𝔽_{p}{y_{1,m}, y_{0}^{m+1}} .

Additively, H_{ku}^{2m+1}(P; ℤ/p) ≅ 𝔽_{p}{Q_{1,m}} is generated by a class dual to τ_{1,m}, which is the image of y_{1,m} under left composition with pr_{2}. It equals the ku-module k-invariant of ku/p. Thus there are precisely p choices d = y_{1,m} + αy_{0}^{m+1}, with α ∈ 𝔽_{p}, for how to extend any given associative ku-algebra structure on P = P^{2m−2} to one on P^{2m} = ku/(p, u^{m+1}). In the limit, we find that there are an uncountable number of associative ku-algebra structures on ku/p = holim_{m} P^{2m}, each indexed by a sequence of choices α ∈ 𝔽_{p} for all m ≥ 1.

The periodic spectrum KU/p can be obtained from ku/p by Bousfield KU-locali-zation in the category of ku-modules [EKMM97, VIII.4], which makes it an associative KU-algebra. The classification of periodic S-algebra structures is the same as in the connective case, since the original ku-algebra structure on ku/p can be recovered from that on KU/p by a functorial passage to the connective cover. To construct ℓ/p as an associative ℓ-algebra, or L/p as an associative L-algebra, replace u by v_{1} in these arguments.

By varying the ground S-algebra, we obtain the same conclusions about ku/p as a ku_{(p)}-algebra or ku_{p}-algebra, and about ℓ/p as an ℓ_{p}-algebra.

For each choice of ku-algebra structure on ku/p, the zeroth Postnikov section

π: ku/p → Hℤ/p

is a ku-algebra map, with the unique ku-algebra structure on the target. Hence there is a commutative square of associative ku-algebras

\xymatrix{ku \ar[r]^{i} \ar[d]^{π} & ku/p \ar[d]^{π} \\ Hℤ \ar[r]^{i} & Hℤ/p}

and similarly in the p-local and p-complete cases. In view of the weak equivalence Hℤ∧_{ku} ku/p ≃ Hℤ/p, this square expresses the associative Hℤ-algebra Hℤ/p as the base change of the associative ku-algebra ku/p along π: ku → Hℤ. Likewise, there is a commutative square of associative ℓ_{p}-algebras

\xymatrix{ℓ_{p} \ar[r]^{i} \ar[d]^{π} & ℓ/p \ar[d]^{π} \\ Hℤ_{p} \ar[r]^{i} & Hℤ/p}  (2.1)

that expresses Hℤ/p as the base change of ℓ/p along ℓ_{p} → Hℤ_{p}, and similarly in the p-local case. By omission of structure, these squares are also diagrams of S-algebras and S-algebra maps.

We end this section by formulating the mod p analogue of the localization cofibre sequence in algebraic K-theory

K(ℤ_{p}) → K(ℓ_{p}) → K(L_{p}) → ΣK(ℤ_{p})  (2.2)

conjectured by the second author and established by Blumberg and Mandell [BM08, p. 157]

Proposition 2.2. There is a localization cofibre sequence of spectra

K(ℤ/p) → K(ℓ/p) → K(L/p) → ΣK(ℤ/p)

where the first map is the transfer and the second map is induced by the localization ℓ/p → ℓ/p[v_{1}^{−1}] = L/p.

Proof. The proof of the existence of the localization sequence (2.2) given in [BM08, p. 160–163] and the identification of the transfer map adapt without change to cover the mod p analogue stated in this proposition. Here we use that a finite cell ℓ/p-module that is v_{1}-torsion has finite homotopy groups, and the non-zero groups are concentrated in a finite range of degrees.

3 Topological Hochschild homology

We shall compute the V (1)-homotopy of the topological Hochschild homology THH(−) and topological cyclic homology T C(−; p) of the S-algebras in diagram (2.1), for primes p ≥ 5. Passing to connective covers, this also computes the V (1)-homotopy of the algebraic K-theory spectra appearing in that square. With these coefficients, or more generally, after p-adic completion, the functors THH and T C are insensitive to p-completion in the argument, so we shall simplify the notation slightly by working with the associative S-algebras ℓ and Hℤ_{(p)} in place of ℓ_{p} and Hℤ_{p}. For ordinary rings R we almost always shorten notations like THH(HR) to THH(R).

The computations follow the strategy of [Bök], [BM94], [BM95] and [HM97] for Hℤ/p and Hℤ, and of [MS93] and [AR02] for ℓ. See also [AR05, §§4–7] for further discussion of the THH-part of such computations. In this section we shall compute the mod p homology of the topological Hochschild homology of ℓ/p as a module over the corresponding homology for ℓ, for any odd prime p.

Remark 3.1. Our computations are based on comparisons, using the maps displayed in diagram (2.1) above. We will abuse notation and use the same name for classes in the homology or V (1)-homotopy of THH(ℓ_{p}), THH(ℓ/p), THH(ℤ_{p}) or THH(ℤ/p), when these classes unambiguously correspond to each other under the homomorphisms induced by the maps i and π in (2.1). We also use this abuse of notations in later sections for the V (1)-homotopy of T C, etc.

We write H_{∗}(−) for homology with mod p coefficients. It takes values in graded A_{∗}- comodules, where A_{∗} is the dual Steenrod algebra [Mil58, Th 2]. Explicitly (for p odd),

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

with coproduct

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

and

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

Here ξ̄_{0} = 1, ξ̄_{k} = χ(ξ_{k}) has degree 2(p^{k} − 1) and τ̄_{k} = χ(τ_{k}) has degree 2p^{k} − 1, where χ is the canonical conjugation [MM65, 8.4]. Then the maps i and the zeroth Postnikov sections π of (2.1) induce identifications

\begin{aligned}H_{∗}(Hℤ_{(p)}) & = P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ 1) \\ H_{∗}(ℓ) & = P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ 2) \\ H_{∗}(ℓ/p) & = P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{0}, τ̄_{k} | k ≥ 2) \end{aligned}

as A_{∗}-comodule subalgebras of H_{∗}(Hℤ/p) = A_{∗}. We often make use of the following A_{∗}-comodule coactions

\begin{aligned}ν(τ̄_{0}) & = 1 ⊗ τ̄_{0} + τ̄_{0} ⊗ 1 \\ ν(ξ̄_{1}) & = 1 ⊗ ξ̄_{1} + ξ̄_{1} ⊗ 1 \\ ν(τ̄_{1}) & = 1 ⊗ τ̄_{1} + τ̄_{0} ⊗ ξ̄_{1} + τ̄_{1} ⊗ 1 \\ ν(ξ̄_{2}) & = 1 ⊗ ξ̄_{2} + ξ̄_{1} ⊗ ξ̄_{1}^{p} + ξ̄_{2} ⊗ 1 \\ ν(τ̄_{2}) & = 1 ⊗ τ̄_{2} + τ̄_{0} ⊗ ξ̄_{2} + τ̄_{1} ⊗ ξ̄_{1}^{p} + τ̄_{2} ⊗ 1 . \end{aligned}

The Bökstedt spectral sequences

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

for the commutative S-algebras B = Hℤ/p, Hℤ_{(p)} and ℓ begin

\begin{aligned}E^{2}(ℤ/p) & = A_{∗} ⊗ E(σξ̄_{k} | k ≥ 1) ⊗ Γ(στ̄_{k} | k ≥ 0) \\ E^{2}(ℤ_{(p)}) & = H_{∗}(Hℤ_{(p)}) ⊗ E(σξ̄_{k} | k ≥ 1) ⊗ Γ(στ̄_{k} | k ≥ 1) \\ E^{2}(ℓ) & = H_{∗}(ℓ) ⊗ E(σξ̄_{k} | k ≥ 1) ⊗ Γ(στ̄_{k} | k ≥ 2) . \end{aligned}

Here HH_{∗}(H_{∗}(B)) denotes the Hochschild homology of the graded 𝔽_{p}-algebra H_{∗}(B). In the above formula we made use of the 𝔽_{p}-linear operator σ: H_{∗}(B) → HH_{1}(H_{∗}(B)), x ↦ σx, where σx is the class represented by 1 ⊗ x − x ⊗ 1 in the Hochschild complex. Notice that σ is the restriction of Connes’ operator d to HH_{0}(H_{∗}(B)) = H_{∗}(B), and is a derivation in the sense that

σ(xy) = xσ(y) + (−1)^{|x||y|}yσ(x)

for all x, y ∈ H_{∗}(B). These spectral sequences are (graded) commutative A_{∗}-comodule algebra spectral sequences, and there are differentials

d^{p−1}(γ_{j}στ̄_{k}) ≐ σξ̄_{k+1} · γ_{j−p}στ̄_{k}

for j ≥ p and k ≥ 0, see [Bök, Lem. 1.3], [Hun96, Th. 1] or [Aus05, Lem. 5.3], leaving

\begin{aligned}E^{∞}(ℤ/p) & = A_{∗} ⊗ P_{p}(στ̄_{k} | k ≥ 0) \\ E^{∞}(ℤ_{(p)}) & = H_{∗}(Hℤ_{(p)}) ⊗ E(σξ̄_{1}) ⊗ P_{p}(στ̄_{k} | k ≥ 1) \\ E^{∞}(ℓ) & = H_{∗}(ℓ) ⊗ E(σξ̄_{1}, σξ̄_{2}) ⊗ P_{p}(στ̄_{k} | k ≥ 2) . \end{aligned}

The inclusion of 0-simplices η: B → THH(B) is split for commutative B by the augmentation ϵ: THH(B) → B. Thus there are unique representatives in Bökstedt filtration 1, with zero augmentation, for each of the classes σx. There are multiplicative extensions (στ̄_{k})^{p} = στ̄_{k+1} for k ≥ 0, see [AR05, Prop. 5.9], so

\begin{aligned}H_{∗}(THH(ℤ/p)) & = A_{∗} ⊗ P (στ̄_{0}) \\ H_{∗}(THH(ℤ_{(p)})) & = H_{∗}(Hℤ_{(p)}) ⊗ E(σξ̄_{1}) ⊗ P (στ̄_{1}) \\ H_{∗}(THH(ℓ)) & = H_{∗}(ℓ) ⊗ E(σξ̄_{1}, σξ̄_{2}) ⊗ P (στ̄_{2}) \end{aligned}

as A_{∗}-comodule algebras. The A_{∗}-comodule coactions are given by

\begin{aligned}ν(στ̄_{0}) & = 1 ⊗ στ̄_{0} \\ ν(σξ̄_{1}) & = 1 ⊗ σξ̄_{1} \\ ν(στ̄_{1}) & = 1 ⊗ στ̄_{1} + τ̄_{0} ⊗ σξ̄_{1} \\ ν(σξ̄_{2}) & = 1 ⊗ σξ̄_{2} \\ ν(στ̄_{2}) & = 1 ⊗ στ̄_{2} + τ̄_{0} ⊗ σξ̄_{2} . \end{aligned}  (3.1)

The natural map π_{∗}: THH(ℓ) → THH(ℤ_{(p)}) induced by π: ℓ → ℤ_{(p)} takes σξ̄_{2} to 0 and στ̄_{2} to (στ̄_{1})^{p}. The natural map i_{∗}: THH(ℤ_{(p)}) → THH(ℤ/p) induced by i: ℤ_{(p)} → ℤ/p takes σξ̄_{1} to 0 and στ̄_{1} to (στ̄_{0})^{p}.

The Bökstedt spectral sequence for the associative S-algebra B = ℓ/p begins

E^{2}(ℓ/p) = H_{∗}(ℓ/p) ⊗ E(σξ̄_{k} | k ≥ 1) ⊗ Γ(στ̄_{0}, στ̄_{k} | k ≥ 2) .

It is an A_{∗}-comodule module spectral sequence over the Bökstedt spectral sequence for ℓ, since the ℓ-algebra multiplication ℓ ∧ ℓ/p → ℓ/p is a map of associative S-algebras. However, it is not itself an algebra spectral sequence, since the product on ℓ/p is not commutative enough to induce a natural product structure on THH(ℓ/p). Nonetheless, we will use the algebra structure present at the E^{2}-term to help in naming classes.

The map π: ℓ/p → Hℤ/p induces an injection of Bökstedt spectral sequence E^{2}- terms, so there are differentials generated algebraically by

d^{p−1}(γ_{j}στ̄_{k}) ≐ σξ̄_{k+1} · γ_{j−p}στ̄_{k}

for j ≥ p, k = 0 or k ≥ 2, leaving

E^{∞}(ℓ/p) = H_{∗}(ℓ/p) ⊗ E(σξ̄_{2}) ⊗ P_{p}(στ̄_{0}, στ̄_{k} | k ≥ 2)  (3.2)

as an A_{∗}-comodule module over E^{∞}(ℓ). In order to obtain H_{∗}(THH(ℓ/p)), we need to resolve the A_{∗}-comodule and H_{∗}(THH(ℓ))-module extensions. This is achieved in Lemma 3.3 below.

The natural map π_{∗}: E^{∞}(ℓ/p) → E^{∞}(ℤ/p) is an isomorphism in total degrees ≤ (2p − 2) and injective in total degrees ≤ (2p^{2} − 2). The first class in the kernel is σξ̄_{2}. Hence there are unique classes

1 , τ̄_{0} , στ̄_{0} , τ̄_{0}στ̄_{0} , …, (στ̄_{0})^{p−1}

in degrees 0 ≤ ∗ ≤ 2p − 2 of H_{∗}(THH(ℓ/p)), mapping to classes with the same names in H_{∗}(THH(ℤ/p)). More concisely, these are the monomials τ̄_{0}^{δ}(στ̄_{0})^{i} for 0 ≤ δ ≤ 1 and 0 ≤ i ≤ p − 1, except that the degree (2p − 1) case (δ, i) = (1, p − 1) is omitted. The A_{∗}-comodule coaction on these classes is given by the same formulas in H_{∗}(THH(ℓ/p)) as in H_{∗}(THH(ℤ/p)), cf. (3.1).

There is also a class ξ̄_{1} in degree (2p − 2) of H_{∗}(THH(ℓ/p)) mapping to a class with the same name, and same A_{∗}-coaction, in H_{∗}(THH(ℤ/p)).

In degree (2p − 1), π_{∗} is a map of extensions from

0 → 𝔽_{p}{ξ̄_{1}τ̄_{0}} → H_{2p−1}(THH(ℓ/p)) → 𝔽_{p}{τ̄_{0}(στ̄_{0})^{p−1}} → 0

to

0 → 𝔽_{p}{τ̄_{1}, ξ̄_{1}τ̄_{0}} → H_{2p−1}(THH(ℤ/p)) → 𝔽_{p}{τ̄_{0}(στ̄_{0})^{p−1}} → 0 .

The latter extension is canonically split by the augmentation ϵ: THH(ℤ/p) → Hℤ/p, which uses the commutativity of the S-algebra Hℤ/p.

In degree 2p, the map π_{∗} goes from

H_{2p}(THH(ℓ/p)) = 𝔽_{p}{ξ̄_{1}στ̄_{0}}

to

0 → 𝔽_{p}{τ̄_{0}τ̄_{1}} → H_{2p}(THH(ℤ/p)) → 𝔽_{p}{στ̄_{1}, ξ̄_{1}στ̄_{0}} → 0 .

Again the latter extension is canonically split.

Lemma 3.2. There is a unique class y in H_{2p−1}(THH(ℓ/p)) represented by τ̄_{0}(στ̄_{0})^{p−1} in E_{p−1,p}^{∞}(ℓ/p) and mapped by π_{∗} to τ̄_{0}(στ̄_{0})^{p−1} − τ̄_{1} in H_{∗}(THH(ℤ/p)).

Proof. This follows from naturality of the suspension operator σ and the multiplicative relation (στ̄_{0})^{p} = στ̄_{1} in H_{∗}(THH(ℤ/p)). A class y in H_{2p−1}(THH(ℓ/p)) represented by τ̄_{0}(στ̄_{0})^{p−1} is determined modulo ξ̄_{1}τ̄_{0}. Its image in H_{2p−1}(THH(ℤ/p)) thus has the form ατ̄_{1} + τ̄_{0}(στ̄_{0})^{p−1} modulo ξ̄_{1}τ̄_{0}, for some α ∈ 𝔽_{p}. The suspension σy lies in H_{2p}(THH(ℓ/p)) = 𝔽_{p}{ξ̄_{1}στ̄_{0}}, so its image in H_{2p}(THH(ℤ/p)) is 0 modulo τ̄_{0}τ̄_{1} and ξ̄_{1}στ̄_{0}. It is also the suspension of ατ̄_{1} + τ̄_{0}(στ̄_{0})^{p−1} modulo ξ̄_{1}τ̄_{0}, which equals σ(ατ̄_{1}) + (στ̄_{0})^{p} = (α + 1)στ̄_{1}. In particular, the coefficient (α + 1) of στ̄_{1} is 0, so α = −1.

Let

H_{∗}(THH(ℓ))/(σξ̄_{1}) = H_{∗}(ℓ) ⊗ E(σξ̄_{2}) ⊗ P (στ̄_{2})

denote the quotient algebra of H_{∗}(THH(ℓ)) by the ideal generated by σξ̄_{1}.

Lemma 3.3. The classes

1 , τ̄_{0} , στ̄_{0} , τ̄_{0}στ̄_{0} , …, (στ̄_{0})^{p−1} , τ̄_{0}(στ̄_{0})^{p−1} ,

in E^{∞}(ℓ/p) represent unique homology classes in H_{∗}(THH(ℓ/p)), which by abuse of notation will be denoted

1 , τ̄_{0} , στ̄_{0} , τ̄_{0}στ̄_{0} , …, (στ̄_{0})^{p−1} , y ,

mapping under π_{∗} to classes with the same names in H_{∗}(THH(ℤ/p)), except for y, which maps to

τ̄_{0}(στ̄_{0})^{p−1} − τ̄_{1} .

The graded H_{∗}(THH(ℓ))-module H_{∗}(THH(ℓ/p)) is a free H_{∗}(THH(ℓ))/(σξ̄_{1})-module of rank 2p generated by these classes in degrees 0 through 2p − 1:

H_{∗}(THH(ℓ/p)) = H_{∗}(THH(ℓ))/(σξ̄_{1}) ⊗ 𝔽_{p}{1, τ̄_{0}, στ̄_{0}, τ̄_{0}στ̄_{0}, …, (στ̄_{0})^{p−1}, y} .

The A_{∗}-comodule coactions are given by

ν((στ̄_{0})^{i}) = 1 ⊗ (στ̄_{0})^{i}

for 0 ≤ i ≤ p − 1,

ν(τ̄_{0}(στ̄_{0})^{i}) = 1 ⊗ τ̄_{0}(στ̄_{0})^{i} + τ̄_{0} ⊗ (στ̄_{0})^{i}

for 0 ≤ i ≤ p − 2, and

ν(y) = 1 ⊗ y + τ̄_{0} ⊗ (στ̄_{0})^{p−1} − τ̄_{0} ⊗ ξ̄_{1} − τ̄_{1} ⊗ 1 .

Proof. H_{∗}(ℓ/p) is freely generated as a module over H_{∗}(ℓ) by 1 and τ̄_{0}, and the classes σξ̄_{2} and στ̄_{2} in H_{∗}(THH(ℓ)) induce multiplication by the same symbols in E^{∞}(ℓ/p), as given in (3.2). This generates all of E^{∞}(ℓ/p) from the 2p classes τ̄_{0}^{δ}(στ̄_{0})^{i} for 0 ≤ δ ≤ 1 and 0 ≤ i ≤ p − 1.

We claim that multiplication by σξ̄_{1} acts trivially on H_{∗}(THH(ℓ/p)). It suffices to verify this on the module generators τ̄_{0}^{δ}(στ̄_{0})^{i}, for which the product with σξ̄_{1} remains in the range of degrees where the map to H_{∗}(THH(ℤ/p)) is injective. The action of σξ̄_{1} is trivial on H_{∗}(THH(ℤ/p)), since d^{p−1}(γ_{p}στ̄_{0}) ≐ σξ̄_{1} and ϵ(σξ̄_{1}) = 0, and this implies the claim.

The A_{∗}-comodule coaction on each module generator, including y, is determined by that on its image under π_{∗}. In the latter case, for example, we have

(1 ⊗ π_{∗})(ν(y)) = ν(π_{∗}(y)) = ν(τ̄_{0}(στ̄_{0})^{p−1} − τ̄_{1})
= 1 ⊗ τ̄_{0}(στ̄_{0})^{p−1} + τ̄_{0} ⊗ (στ̄_{0})^{p−1} − 1 ⊗ τ̄_{1} − τ̄_{0} ⊗ ξ̄_{1} − τ̄_{1} ⊗ 1
= (1 ⊗ π_{∗})(1 ⊗ y + τ̄_{0} ⊗ (στ̄_{0})^{p−1} − τ̄_{0} ⊗ ξ̄_{1} − τ̄_{1} ⊗ 1) ,

and this proves our formula for ν(y) since 1 ⊗ π_{∗} is injective in this degree.

Remark 3.4. Notice that Lemma 3.3 implies that for different choices of ℓ-module structure on ℓ/p, the resulting homology groups H_{∗}(THH(ℓ/p)) are (abstractly) isomorphic as graded H_{∗}(THH(ℓ))-modules and A_{∗}-comodules.

4 Passage to V (1)-homotopy

For p ≥ 5 the Smith–Toda complex V (1) = S ∪_{p} e^{1} ∪_{α_{1}} e^{2p−1} ∪_{p} e^{2p} is a homotopy commutative ring spectrum [Smi70, Th 5.1], [Oka84, Ex. 4.5]. It is defined as the mapping cone of the Adams self-map v_{1} : Σ^{2p−2}V (0) → V (0) of the mod p Moore spectrum V (0) = S ∪_{p} e^{1}. Hence there is a cofiber sequence

Σ^{2p−2}V (0) ⟶^{v_{1}} V (0) ⟶^{i_{1}} V (1) ⟶^{j_{1}} Σ^{2p−1}V (0) .

There are some choices of orientations involved in fixing such an exact triangle, compare for instance with [HM03, Sect. 2.1]. The composite map β_{1,1} = i_{1}j_{1} : V (1) → Σ^{2p−1}V (1) defines the primary v_{1}-Bockstein homomorphism, acting naturally on V (1)_{∗}(X).

In this section we compute V (1)_{∗}THH(ℓ/p) as a module over V (1)_{∗}THH(ℓ), for any prime p ≥ 5. The unique ring spectrum map from V (1) to Hℤ/p induces the identification

H_{∗}(V (1)) = E(τ_{0}, τ_{1})

(no conjugations) as A_{∗}-comodule subalgebras of A_{∗}, see [Tod71, §4]. Here

\begin{aligned}ν(τ_{0}) & = 1 ⊗ τ_{0} + τ_{0} ⊗ 1 \\ ν(τ_{1}) & = 1 ⊗ τ_{1} + ξ_{1} ⊗ τ_{0} + τ_{1} ⊗ 1 . \end{aligned}

A form of the following lemma goes back to [Whi62, p. 271].

Lemma 4.1. Let M be any Hℤ/p-module spectrum. Then M is equivalent to a wedge sum of suspensions of Hℤ/p. Hence H_{∗}(M) is a sum of shifted copies of A_{∗} as an A_{∗}- comodule, and the Hurewicz homomorphism π_{∗}(M) → H_{∗}(M) identifies π_{∗}(M) with the A_{∗}-comodule primitives in H_{∗}(M).

Proof. The module action map λ: Hℤ/p ∧ M → M is a retraction, so π_{∗}(M) is a direct summand of π_{∗}(Hℤ/p ∧ M) = H_{∗}(M), hence is a graded ℤ/p-vector space. Choose maps α: S^{n} → M that represent a basis for this vector space. The wedge sum of the maps λ∘(1∧α): Σ^{n}Hℤ/p = Hℤ/p∧S^{n} → M is the desired π_{∗}-isomorphism ⋁_{α} Σ^{n}Hℤ/p → M.

For each ℓ-algebra B, V (1) ∧ THH(B) is a module spectrum over V (1) ∧ THH(ℓ) and thus over V (1) ∧ ℓ ≃ Hℤ/p, so H_{∗}(V (1) ∧ THH(B)) is a sum of copies of A_{∗} as an A_{∗}-comodule, by Lemma 4.1. In particular, V (1)_{∗}THH(B) = π_{∗}(V (1) ∧ THH(B)) is naturally identified with the subgroup of A_{∗}-comodule primitives in

H_{∗}(V (1) ∧ THH(B)) ≅ H_{∗}(V (1)) ⊗ H_{∗}(THH(B))

with the diagonal A_{∗}-comodule coaction. We write v ∧ x for the image of v ⊗ x under this identification, with v ∈ H_{∗}(V (1)) and x ∈ H_{∗}(THH(B)). Let

\begin{aligned}ϵ_{0} & = 1 ∧ τ̄_{0} + τ_{0} ∧ 1 \\ ϵ_{1} & = 1 ∧ τ̄_{1} + τ_{0} ∧ ξ̄_{1} + τ_{1} ∧ 1 \\ λ_{1} & = 1 ∧ σξ̄_{1} \\ λ_{2} & = 1 ∧ σξ̄_{2} \\ μ_{0} & = 1 ∧ στ̄_{0} \\ μ_{1} & = 1 ∧ στ̄_{1} + τ_{0} ∧ σξ̄_{1} \\ μ_{2} & = 1 ∧ στ̄_{2} + τ_{0} ∧ σξ̄_{2} . \end{aligned}  (4.1)

These are all A_{∗}-comodule primitive, when defined, in H_{∗}(V (1) ∧ THH(B)) for B = ℓ, ℓ/p, Hℤ_{p} or Hℤ/p (see Remark 3.1). By a dimension count,

\begin{aligned}V (1)_{∗}THH(ℤ/p) & = E(ϵ_{0}, ϵ_{1}) ⊗ P (μ_{0}) \\ V (1)_{∗}THH(ℤ_{(p)}) & = E(ϵ_{1}) ⊗ E(λ_{1}) ⊗ P (μ_{1}) \\ V (1)_{∗}THH(ℓ) & = E(λ_{1}, λ_{2}) ⊗ P (μ_{2}) \end{aligned}

as commutative 𝔽_{p}-algebras. The map π: ℓ → Hℤ_{(p)} takes λ_{2} to 0 and μ_{2} to μ_{1}^{p}. The map i: Hℤ_{(p)} → Hℤ/p takes λ_{1} to 0 and μ_{1} to μ_{0}^{p}. Note that μ_{2} ∈ V (1)_{2p^{2}}THH(ℓ) was simply denoted μ in [AR02].

In degrees ≤ (2p − 2) of H_{∗}(V (1) ∧ THH(ℓ/p)) the classes

μ_{0}^{i} := 1 ∧ (στ̄_{0})^{i}  (4.2)

for 0 ≤ i ≤ p − 1 and

ϵ_{0}μ_{0}^{i} := 1 ∧ τ̄_{0}(στ̄_{0})^{i} + τ_{0} ∧ (στ̄_{0})^{i}  (4.3)

for 0 ≤ i ≤ p − 2 are A_{∗}-comodule primitive, hence lift uniquely to V (1)_{∗}THH(ℓ/p). These map to the classes ϵ_{0}^{δ}μ_{0}^{i} in V (1)_{∗}THH(ℤ/p) for 0 ≤ δ ≤ 1 and 0 ≤ i ≤ p − 1, except that the degree bound excludes the top case of ϵ_{0}μ_{0}^{p−1}.

In degree (2p−1) of H_{∗}(V (1)∧THH(ℓ/p)) we have generators 1∧ξ̄_{1}τ̄_{0}, τ_{0}∧(στ̄_{0})^{p−1}, τ_{0} ∧ ξ̄_{1}, τ_{1} ∧ 1 and 1 ∧ y. These have coactions

\begin{aligned}ν(1 ∧ ξ̄_{1}τ̄_{0}) & = 1 ⊗ 1 ∧ ξ̄_{1}τ̄_{0} + τ̄_{0} ⊗ 1 ∧ ξ̄_{1} + ξ̄_{1} ⊗ 1 ∧ τ̄_{0} + ξ̄_{1}τ̄_{0} ⊗ 1 ∧ 1 \\ ν(τ_{0} ∧ (στ̄_{0})^{p−1}) & = 1 ⊗ τ_{0} ∧ (στ̄_{0})^{p−1} + τ_{0} ⊗ 1 ∧ (στ̄_{0})^{p−1} \\ ν(τ_{0} ∧ ξ̄_{1}) & = 1 ⊗ τ_{0} ∧ ξ̄_{1} + τ_{0} ⊗ 1 ∧ ξ̄_{1} + ξ̄_{1} ⊗ τ_{0} ∧ 1 + ξ̄_{1}τ_{0} ⊗ 1 ∧ 1 \\ ν(τ_{1} ∧ 1) & = 1 ⊗ τ_{1} ∧ 1 + ξ_{1} ⊗ τ_{0} ∧ 1 + τ_{1} ⊗ 1 ∧ 1 \end{aligned}

and

ν(1 ∧ y) = 1 ⊗ 1 ∧ y + τ̄_{0} ⊗ 1 ∧ (στ̄_{0})^{p−1} − τ̄_{0} ⊗ 1 ∧ ξ̄_{1} − τ̄_{1} ⊗ 1 ∧ 1 .

Hence the sum

ϵ̄_{1} := 1 ∧ y + τ_{0} ∧ (στ̄_{0})^{p−1} − τ_{0} ∧ ξ̄_{1} − τ_{1} ∧ 1  (4.4)

is A_{∗}-comodule primitive. Its image under π_{∗} in H_{∗}(V (1) ∧ THH(ℤ/p)) is

ϵ_{0}μ_{0}^{p−1} − ϵ_{1} = 1 ∧ τ̄_{0}(στ̄_{0})^{p−1} + τ_{0} ∧ (στ̄_{0})^{p−1} − 1 ∧ τ̄_{1} − τ_{0} ∧ ξ̄_{1} − τ_{1} ∧ 1 .

Let

V (1)_{∗}THH(ℓ)/(λ_{1}) = E(λ_{2}) ⊗ P (μ_{2})

be the quotient algebra of V (1)_{∗}THH(ℓ) by the ideal generated by λ_{1}.

Proposition 4.2. The classes

1 , ϵ_{0} , μ_{0} , ϵ_{0}μ_{0} , …, μ_{0}^{p−1} , ϵ̄_{1} ∈ H_{∗}(V (1) ∧ THH(ℓ/p))

defined in (4.2), (4.3) and (4.4) have unique lifts with same names in V (1)_{∗}THH(ℓ/p). The graded V (1)_{∗}THH(ℓ)-module V (1)_{∗}THH(ℓ/p) is a free V (1)_{∗}THH(ℓ)/(λ_{1})-mod-ule generated by these 2p classes:

V (1)_{∗}THH(ℓ/p) = V (1)_{∗}THH(ℓ)/(λ_{1}) ⊗ 𝔽_{p}{1, ϵ_{0}, μ_{0}, ϵ_{0}μ_{0}, …, μ_{0}^{p−1}, ϵ̄_{1}} .

The map π_{∗} to V (1)_{∗}THH(ℤ/p) takes ϵ_{0}^{δ}μ_{0}^{i} in degree 0 ≤ δ + 2i ≤ 2p − 2 to ϵ_{0}^{δ}μ_{0}^{i}, and takes ϵ̄_{1} in degree (2p − 1) to ϵ_{0}μ_{0}^{p−1} − ϵ_{1}.

Proof. Additively, this follows by another dimension count, and the description of π_{∗} follows from the definition of the classes in question. It remains to prove that the action of V (1)_{∗}THH(ℓ) is as claimed.

The action of μ_{2}^{i} and λ_{2}μ_{2}^{i} in V (1)_{∗}THH(ℓ) on the generators

1, ϵ_{0}, μ_{0}, ϵ_{0}μ_{0}, …, μ_{0}^{p−1}, ϵ̄_{1}

of V (1)_{∗}THH(ℓ/p) is non-trivial for all i ≥ 0, since the corresponding statement holds for the images of these classes in H_{∗}(V (1) ∧ THH(ℓ)) and H_{∗}(V (1) ∧ THH(ℓ/p)). This follows from Lemma 3.3 and the definition these classes. It remains to show that λ_{1} acts trivially on V (1)_{∗}THH(ℓ/p). For degree reasons, multiplication by λ_{1} is zero on all classes except possibly μ_{2}^{i} and λ_{2}μ_{2}^{i}, for i ≥ 0. Because of the module structure, it suffices to shows that λ_{1} = λ_{1}·1 = 0 in V (1)_{∗}THH(ℓ/p). This follows from the statement that the image of λ_{1} in H_{∗}(V (1) ∧ THH(ℓ/p)) is equal to 1 ∧ σξ̄_{1} = 0, as implied by Lemma 3.3.

5 The C_{p}-Tate construction

For the remainder of this paper, let p be a prime with p ≥ 5. We briefly recall the terminology on equivariant stable homotopy theory used in the sequel, and refer to [GM95], [HM97, §1], [HM03, §4] and [AR02, §3] for more details. Let C_{p^{n}} denote the cyclic group of order p^{n}, considered as a closed subgroup of the circle group S^{1}, and let G = S^{1} or C_{p^{n}}. For each spectrum X with S^{1}-action, let X_{hG} = EG_{+} ∧_{G} X and X^{hG} = F (EG_{+}, X)^{G} denote its homotopy orbit and homotopy fixed point spectra, as usual. We now write X^{tG} = [ẼG ∧ F (EG_{+}, X)]^{G} for the G-Tate construction on X, which was denoted t_{G}(X)^{G} in [GM95] and ℍ̂(G, X) in [HM97, HM03, AR02].

We denote by F the Frobenius map X^{C_{p^{n}}} → X^{C_{p^{n−1}}} given by the inclusion of fixed-point spectra, and by V the Verschiebung map X^{C_{p^{n−1}}} → X^{C_{p^{n}}} given by transfer. We shall also consider the homotopy Frobenius, Tate Frobenius and homotopy Verschiebung maps F^{h}: X^{hS^{1}} → X^{hC_{p^{n}}}, F^{h}: X^{hC_{p^{n}}} → X^{hC_{p^{n−1}}}, F^{t}: X^{tS^{1}} → X^{tC_{p^{n}}} and V^{h}: X^{hC_{p^{n−1}}} → X^{hC_{p^{n}}}.

There are conditionally convergent G-homotopy fixed point and G-Tate spectral sequences in V (1)-homotopy for X, with

E_{s,t}^{2}(G, X) = H_{gp}^{−s}(G; V (1)_{t}(X)) ⟹ V (1)_{s+t}(X^{hG})

and

Ê_{s,t}^{2}(G, X) = Ĥ_{gp}^{−s}(G; V (1)_{t}(X)) ⟹ V (1)_{s+t}(X^{tG}) .

Here H_{gp}^{∗}(G; V (1)_{∗}(X)) denotes the group cohomology of G and Ĥ_{gp}^{∗}(G; V (1)_{∗}(X)) the Tate cohomology of G, with coefficients in V (1)_{∗}(X). Notice that in our case, with X = THH(B), the action of G on V (1)_{∗}(X) is trivial, since it is the restriction of an S^{1}-action. We write H_{gp}^{∗}(C_{p^{n}}; 𝔽_{p}) = E(u_{n}) ⊗ P (t) and Ĥ_{gp}^{∗}(C_{p^{n}}; 𝔽_{p}) = E(u_{n}) ⊗ P (t^{±1}) with u_{n} in degree 1 and t in degree 2, see for example [Ben98, Prop. 3.5.5] and [HM03, Lem. 4.2.1]. So u_{n}, t and x ∈ V (1)_{t}(X) have bidegree (−1, 0), (−2, 0) and (0, t) in either spectral sequence, respectively. See [HM03, §4.3] for proofs of the multiplicative properties of these spectral sequences. Similarly, we write H_{gp}^{∗}(S^{1}; 𝔽_{p}) = P (t) and Ĥ_{gp}^{∗}(S^{1}; 𝔽_{p}) = P (t^{±1}). We have morphisms of spectral sequences induced by the homotopy and Tate Frobenii, which on the E^{2}-terms map t to t and u_{n} to zero.

We are principally interested in the case when X = THH(B), with the S^{1}-action given by the cyclic structure [Lod98, Def. 7.1.9], [HM03, §1.2]. It is a cyclotomic spectrum, in the sense of [HM97, §1], leading to the commutative diagram

[figure: THH(B)_{hC_{p^{n}}}; N; THH(B)^{C_{p^{n}}}; R; Γ_{n}; THH(B)^{C_{p^{n−1}}}; Γ̂_{n}; ΣTHH(B)_{hC_{p^{n}}}; THH(B)_{hC_{p^{n}}}; N^{h}; THH(B)^{hC_{p^{n}}}; R^{h}; THH(B)^{tC_{p^{n}}}; ΣTHH(B)_{hC_{p^{n}}}]

of horizontal cofiber sequences. We abbreviate Ê^{2}(G, THH(B)) to Ê^{2}(G, B), etc. When B is a commutative S-algebra, this is a commutative algebra spectral sequence, and when B is an associative A-algebra, with A commutative, then Ê^{∗}(G, B) is a module spectral sequence over Ê^{∗}(G, A). The map R^{h} corresponds to the inclusion E^{2}(G, B) → Ê^{2}(G, B) from the second quadrant to the upper half-plane, for connective B.

Definition 5.1. We call a homomorphism of graded groups k-coconnected if it is an isomorphism in all dimensions greater than k and injective in dimension k.

In this section we compute V (1)_{∗}THH(ℓ/p)^{tC_{p}} by means of the C_{p}-Tate spectral sequence in V (1)-homotopy for THH(ℓ/p). In Propositions 5.7 and 5.8 we show that the comparison map Γ̂_{1} : V (1)_{∗}THH(ℓ/p) → V (1)_{∗}THH(ℓ/p)^{tC_{p}} is (2p − 2)-coconnected and can be identified with the algebraic localization homomorphism that inverts μ_{2}.

First we recall the structure of the C_{p}-Tate spectral sequence for THH(ℤ/p), with V (0)- and V (1)-coefficients. We have V (0)_{∗}THH(ℤ/p) = E(ϵ_{0}) ⊗ P (μ_{0}), and (with an obvious notation for the case of V (0)-homotopy) the E^{2}-terms are

\begin{aligned}Ê^{2}(C_{p}, ℤ/p; V (0)) & = E(u_{1}) ⊗ P (t^{±1}) ⊗ E(ϵ_{0}) ⊗ P (μ_{0}) \\ Ê^{2}(C_{p}, ℤ/p) & = E(u_{1}) ⊗ P (t^{±1}) ⊗ E(ϵ_{0}, ϵ_{1}) ⊗ P (μ_{0}) . \end{aligned}

In each G-Tate spectral sequence we have a first differential

d^{2}(x) = t · σx ,

see e.g. [Rog98, §3.3]. We easily deduce σϵ_{0} = μ_{0} and σϵ_{1} = μ_{0}^{p} from (4.1), so

\begin{aligned}Ê^{3}(C_{p}, ℤ/p; V (0)) & = E(u_{1}) ⊗ P (t^{±1}) \\ Ê^{3}(C_{p}, ℤ/p) & = E(u_{1}) ⊗ P (t^{±1}) ⊗ E(ϵ_{0}μ_{0}^{p−1} − ϵ_{1}) . \end{aligned}

Thus the V (0)-homotopy spectral sequence collapses at Ê^{3} = Ê^{∞}. By naturality with respect to the map i_{1} : V (0) → V (1), all the classes on the horizontal axis of Ê^{3}(C_{p}, ℤ/p) are infinite cycles, so also the latter spectral sequence collapses at Ê^{3}(C_{p}, ℤ/p).

We know from [HM03, Cor. 4.4.2] that the comparison map

Γ̂_{1} : V (0)_{∗}THH(ℤ/p) → V (0)_{∗}THH(ℤ/p)^{tC_{p}}

takes ϵ_{0}^{δ}μ_{0}^{i} to (u_{1}t^{−1})^{δ}t^{−i}, for all 0 ≤ δ ≤ 1, i ≥ 0. In particular, the integral map Γ̂_{1} : π_{∗}THH(ℤ/p) → π_{∗}THH(ℤ/p)^{tC_{p}} is (−2)-coconnected. From this we can deduce the following behavior of the comparison map Γ̂_{1} in V (1)-homotopy.

Lemma 5.2. The map

Γ̂_{1} : V (1)_{∗}THH(ℤ/p) → V (1)_{∗}THH(ℤ/p)^{tC_{p}}

takes the classes ϵ_{0}^{δ}μ_{0}^{i} from V (0)_{∗}THH(ℤ/p), for 0 ≤ δ ≤ 1 and i ≥ 0, to classes represented in Ê^{∞}(C_{p}, ℤ/p) by (u_{1}t^{−1})^{δ}t^{−i} (on the horizontal axis). Furthermore, it takes the class ϵ_{0}μ_{0}^{p−1}−ϵ_{1} in degree (2p−1) to a class represented by ϵ_{0}μ_{0}^{p−1}−ϵ_{1} (on the vertical axis).

Proof. The classes ϵ_{0}^{δ}μ_{0}^{i} are in the image from V (0)-homotopy, and we recalled above that they are detected by (u_{1}t^{−1})^{δ}t^{−i} in the V (0)-homotopy C_{p}-Tate spectral sequence for THH(ℤ/p). By naturality along i_{1} : V (0) → V (1), they are detected by the same (nonzero) classes in the V (1)-homotopy spectral sequence Ê^{∞}(C_{p}, ℤ/p).

To find the representative for Γ̂_{1}(ϵ_{0}μ_{0}^{p−1} − ϵ_{1}) in degree (2p − 1), we appeal to the cyclotomic trace map from algebraic K-theory, or more precisely, to the commutative diagram

\xymatrix{ & K(B) \ar[dr]^{tr} \ar[d]^{tr_{1}} \ar[dl]_{tr} \\ THH(B) & THH(B)^{C_{p}} \ar[r]^{R} \ar[d]^{Γ_{1}} \ar[l]_{F} & THH(B) \ar[d]^{Γ̂_{1}} \\  & THH(B)^{hC_{p}} \ar[r]^{R^{h}} \ar@{-->}[ul] & THH(B)^{tC_{p}} .}  (5.1)

The Bökstedt trace map tr: K(B) → THH(B) admits a preferred lift tr_{n} through each fixed point spectrum THH(B)^{C_{p^{n}}}, which homotopy equalizes the iterated restriction and Frobenius maps R^{n} and F^{n} to THH(B), see [Dun04, §3]. In particular, the σ-operator on V (1)_{∗}THH(B) is zero on classes in the image of tr.

In the case B = Hℤ/p we know that K(ℤ/p)_{p} ≃ Hℤ_{p}, so V (1)_{∗}K(ℤ/p) = E(ϵ̄_{1}), where the v_{1}-Bockstein of ϵ̄_{1} is −1. The Bökstedt trace image tr(ϵ̄_{1}) ∈ V (1)_{∗}THH(ℤ/p) lies in 𝔽_{p}{ϵ_{1}, ϵ_{0}μ_{0}^{p−1}}, has v_{1}-Bockstein tr(−1) = −1 and suspends by σ to 0. Hence

tr(ϵ̄_{1}) = ϵ_{0}μ_{0}^{p−1} − ϵ_{1} .

As we recalled above, the map Γ̂_{1} : π_{∗}THH(ℤ/p) → π_{∗}THH(ℤ/p)^{tC_{p}} is (−2)-coconnec-ted, so the corresponding map in V (1)-homotopy is at least (2p − 2)-coconnected. Thus it takes ϵ_{0}μ_{0}^{p−1} − ϵ_{1} to a nonzero class in V (1)_{∗}THH(ℤ/p)^{tC_{p}}, represented somewhere in total degree (2p − 1) of Ê^{∞}(C_{p}, ℤ/p), in the lower right hand corner of the diagram.

Going down the middle part of the diagram, we reach a class (Γ_{1}∘tr_{1})(ϵ̄_{1}), represented in total degree (2p − 1) in the left half-plane C_{p}-homotopy fixed point spectral sequence E^{∞}(C_{p}, ℤ/p). Its image under the edge homomorphism to V (1)_{∗}THH(ℤ/p) equals (F ∘ tr_{1})(ϵ̄_{1}) = tr(ϵ̄_{1}), hence (Γ_{1}∘tr_{1})(ϵ̄_{1}) is represented by ϵ_{0}μ_{0}^{p−1}−ϵ_{1} in E_{0,2p−1}^{∞}(C_{p}, ℤ/p). Its image under R^{h} in the C_{p}-Tate spectral sequence is the generator of Ê_{0,2p−1}^{∞}(C_{p}, ℤ/p) = 𝔽_{p}{ϵ_{0}μ_{0}^{p−1} − ϵ_{1}}, hence that generator is the E^{∞}-representative of Γ̂_{1}(ϵ_{0}μ_{0}^{p−1} − ϵ_{1}).

The (2p − 2)-connected map π: ℓ/p → Hℤ/p induces a (2p − 1)-connected map V (1)_{∗}K(ℓ/p) → V (1)_{∗}K(ℤ/p) = E(ϵ̄_{1}), by [BM94, Prop. 10.9]. We can lift the algebraic K-theory class ϵ̄_{1} to ℓ/p. This lift is not unique, but we fix one choice.

Definition 5.3. We call

ϵ̄_{1}^{K} ∈ V (1)_{2p−1}K(ℓ/p)

a chosen class that maps to the generator ϵ̄_{1} in V (1)_{2p−1}K(ℤ/p) ≅ ℤ/p.

Lemma 5.4. The Bökstedt trace tr: V (1)_{∗}K(ℓ/p) → V (1)_{∗}THH(ℓ/p) takes ϵ̄_{1}^{K} to ϵ̄_{1}.

Proof. In the commutative square

\xymatrix{V (1)_{∗}K(ℓ/p) \ar[d]^{π_{∗}} \ar[r]^{tr} & V (1)_{∗}THH(ℓ/p) \ar[d]^{π_{∗}} \\ V (1)_{∗}K(ℤ/p) \ar[r]^{tr} & V (1)_{∗}THH(ℤ/p)}

the trace image tr(ϵ̄_{1}^{K}) in V (1)_{∗}THH(ℓ/p) must map under π_{∗} to tr(ϵ̄_{1}) = ϵ_{0}μ_{0}^{p−1} − ϵ_{1} in V (1)_{∗}THH(ℤ/p), which by Proposition 4.2 characterizes it as being equal to the class ϵ̄_{1}. Hence tr(ϵ̄_{1}^{K}) = ϵ̄_{1}.

Next we turn to the C_{p}-Tate spectral sequence Ê^{∗}(C_{p}, ℓ/p) in V (1)-homotopy for THH(ℓ/p). Its E^{2}-term is

Ê^{2}(C_{p}, ℓ/p) = E(u_{1}) ⊗ P (t^{±1}) ⊗ 𝔽_{p}{1, ϵ_{0}, μ_{0}, ϵ_{0}μ_{0}, …, μ_{0}^{p−1}, ϵ̄_{1}} ⊗ E(λ_{2}) ⊗ P (μ_{2}) .

We have d^{2}(x) = t · σx, where

σ(ϵ_{0}^{δ}μ_{0}^{i−1}) = \begin{cases}μ_{0}^{i} & for δ = 1, 0 < i < p, \\ 0 & otherwise \end{cases}

is readily deduced from (4.1), and σ(ϵ̄_{1}) = 0 since ϵ̄_{1} is in the image of tr. Thus

Ê^{3}(C_{p}, ℓ/p) = E(u_{1}) ⊗ P (t^{±1}) ⊗ E(ϵ̄_{1}) ⊗ E(λ_{2}) ⊗ P (tμ_{2}) .  (5.2)

We prefer to use tμ_{2} rather than μ_{2} as a generator, since it represents multiplication by v_{2} (up to a unit factor in 𝔽_{p}) in all module spectral sequences over E^{∗}(S^{1}, ℓ), by [AR02, Prop. 4.8].

To proceed, we shall use that Ê^{∗}(C_{p}, ℓ/p) is a module over the spectral sequence for THH(ℓ). We therefore recall the structure of the latter spectral sequence, from [AR02, Th. 5.5]. It begins

Ê^{2}(C_{p}, ℓ) = E(u_{1}) ⊗ P (t^{±1}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ_{2}) .

The classes λ_{1}, λ_{2} and tμ_{2} are infinite cycles, and the differentials

\begin{matrix}d^{2p}(t^{1−p}) & ≐ tλ_{1} \\ d^{2p^{2}}(t^{p−p^{2}}) & ≐ t^{p}λ_{2} \\ d^{2p^{2}+1}(u_{1}t^{−p^{2}}) & ≐ tμ_{2} \end{matrix}

leave the terms

\begin{aligned}Ê^{2p+1}(C_{p}, ℓ) & = E(u_{1}, λ_{1}, λ_{2}) ⊗ P (t^{±p}, tμ_{2}) \\ Ê^{2p^{2}+1}(C_{p}, ℓ) & = E(u_{1}, λ_{1}, λ_{2}) ⊗ P (t^{±p^{2}}, tμ_{2}) \\ Ê^{2p^{2}+2}(C_{p}, ℓ) & = E(λ_{1}, λ_{2}) ⊗ P (t^{±p^{2}}) \end{aligned}

with Ê^{2p^{2}+2} = Ê^{∞}, converging to V (1)_{∗}THH(ℓ)^{tC_{p}}. The comparison map Γ̂_{1} takes λ_{1}, λ_{2} and μ_{2} to λ_{1}, λ_{2} and t^{−p^{2}} (up to a unit factor in 𝔽_{p}), respectively, inducing the algebraic localization map and identification

Γ̂_{1} : V (1)_{∗}THH(ℓ) → V (1)_{∗}THH(ℓ)[μ_{2}^{−1}] ≅ V (1)_{∗}THH(ℓ)^{tC_{p}} .

Lemma 5.5. In Ê^{∗}(C_{p}, ℓ/p), the class u_{1}t^{−p} supports the nonzero differential

d^{2p^{2}}(u_{1}t^{−p}) ≐ u_{1}t^{p^{2}−p}λ_{2} ,

and does not survive to the E^{∞}-term.

Proof. In Ê^{∗}(C_{p}, ℓ), there is such a differential. By naturality along i: ℓ → ℓ/p, it follows that there is also such a differential in Ê^{∗}(C_{p}, ℓ/p). It remains to argue that the target class is nonzero at the E^{2p^{2}}-term. Considering the E^{3}-term in (5.2), the only possible source of a previous differential hitting u_{1}t^{p^{2}−p}λ_{2} is ϵ̄_{1}, supporting a d^{2p^{2}−2p+1}-differential. But ϵ̄_{1} is in an even column and u_{1}t^{p^{2}−p}λ_{2} is in an odd column. By naturality with respect to the Tate Frobenius map F^{t}: THH(ℓ/p)^{tS^{1}} → THH(ℓ/p)^{tC_{p}}, any such differential from an even to an odd column must be zero. Indeed, the S^{1}-Tate spectral sequence has E^{2}-term given by P (t^{±1}) ⊗ V (1)_{∗}THH(ℓ/p), and F^{t} induces the injective homomorphism that takes Ê^{2}(S^{1}, ℓ/p) isomorphically to the even columns of Ê^{2}(C_{p}, ℓ/p). Since Ê^{∗}(S^{1}, ℓ/p) is concentrated in even columns, all differentials of odd length are zero. By naturality, classes of Ê^{r}(C_{p}, ℓ/p) that lie in the image of Ê^{r}(F^{t}) cannot support a differential of odd length; compare with [AR02, Lemma 5.2]. In the present situation, the d^{2}-differential of Ê^{∗}(C_{p}, ℓ/p) leading to (5.2) is also non-zero in Ê^{∗}(S^{1}, ℓ/p), so that we have

Ê^{3}(S^{1}, ℓ/p) = P (t^{±1}) ⊗ E(ϵ̄_{1}) ⊗ E(λ_{2}) ⊗ P (tμ_{2}) .

By inspection, if the class ϵ̄_{1} ∈ Ê^{2}(C_{p}, ℓ/p) survives to Ê^{2p^{2}−2p+1}(C_{p}, ℓ/p), then it will lie in the image of Ê^{2p^{2}−2p+1}(F^{t}).

To determine the map Γ̂_{1} we use naturality with respect to the map π: ℓ/p → Hℤ/p.

Lemma 5.6. The classes 1, ϵ_{0}, μ_{0}, ϵ_{0}μ_{0}, …, μ_{0}^{p−1} and ϵ̄_{1} in V (1)_{∗}THH(ℓ/p) map under Γ̂_{1} to classes in V (1)_{∗}THH(ℓ/p)^{tC_{p}} that are represented in Ê^{∞}(C_{p}, ℓ/p) by the permanent cycles (u_{1}t^{−1})^{δ}t^{−i} (on the horizontal axis) in degrees ≤ (2p − 2), and by the permanent cycle ϵ̄_{1} (on the vertical axis) in degree (2p − 1).

Proof. In the commutative square

\xymatrix{V (1)_{∗}THH(ℓ/p) \ar[r]^{Γ̂_{1}} \ar[d]^{π_{∗}} & V (1)_{∗}THH(ℓ/p)^{tC_{p}} \ar[d]^{π_{∗}} \\ V (1)_{∗}THH(ℤ/p) \ar[r]^{Γ̂_{1}} & V (1)_{∗}THH(ℤ/p)^{tC_{p}}}

the classes ϵ_{0}^{δ}μ_{0}^{i} in the upper left hand corner map to classes in the lower right hand corner that are represented by (u_{1}t^{−1})^{δ}t^{−i} in degrees ≤ (2p − 2), and ϵ̄_{1} maps to ϵ_{0}μ_{0}^{p−1} − ϵ_{1} in degree (2p − 1). This follows by combining Proposition 4.2 and Lemma 5.2.

The first (2p−1) of these are represented in maximal filtration (on the horizontal axis), so their images in the upper right hand corner must be represented by permanent cycles (u_{1}t^{−1})^{δ}t^{−i} in the Tate spectral sequence Ê^{∞}(C_{p}, ℓ/p).

The image of the last class, ϵ̄_{1}, in the upper right hand corner could either be represented by ϵ̄_{1} in bidegree (0, 2p − 1) or by u_{1}t^{−p} in bidegree (2p − 1, 0). However, the last class supports a differential d^{2p^{2}}(u_{1}t^{−p}) ≐ u_{1}t^{p^{2}−p}λ_{2}, by Lemma 5.5 above. This only leaves the other possibility, that Γ̂_{1}(ϵ̄_{1}) is represented by ϵ̄_{1} in Ê^{∞}(C_{p}, ℓ/p).

We proceed to determine the differential structure in Ê^{∗}(C_{p}, ℓ/p), making use of the permanent cycles identified above.

Proposition 5.7. The C_{p}-Tate spectral sequence in V (1)-homotopy for THH(ℓ/p) has

Ê^{3}(C_{p}, ℓ/p) = E(u_{1}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±1}, tμ_{2}) .

It has differentials generated by

d^{2p^{2}−2p+2}(t^{p−p^{2}} · t^{−i}ϵ̄_{1}) ≐ tμ_{2} · t^{−i}

for 0 < i < p, d^{2p^{2}}(t^{p−p^{2}}) ≐ t^{p}λ_{2} and d^{2p^{2}+1}(u_{1}t^{−p^{2}}) ≐ tμ_{2}. The subsequent terms are

Ê^{2p^{2}−2p+3}(C_{p}, ℓ/p) = E(u_{1}, λ_{2}) ⊗ 𝔽_{p}{t^{−i} | 0 < i < p} ⊗ P (t^{±p})
⊕ E(u_{1}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p}, tμ_{2})
Ê^{2p^{2}+1}(C_{p}, ℓ/p) = E(u_{1}, λ_{2}) ⊗ 𝔽_{p}{t^{−i} | 0 < i < p} ⊗ P (t^{±p^{2}})
⊕ E(u_{1}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2}}, tμ_{2})
Ê^{2p^{2}+2}(C_{p}, ℓ/p) = E(u_{1}, λ_{2}) ⊗ 𝔽_{p}{t^{−i} | 0 < i < p} ⊗ P (t^{±p^{2}})
⊕ E(ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2}}) .

The last term can be rewritten as

Ê^{∞}(C_{p}, ℓ/p) = (E(u_{1}) ⊗ 𝔽_{p}{t^{−i} | 0 < i < p} ⊕ E(ϵ̄_{1})) ⊗ E(λ_{2}) ⊗ P (t^{±p^{2}}) .

Proof. We have already identified the E^{2}- and E^{3}-terms above. The E^{3}-term (5.2)is generated over Ê^{3}(C_{p}, ℓ) by an 𝔽_{p}-basis for E(ϵ̄_{1}), so the next possible differential is induced by d^{2p}(t^{1−p}) ≐ tλ_{1}. But multiplication by λ_{1} is trivial in V (1)_{∗}THH(ℓ/p), by Proposition 4.2, so Ê^{3}(C_{p}, ℓ/p) = Ê^{2p+1}(C_{p}, ℓ/p). This term is generated over Ê^{2p+1}(C_{p}, ℓ) by P_{p}(t^{−1}) ⊗ E(ϵ̄_{1}). Here 1, t^{−1}, …, t^{1−p} and ϵ̄_{1} are permanent cycles, by Lemma 5.6. Any d^{r}-differential before d^{2p^{2}} must therefore originate on a class t^{−i}ϵ̄_{1} for 0 < i < p, and be of even length r, since these classes lie in even columns. For bidegree reasons, the first possibility is r = 2p^{2} − 2p + 2, so Ê^{3}(C_{p}, ℓ/p) = Ê^{2p^{2}−2p+2}(C_{p}, ℓ/p).

Multiplication by v_{2} acts trivially on V (1)_{∗}THH(ℓ) and V (1)_{∗}THH(ℓ)^{tC_{p}} for degree reasons, and therefore also on V (1)_{∗}THH(ℓ/p) and V (1)_{∗}THH(ℓ/p)^{tC_{p}} by the module structure. The class v_{2} maps to tμ_{2} in the S^{1}-Tate spectral sequence for ℓ, as recalled above, so multiplication by v_{2} is represented by multiplication by tμ_{2} in the C_{p}-Tate spectral sequence for ℓ/p. Applied to the permanent cycles (u_{1}t^{−1})^{δ}t^{−i} in degrees ≤ (2p − 2), this implies that the products

tμ_{2} · (u_{1}t^{−1})^{δ}t^{−i}

must be infinite cycles representing zero, i.e., they must be hit by differentials. In the cases δ = 1, 0 ≤ i ≤ p − 2, these classes in odd columns cannot be hit by differentials of odd length, such as d^{2p^{2}+1}, so the only possibility is

d^{2p^{2}−2p+2}(t^{p−p^{2}} · (u_{1}t^{−1})t^{−i}ϵ̄_{1}) ≐ tμ_{2} · (u_{1}t^{−1})t^{−i}

for 0 ≤ i ≤ p − 2. By the module structure (consider multiplication by u_{1}) it follows that

d^{2p^{2}−2p+2}(t^{p−p^{2}} · t^{−i}ϵ̄_{1}) ≐ tμ_{2} · t^{−i}

for 0 < i < p. Hence we can compute from (5.2) that

\begin{aligned}Ê^{2p^{2}−2p+3}(C_{p}, ℓ/p) & = E(u_{1}) ⊗ P (t^{±p}) ⊗ 𝔽_{p}\{t^{−i} | 0 < i < p\} ⊗ E(λ_{2}) \\ & ⊕ E(u_{1}) ⊗ P (t^{±p}) ⊗ E(ϵ̄_{1}) ⊗ E(λ_{2}) ⊗ P (tμ_{2}) . \end{aligned}

This is generated over Ê^{2p+1}(C_{p}, ℓ) by the permanent cycles 1, t^{−1}, …, t^{1−p} and ϵ̄_{1}, so the next differential is induced by d^{2p^{2}}(t^{p−p^{2}}) ≐ t^{p}λ_{2}. This leaves

\begin{aligned}Ê^{2p^{2}+1}(C_{p}, ℓ/p) & = E(u_{1}) ⊗ P (t^{±p^{2}}) ⊗ 𝔽_{p}\{t^{−i} | 0 < i < p\} ⊗ E(λ_{2}) \\ & ⊕ E(u_{1}) ⊗ P (t^{±p^{2}}) ⊗ E(ϵ̄_{1}) ⊗ E(λ_{2}) ⊗ P (tμ_{2}) . \end{aligned}

Finally, d^{2p^{2}+1}(u_{1}t^{−p^{2}}) ≐ tμ_{2} applies, and leaves

\begin{aligned}Ê^{2p^{2}+2}(C_{p}, ℓ/p) & = E(u_{1}) ⊗ P (t^{±p^{2}}) ⊗ 𝔽_{p}\{t^{−i} | 0 < i < p\} ⊗ E(λ_{2}) \\ & ⊕ P (t^{±p^{2}}) ⊗ E(ϵ̄_{1}) ⊗ E(λ_{2}) . \end{aligned}

For bidegree reasons, Ê^{2p^{2}+2} = Ê^{∞}.

Proposition 5.8. The comparison map Γ̂_{1} takes the classes

ϵ_{0}^{δ}μ_{0}^{i}, ϵ̄_{1}, λ_{2} and μ_{2} in V (1)_{∗}THH(ℓ/p)

to classes in V (1)_{∗}THH(ℓ/p)^{tC_{p}} represented by

(u_{1}t^{−1})^{δ}t^{−i}, ϵ̄_{1}, λ_{2} and t^{−p^{2}} in Ê^{∞}(C_{p}, ℓ/p),

up to a unit factor in 𝔽_{p}, respectively. Thus

V (1)_{∗}THH(ℓ/p)^{tC_{p}} ≅ 𝔽_{p}{1, ϵ_{0}, μ_{0}, ϵ_{0}μ_{0}, …, μ_{0}^{p−1}, ϵ̄_{1}} ⊗ E(λ_{2}) ⊗ P (μ_{2}^{±1})

and Γ̂_{1} induces an identification V (1)_{∗}THH(ℓ/p)[μ_{2}^{−1}] ≅ V (1)_{∗}THH(ℓ/p)^{tC_{p}}. In particular, Γ̂_{1} factors as the algebraic localization map and identification

Γ̂_{1} : V (1)_{∗}THH(ℓ/p) → V (1)_{∗}THH(ℓ/p)[μ_{2}^{−1}] ≅ V (1)_{∗}THH(ℓ/p)^{tC_{p}} ,

and is (2p − 2)-coconnected.

Proof. The image under Γ̂_{1} of the classes 1, ϵ_{0}, μ_{0}, ϵ_{0}μ_{0}, …, μ_{0}^{p−1} and ϵ̄_{1} was given in Lemma 5.6, and the action on the classes λ_{2} and μ_{2} is given in the proof of [AR02, Th. 5.5]. The structure of V (1)_{∗}THH(ℓ/p)^{tC_{p}} is then immediate from the E^{∞}-term in Proposition 5.7. The top class not in the image of Γ̂_{1} is ϵ̄_{1}λ_{2}μ_{2}^{−1}, in degree (2p − 2).

Recall that

T F (B; p) = holim_{n,F} THH(B)^{C_{p^{n}}}
T R(B; p) = holim_{n,R} THH(B)^{C_{p^{n}}}

are defined as the homotopy limits over the Frobenius and the restriction maps

F, R: THH(B)^{C_{p^{n}}} → THH(B)^{C_{p^{n−1}}} ,

respectively.

Corollary 5.9. The comparison maps

\begin{aligned}Γ_{n} & : THH(ℓ/p)^{C_{p^{n}}} → THH(ℓ/p)^{hC_{p^{n}}} \\ Γ̂_{n} & : THH(ℓ/p)^{C_{p^{n−1}}} → THH(ℓ/p)^{tC_{p^{n}}} \end{aligned}

for n ≥ 1, and

\begin{aligned}Γ & : T F (ℓ/p; p) → THH(ℓ/p)^{hS^{1}} \\ Γ̂ & : T F (ℓ/p; p) → THH(ℓ/p)^{tS^{1}} \end{aligned}

all induce (2p − 2)-coconnected homomorphisms on V (1)-homotopy.

Proof. This follows from a theorem of Tsalidis [Tsa98, Th. 2.4] and Proposition 5.8 above, just like in [AR02, Th. 5.7]. See also [BBLNR, Ex. 10.2]

6 Higher fixed points

Let n ≥ 1. Write v_{p}(i) for the p-adic valuation of i. Define a numerical function ρ(−) by

\begin{aligned}ρ(2k − 1) & = (p^{2k+1} + 1)/(p + 1) = p^{2k} − p^{2k−1} + ⋯ − p + 1 \\ ρ(2k) & = (p^{2k+2} − p^{2})/(p^{2} − 1) = p^{2k} + p^{2k−2} + ⋯ + p^{2} \end{aligned}

for k ≥ 0, so ρ(−1) = 1 and ρ(0) = 0. For even arguments, ρ(2k) = r(2k) as defined in [AR02, Def. 2.5].

In all of the following spectral sequences we know that λ_{2}, tμ_{2} and ϵ̄_{1} are infinite cycles. For λ_{2} and ϵ̄_{1} this follows from the C_{p^{n}}-fixed point analogue of diagram (5.1), by [AR02, Prop. 2.8] and Lemma 5.4. For tμ_{2} it follows from [AR02, Prop. 4.8], by naturality.

Theorem 6.1. The C_{p^{n}}-Tate spectral sequence in V (1)-homotopy for THH(ℓ/p) begins

Ê^{2}(C_{p^{n}}, ℓ/p) = E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{1, ϵ_{0}, μ_{0}, ϵ_{0}μ_{0}, …, μ_{0}^{p−1}, ϵ̄_{1}} ⊗ P (t^{±1}, μ_{2})

and converges to V (1)_{∗}THH(ℓ/p)^{tC_{p^{n}}}. It is a module spectral sequence over the algebra spectral sequence Ê^{∗}(C_{p^{n}}, ℓ) converging to V (1)_{∗}THH(ℓ)^{tC_{p^{n}}}.

There is an initial d^{2}-differential generated by

d^{2}(ϵ_{0}μ_{0}^{i−1}) = tμ_{0}^{i}

for 0 < i < p. Next, there are 2n families of even length differentials generated by

d^{2ρ(2k−1)}(t^{p^{2k−1}−p^{2k}+i} · ϵ̄_{1}) ≐ (tμ_{2})^{ρ(2k−3)} · t^{i}

for v_{p}(i) = 2k − 2, for each k = 1, …, n, and

d^{2ρ(2k)}(t^{p^{2k−1}−p^{2k}}) ≐ λ_{2} · t^{p^{2k−1}} · (tμ_{2})^{ρ(2k−2)}

for each k = 1, …, n. Finally, there is a differential of odd length generated by

d^{2ρ(2n)+1}(u_{n} · t^{−p^{2n}}) ≐ (tμ_{2})^{ρ(2n−2)+1} .

We shall prove Theorem 6.1 by induction on n. The base case n = 1 was covered by Proposition 5.7. We can therefore assume that Theorem 6.1 holds for some fixed n ≥ 1, and must prove the corresponding statement for n + 1. First we make the following deduction.

Corollary 6.2. The initial differential in the C_{p^{n}}-Tate spectral sequence in V (1)-homoto-py for THH(ℓ/p) leaves

Ê^{3}(C_{p^{n}}, ℓ/p) = E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±1}, tμ_{2}) .

The next 2n families of differentials leave the intermediate terms

\begin{aligned}Ê^{2ρ(1)+1}(C_{p^{n}} & , ℓ/p) = E(u_{n}, λ_{2}) ⊗ 𝔽_{p}\{t^{−i} | 0 < i < p\} ⊗ P (t^{±p}) \\ & ⊕ E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p}, tμ_{2}) \end{aligned}

(for m = 1),

Ê^{2ρ(2m−1)+1}(C_{p^{n}}, ℓ/p) = E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{t^{−i} | 0 < i < p} ⊗ P (t^{±p^{2}})

⊕ ⨁_{k=2}^{m} E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{t^{j} | j ∈ ℤ, v_{p}(j) = 2k − 2} ⊗ P_{ρ(2k−3)}(tμ_{2})

⊕ ⨁_{k=2}^{m−1} E(u_{n}, ϵ̄_{1}) ⊗ 𝔽_{p}{t^{j}λ_{2} | j ∈ ℤ, v_{p}(j) = 2k − 1} ⊗ P_{ρ(2k−2)}(tμ_{2})

⊕ E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2m−1}}, tμ_{2})

for m = 2, …, n, and

Ê^{2ρ(2m)+1}(C_{p^{n}}, ℓ/p) = E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{t^{−i} | 0 < i < p} ⊗ P (t^{±p^{2}})

⊕ ⨁_{k=2}^{m} E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{t^{j} | j ∈ ℤ, v_{p}(j) = 2k − 2} ⊗ P_{ρ(2k−3)}(tμ_{2})

⊕ ⨁_{k=2}^{m} E(u_{n}, ϵ̄_{1}) ⊗ 𝔽_{p}{t^{j}λ_{2} | j ∈ ℤ, v_{p}(j) = 2k − 1} ⊗ P_{ρ(2k−2)}(tμ_{2})

⊕ E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2m}}, tμ_{2})

for m = 1, …, n. The final differential leaves the E^{2ρ(2n)+2} = E^{∞}-term, equal to

Ê^{∞}(C_{p^{n}}, ℓ/p) = E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{t^{−i} | 0 < i < p} ⊗ P (t^{±p^{2}})

⊕ ⨁_{k=2}^{n} E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{t^{j} | j ∈ ℤ, v_{p}(j) = 2k − 2} ⊗ P_{ρ(2k−3)}(tμ_{2})

⊕ ⨁_{k=2}^{n} E(u_{n}, ϵ̄_{1}) ⊗ 𝔽_{p}{t^{j}λ_{2} | j ∈ ℤ, v_{p}(j) = 2k − 1} ⊗ P_{ρ(2k−2)}(tμ_{2})

⊕ E(ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2n}}) ⊗ P_{ρ(2n−2)+1}(tμ_{2}) .

Proof. The statements about the E^{3}-, E^{2ρ(1)+1}- and E^{2ρ(2)+1}-terms are clear from Proposition 5.7. For each m = 2, …, n we proceed by a secondary induction. The differential

d^{2ρ(2m−1)}(t^{p^{2m−1}−p^{2m}+i} · ϵ̄_{1}) ≐ (tμ_{2})^{ρ(2m−3)} · t^{i}

for v_{p}(i) = 2m − 2 is non-trivial only on the summand

E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2m−2}}, tμ_{2})

of the E^{2ρ(2m−2)+1} = E^{2ρ(2m−1)}-term, with homology

\begin{matrix}E(u_{n}, λ_{2}) ⊗ 𝔽_{p}\{t^{j} | j ∈ ℤ, v_{p}(j) = 2m − 2\} ⊗ P_{ρ(2m−3)}(tμ_{2}) \\ ⊕ E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2m−1}}, tμ_{2}) . \end{matrix}

This gives the stated E^{2ρ(2m−1)+1}-term. Similarly, the differential

d^{2ρ(2m)}(t^{p^{2m−1}−p^{2m}}) ≐ λ_{2} · t^{p^{2m−1}} · (tμ_{2})^{ρ(2m−2)}

is non-trivial only on the summand

E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2m−1}}, tμ_{2})

of the E^{2ρ(2m−1)+1} = E^{2ρ(2m)}-term, with homology

\begin{matrix}E(u_{n}, ϵ̄_{1}) ⊗ 𝔽_{p}\{t^{j}λ_{2} | j ∈ ℤ, v_{p}(j) = 2m − 1\} ⊗ P_{ρ(2m−2)}(tμ_{2}) \\ ⊕ E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2m}}, tμ_{2}) . \end{matrix}

This gives the stated E^{2ρ(2m)+1}-term. The final differential

d^{2ρ(2n)+1}(u_{n} · t^{−p^{2n}}) ≐ (tμ_{2})^{ρ(2n−2)+1}

is non-trivial only on the summand

E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2n}}, tμ_{2})

of the E^{2ρ(2n)+1}-term, with homology

E(ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2n}}) ⊗ P_{ρ(2n−2)+1}(tμ_{2}) .

This gives the stated E^{2ρ(2n)+2}-term. At this stage there is no room for any further differentials, since the spectral sequence is concentrated in a narrower horizontal band than the vertical height of the following differentials.

Next we compare the C_{p^{n}}-Tate spectral sequence with the C_{p^{n}}-homotopy fixed point spectral sequence obtained by restricting the E^{2}-term to the second quadrant (s ≤ 0, t ≥ 0). It is algebraically easier to handle the latter after inverting μ_{2}, which can be interpreted as comparing THH(ℓ/p) with its C_{p}-Tate construction.

In general, there is a commutative diagram

\xymatrix{THH(B)^{C_{p^{n}}} \ar[r]^{R} \ar[d]^{Γ_{n}} & THH(B)^{C_{p^{n−1}}} \ar[r]^{Γ_{n−1}} \ar[d]^{Γ̂_{n}} & THH(B)^{hC_{p^{n−1}}} \ar[d]^{Γ̂_{1}^{hC_{p^{n−1}}}} \\ THH(B)^{hC_{p^{n}}} \ar[r]^{R^{h}} & THH(B)^{tC_{p^{n}}} \ar[r]^{G_{n−1}} & (ρ_{p}^{∗}THH(B)^{tC_{p}})^{hC_{p^{n−1}}} .}  (6.1)

Here ρ_{p}^{∗}THH(B)^{tC_{p}} is a notation for the S^{1}-spectrum obtained from the S^{1}/C_{p}-spectrum THH(B)^{tC_{p}} via the p-th root isomorphism ρ_{p}: S^{1} → S^{1}/C_{p}, and G_{n−1} is the comparison map from the C_{p^{n−1}}-fixed points to the C_{p^{n−1}}-homotopy fixed points of ρ_{p}^{∗}THH(B)^{tC_{p}}, in view of the identification

(ρ_{p}^{∗}THH(B)^{tC_{p}})^{C_{p^{n−1}}} = THH(B)^{tC_{p^{n}}} .

We are of course considering the case B = ℓ/p. In V (1)-homotopy all four maps with labels containing Γ are (2p − 2)-coconnected, by Corollary 5.9, so G_{n−1} is at least (2p − 1)-coconnected. (We shall see in Lemma 6.8 that V (1)_{∗}G_{n−1} is an isomorphism in all degrees.) By Proposition 5.8 the map Γ̂_{1} precisely inverts μ_{2}, so the E^{2}-term of the C_{p^{n}}- homotopy fixed point spectral sequence in V (1)-homotopy for THH(ℓ/p)^{tC_{p}} is obtained by inverting μ_{2} in E^{2}(C_{p^{n}}, ℓ/p). We denote this spectral sequence by μ_{2}^{−1}E^{∗}(C_{p^{n}}, ℓ/p), even though in later terms only a power of μ_{2} is present.

Theorem 6.3. The C_{p^{n}}-homotopy fixed point spectral sequence μ_{2}^{−1}E^{∗}(C_{p^{n}}, ℓ/p) in V (1)- homotopy for THH(ℓ/p)^{tC_{p}} begins

μ_{2}^{−1}E^{2}(C_{p^{n}}, ℓ/p) = E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{1, ϵ_{0}, μ_{0}, ϵ_{0}μ_{0}, …, μ_{0}^{p−1}, ϵ̄_{1}} ⊗ P (t, μ_{2}^{±1})

and converges to V (1)_{∗}(ρ_{p}^{∗}THH(ℓ/p)^{tC_{p}})^{hC_{p^{n}}}, which receives a (2p − 2)-coconnected map (Γ̂_{1})^{hC_{p^{n}}} from V (1)_{∗}THH(ℓ/p)^{hC_{p^{n}}}. There is an initial d^{2}-differential generated by

d^{2}(ϵ_{0}μ_{0}^{i−1}) = tμ_{0}^{i}

for 0 < i < p. Next, there are 2n families of even length differentials generated by

d^{2ρ(2k−1)}(μ_{2}^{p^{2k}−p^{2k−1}+j} · ϵ̄_{1}) ≐ (tμ_{2})^{ρ(2k−1)} · μ_{2}^{j}

for v_{p}(j) = 2k − 2, for each k = 1, …, n, and

d^{2ρ(2k)}(μ_{2}^{p^{2k}−p^{2k−1}}) ≐ λ_{2} · μ_{2}^{−p^{2k−1}} · (tμ_{2})^{ρ(2k)}

for each k = 1, …, n. Finally, there is a differential of odd length generated by

d^{2ρ(2n)+1}(u_{n} · μ_{2}^{p^{2n}}) ≐ (tμ_{2})^{ρ(2n)+1} .

Proof. The differential pattern follows from Theorem 6.1 by naturality with respect to the maps of spectral sequences

μ_{2}^{−1}E^{∗}(C_{p^{n}}, ℓ/p) ⟵^{Γ̂_{1}^{hC_{p^{n}}}} E^{∗}(C_{p^{n}}, ℓ/p) ⟶^{R^{h}} Ê^{∗}(C_{p^{n}}, ℓ/p)

induced by Γ̂_{1}^{hC_{p^{n}}} and R^{h}. The first inverts μ_{2} and the second inverts t, at the level of E^{2}-terms. We are also using that tμ_{2}, the image of v_{2}, multiplies as an infinite cycle in all of these spectral sequences.

Corollary 6.4. The initial differential in the C_{p^{n}}-homotopy fixed point spectral sequence in V (1)-homotopy for THH(ℓ/p)^{tC_{p}} leaves

\begin{aligned}μ_{2}^{−1}E^{3}(C_{p^{n}}, ℓ/p) & = E(u_{n}, λ_{2}) ⊗ 𝔽_{p}\{μ_{0}^{i} | 0 < i < p\} ⊗ P (μ_{2}^{±1}) \\ & ⊕ E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (μ_{2}^{±1}, tμ_{2}) . \end{aligned}

The next 2n families of differentials leave the intermediate terms

μ_{2}^{−1}E^{2ρ(2m−1)+1}(C_{p^{n}}, ℓ/p) = E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{μ_{0}^{i} | 0 < i < p} ⊗ P (μ_{2}^{±1})

⊕ ⨁_{k=1}^{m} E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{μ_{2}^{j} | j ∈ ℤ, v_{p}(j) = 2k − 2} ⊗ P_{ρ(2k−1)}(tμ_{2})

⊕ ⨁_{k=1}^{m−1} E(u_{n}, ϵ̄_{1}) ⊗ 𝔽_{p}{λ_{2}μ_{2}^{j} | j ∈ ℤ, v_{p}(j) = 2k − 1} ⊗ P_{ρ(2k)}(tμ_{2})

⊕ E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (μ_{2}^{±p^{2m−1}}, tμ_{2})

and

μ_{2}^{−1}E^{2ρ(2m)+1}(C_{p^{n}}, ℓ/p) = E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{μ_{0}^{i} | 0 < i < p} ⊗ P (μ_{2}^{±1})

⊕ ⨁_{k=1}^{m} E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{μ_{2}^{j} | j ∈ ℤ, v_{p}(j) = 2k − 2} ⊗ P_{ρ(2k−1)}(tμ_{2})

⊕ ⨁_{k=1}^{m} E(u_{n}, ϵ̄_{1}) ⊗ 𝔽_{p}{λ_{2}μ_{2}^{j} | j ∈ ℤ, v_{p}(j) = 2k − 1} ⊗ P_{ρ(2k)}(tμ_{2})

⊕ E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (μ_{2}^{±p^{2m}}, tμ_{2})

for m = 1, …, n. The final differential leaves the E^{2ρ(2n)+2} = E^{∞}-term, equal to

μ_{2}^{−1}E^{∞}(C_{p^{n}}, ℓ/p) = E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{μ_{0}^{i} | 0 < i < p} ⊗ P (μ_{2}^{±1})

⊕ ⨁_{k=1}^{n} E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{μ_{2}^{j} | j ∈ ℤ, v_{p}(j) = 2k − 2} ⊗ P_{ρ(2k−1)}(tμ_{2})

⊕ ⨁_{k=1}^{n} E(u_{n}, ϵ̄_{1}) ⊗ 𝔽_{p}{λ_{2}μ_{2}^{j} | j ∈ ℤ, v_{p}(j) = 2k − 1} ⊗ P_{ρ(2k)}(tμ_{2})

⊕ E(ϵ̄_{1}, λ_{2}) ⊗ P (μ_{2}^{±p^{2n}}) ⊗ P_{ρ(2n)+1}(tμ_{2}) .

Proof. The computation of the E^{3}-term from the E^{2}-term is straightforward. The rest of the proof goes by a secondary induction on m = 1, …, n, very much like the proof of Corollary 6.2. The differential

d^{2ρ(2m−1)}(μ_{2}^{p^{2m}−p^{2m−1}+j} · ϵ̄_{1}) ≐ (tμ_{2})^{ρ(2m−1)} · μ_{2}^{j}

for v_{p}(j) = 2m − 2 is non-trivial only on the summand

E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (μ_{2}^{±p^{2m−2}}, tμ_{2})

of the E^{3} = E^{2ρ(1)}-term (for m = 1), resp. the E^{2ρ(2m−2)+1} = E^{2ρ(2m−1)}-term (for m = 2, …, n). Its homology is

\begin{matrix}E(u_{n}, λ_{2}) ⊗ 𝔽_{p}\{μ_{2}^{j} | j ∈ ℤ, v_{p}(j) = 2m − 2\} ⊗ P_{ρ(2m−1)}(tμ_{2}) \\ ⊕ E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (μ_{2}^{±p^{2m−1}}, tμ_{2}) , \end{matrix}

which gives the stated E^{2ρ(2m−1)+1}-term. The differential

d^{2ρ(2m)}(μ_{2}^{p^{2m}−p^{2m−1}}) ≐ λ_{2} · μ_{2}^{−p^{2m−1}} · (tμ_{2})^{ρ(2m)}

is non-trivial only on the summand

E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (μ_{2}^{±p^{2m−1}}, tμ_{2})

of the E^{2ρ(2m−1)+1} = E^{2ρ(2m)}-term, leaving

\begin{matrix}E(u_{n}, ϵ̄_{1}) ⊗ 𝔽_{p}\{λ_{2}μ_{2}^{j} | j ∈ ℤ, v_{p}(j) = 2m − 1\} ⊗ P_{ρ(2m)}(tμ_{2}) \\ ⊕ E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (μ_{2}^{±p^{2m}}, tμ_{2}) . \end{matrix}

This gives the stated E^{2ρ(2m)+1}-term. The final differential

d^{2ρ(2n)+1}(u_{n} · μ_{2}^{p^{2n}}) ≐ (tμ_{2})^{ρ(2n)+1}

is non-trivial only on the summand

E(u_{n}, ϵ̄_{1}, λ_{2}) ⊗ P (μ_{2}^{±p^{2n}}, tμ_{2})

of the E^{2ρ(2n)+1}-term, with homology

E(ϵ̄_{1}, λ_{2}) ⊗ P (μ_{2}^{±p^{2n}}) ⊗ P_{ρ(2n)+1}(tμ_{2}) .

This gives the stated E^{2ρ(2n)+2}-term. There is no room for any further differentials, since the spectral sequence is concentrated in a narrower vertical band than the horizontal width of the following differentials, so E^{2ρ(2n)+2} = E^{∞}.

Proof of Theorem 6.1. To make the inductive step to C_{p^{n+1}}, we use that the first d^{r}- differential of odd length in Ê^{∗}(C_{p^{n}}, ℓ/p) occurs for r = r_{0} = 2ρ(2n) + 1. It follows from [AR02, Lem. 5.2] that the terms Ê^{r}(C_{p^{n}}, ℓ/p) and Ê^{r}(C_{p^{n+1}}, ℓ/p) are isomorphic for r ≤ 2ρ(2n) + 1, via the Frobenius map (taking t^{i} to t^{i}) in even columns and the Verschiebung map (taking u_{n}t^{i} to u_{n+1}t^{i}) in odd columns. Furthermore, the differential d^{2ρ(2n)+1} is zero in the latter spectral sequence. This proves the part of Theorem 6.1 for n + 1 that concerns the differentials leading up to the term

\begin{aligned}Ê & ^{2ρ(2n)+2}(C_{p^{n+1}}, ℓ/p) = E(u_{n+1}, λ_{2}) ⊗ 𝔽_{p}\{t^{−i} | 0 < i < p\} ⊗ P (t^{±p^{2}}) \\ & ⊕ ⨁_{k=2}^{n} E(u_{n+1}, λ_{2}) ⊗ 𝔽_{p}\{t^{j} | j ∈ ℤ, v_{p}(j) = 2k − 2\} ⊗ P_{ρ(2k−3)}(tμ_{2}) \\ & ⊕ ⨁_{k=2}^{n} E(u_{n+1}, ϵ̄_{1}) ⊗ 𝔽_{p}\{t^{j}λ_{2} | j ∈ ℤ, v_{p}(j) = 2k − 1\} ⊗ P_{ρ(2k−2)}(tμ_{2}) \\ & ⊕ E(u_{n+1}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2n}}, tμ_{2}) . \end{aligned} (6.2)

Next we use the following commutative diagram, where we abbreviate THH(B) to T (B) for typographical reasons:

[figure: (ρ_{p}^{∗}T (B)^{tC_{p}})^{hC_{p^{n}}}; F; T (B)^{hC_{p^{n}}}; Γ̂_{1}^{hC_{p^{n}}}; F; T (B)^{C_{p^{n}}}; Γ_{n}; Γ̂_{n+1}; F; T (B)^{tC_{p^{n+1}}}; F; ρ_{p}^{∗}T (B)^{tC_{p}}; T (B); Γ̂_{1}; T (B); Γ̂_{1}; ρ_{p}^{∗}T (B)^{tC_{p}} .]  (6.3)

The horizontal maps all induce (2p − 2)-coconnected maps in V (1)-homotopy for B = ℓ/p. Each F is a Frobenius map, forgetting invariance under a C_{p^{n}}-action. Thus the map Γ̂_{n+1} to the right induces an isomorphism of E(λ_{2}) ⊗ P (v_{2})-modules in all degrees ∗ > (2p − 2) from V (1)_{∗}THH(ℓ/p)^{C_{p^{n}}}, implicitly identified to the left with the abutment of μ_{2}^{−1}E^{∗}(C_{p^{n}}, ℓ/p), to V (1)_{∗}THH(ℓ/p)^{tC_{p^{n+1}}}, which is the abutment of Ê^{∗}(C_{p^{n+1}}, ℓ/p). The diagram above ensures that the isomorphism induced by Γ̂_{n+1} is compatible with the one induced by Γ̂_{1}. By Proposition 5.8 it takes ϵ̄_{1}, λ_{2} and μ_{2} to ϵ̄_{1}, λ_{2} and t^{−p^{2}} up to a unit factor in 𝔽_{p}, respectively, and similarly for monomials in these classes.

We focus on the summand

E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{μ_{2}^{j} | j ∈ ℤ, v_{p}(j) = 2n − 2} ⊗ P_{ρ(2n−1)}(tμ_{2})

in μ_{2}^{−1}E^{∞}(C_{p^{n}}, ℓ/p), abutting to V (1)_{∗}THH(ℓ/p)^{C_{p^{n}}} in degrees > (2p−2). In the P (v_{2})- module structure on the abutment, each class μ_{2}^{j} with v_{p}(j) = 2n − 2, j > 0, generates a copy of P_{ρ(2n−1)}(v_{2}), since there are no permanent cycles in the same total degree as y = (tμ_{2})^{ρ(2n−1)} · μ_{2}^{j} that have lower (= more negative) homotopy fixed point filtration. See Lemma 6.5 below for the elementary verification. The P (v_{2})-module isomorphism induced by Γ̂_{n+1} must take this to a copy of P_{ρ(2n−1)}(v_{2}) in V (1)_{∗}THH(ℓ/p)^{tC_{p^{n+1}}}, generated by t^{−p^{2}j}.

Writing i = −p^{2}j, we deduce that for v_{p}(i) = 2n, i < 0, the infinite cycle z = (tμ_{2})^{ρ(2n−1)} · t^{i} must represent zero in the abutment, and must therefore be hit by a differential z = d^{r}(x) in the C_{p^{n+1}}-Tate spectral sequence. Here r ≥ 2ρ(2n) + 2.

Since z generates a free copy of P (tμ_{2}) in the E^{2ρ(2n)+2}-term displayed in (6.2), and d^{r} is P (tμ_{2})-linear, the class x cannot be annihilated by any power of tμ_{2}. This means that x must be contained in the summand

E(u_{n+1}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2n}}, tμ_{2})

of Ê^{2ρ(2n)+2}(C_{p^{n+1}}, ℓ/p). By an elementary check of bidegrees, see Lemma 6.6 below, the only possibility is that x has vertical degree (2p − 1), so that we have differentials

d^{2ρ(2n+1)}(t^{p^{2n+1}−p^{2n+2}+i} · ϵ̄_{1}) ≐ (tμ_{2})^{ρ(2n−1)} · t^{i}

for all i < 0 with v_{p}(i) = 2n. The cases i > 0 follow by the module structure over the C_{p^{n+1}}-Tate spectral sequence for ℓ. The remaining two differentials,

d^{2ρ(2n+2)}(t^{p^{2n+1}−p^{2n+2}}) ≐ λ_{2} · t^{p^{2n+1}} · (tμ_{2})^{ρ(2n)}

and

d^{2ρ(2n+2)+1}(u_{n+1} · t^{−p^{2n+2}}) ≐ (tμ_{2})^{ρ(2n)+1}

are also present in the C_{p^{n+1}}-Tate spectral sequence for ℓ, see [AR02, Th. 6.1], hence follow in the present case by the module structure. With this we have established the complete differential pattern asserted by Theorem 6.1.

Lemma 6.5. For j ∈ ℤ with v_{p}(j) = 2n − 2, where n ≥ 1, there are no classes in μ_{2}^{−1}E^{∞}(C_{p^{n}}, ℓ/p) in the same total degree as y = (tμ_{2})^{ρ(2n−1)} · μ_{2}^{j} that have lower homotopy fixed point filtration.

Proof. The total degree of y is 2(p^{2n+2} − p^{2n+1} + p − 1) + 2p^{2}j ≡ (2p − 2) mod 2p^{2n}, which is even.

Looking at the formula for μ_{2}^{−1}E^{∞}(C_{p^{n}}, ℓ/p) in Corollary 6.4, the classes of lower filtration than y all lie in the terms

E(u_{n}, ϵ̄_{1}) ⊗ 𝔽_{p}{λ_{2}μ_{2}^{i} | j ∈ ℤ, v_{p}(i) = 2n − 1} ⊗ P_{ρ(2n)}(tμ_{2})

and

E(ϵ̄_{1}, λ_{2}) ⊗ P (μ_{2}^{±p^{2n}}) ⊗ P_{ρ(2n)+1}(tμ_{2}) .

Those in even total degree and of lower filtration than y are

u_{n}λ_{2} · μ_{2}^{i}(tμ_{2})^{e}, ϵ̄_{1}λ_{2} · μ_{2}^{i}(tμ_{2})^{e}

with v_{p}(i) = 2n − 1, ρ(2n − 1) < e < ρ(2n), and

μ_{2}^{i}(tμ_{2})^{e}, ϵ̄_{1}λ_{2} · μ_{2}^{i}(tμ_{2})^{e}

with v_{p}(i) ≥ 2n, ρ(2n − 1) < e ≤ ρ(2n).

The total degree of u_{n}λ_{2} · μ_{2}^{i}(tμ_{2})^{e} for v_{p}(i) = 2n − 1 is (−1) + (2p^{2} − 1) + 2p^{2}i + (2p^{2} − 2)e ≡ (2p^{2} − 2)(e + 1) mod 2p^{2n}. For this to agree with the total degree of y, we must have (2p − 2) ≡ (2p^{2} − 2)(e + 1) mod 2p^{2n}, so (e + 1) ≡ 1/(1 + p) mod p^{2n} and e ≡ ρ(2n − 1) − 1 mod p^{2n}. There is no such e with ρ(2n − 1) < e < ρ(2n).

The total degree of ϵ̄_{1}λ_{2} · μ_{2}^{i}(tμ_{2})^{e} for v_{p}(i) = 2n − 1 is (2p − 1) + (2p^{2} − 1) + 2p^{2}i + (2p^{2} − 2)e ≡ 2p + (2p^{2} − 2)(e + 1) mod 2p^{2n}. To agree with that of y, we must have (2p − 2) ≡ 2p + (2p^{2} − 2)(e + 1) mod 2p^{2n}, so (e + 1) ≡ 1/(1 − p^{2}) mod p^{2n} and e ≡ ρ(2n) mod p^{2n}. There is no such e with ρ(2n − 1) < e < ρ(2n).

The total degree of μ_{2}^{i}(tμ_{2})^{e} for v_{p}(i) ≥ 2n is 2p^{2}i + (2p^{2} − 2)e ≡ (2p^{2} − 2)e mod 2p^{2n}. To agree with that of y, we must have (2p − 2) ≡ (2p^{2} − 2)e mod 2p^{2n}, so e ≡ 1/(1 + p) ≡ ρ(2n − 1) mod p^{2n}. There is no such e with ρ(2n − 1) < e ≤ ρ(2n).

The total degree of ϵ̄_{1}λ_{2} · μ_{2}^{i}(tμ_{2})^{e} for v_{p}(i) ≥ 2n is (2p − 1) + (2p^{2} − 1) + 2p^{2}i + (2p^{2} − 2)e. To agree modulo 2p^{2n} with that of y, we must have e ≡ ρ(2n) mod p^{2n}. The only such e with ρ(2n − 1) < e ≤ ρ(2n) is e = ρ(2n). But in that case, the total degree of ϵ̄_{1}λ_{2} · μ_{2}^{i}(tμ_{2})^{e} is 2p + 2p^{2}i + (2p^{2} − 2)(ρ(2n) + 1) = 2(p^{2n+2} + p − 1) + 2p^{2}i. To be equal to that of y, we must have 2p^{2}i + 2p^{2n+1} = 2p^{2}j, which is impossible for v_{p}(i) ≥ 2n and v_{p}(j) = 2n − 2.

Lemma 6.6. For v_{p}(i) = 2n, n ≥ 1 and z = (tμ_{2})^{ρ(2n−1)} · t^{i}, the only class in

E(u_{n+1}, ϵ̄_{1}, λ_{2}) ⊗ P (t^{±p^{2n}}, tμ_{2})

that can support a non-zero differential d^{r}(x) = z for r ≥ 2ρ(2n) + 2 is (a unit times)

x = t^{p^{2n+1}−p^{2n+2}+i} · ϵ̄_{1} .

Proof. The class z has total degree (2p^{2} − 2)ρ(2n − 1) − 2i = 2p^{2n+2} − 2p^{2n+1} + 2p − 2 − 2i ≡ (2p − 2) mod 2p^{2n}, which is even, and vertical degree 2p^{2}ρ(2n − 1). Hence x has odd total degree, and vertical degree at most 2p^{2}ρ(2n − 1) − 2ρ(2n) − 1 = 2p^{2n+2} − 2p^{2n+1} − ⋯ − 2p^{3} − 1. This leaves the possibilities

u_{n+1} · t^{j}(tμ_{2})^{e}, ϵ̄_{1} · t^{j}(tμ_{2})^{e}, λ_{2} · t^{j}(tμ_{2})^{e}

with v_{p}(j) ≥ 2n and 0 ≤ e < p^{2n} − p^{2n−1} − ⋯ − p = ρ(2n − 1) − ρ(2n − 2) − 1, and

u_{n+1}ϵ̄_{1}λ_{2} · t^{j}(tμ_{2})^{e}

with v_{p}(j) ≥ 2n and 0 ≤ e < p^{2n} − p^{2n−1} − ⋯ − p − 1 = ρ(2n − 1) − ρ(2n − 2) − 2.

The total degree of x must be one more than the total degree of z, hence is congruent to (2p − 1) modulo 2p^{2n}.

The total degree of u_{n+1}·t^{j}(tμ_{2})^{e} is −1−2j+(2p^{2}−2)e ≡ −1+(2p^{2}−2)e mod 2p^{2n}. To have (2p − 1) ≡ −1 + (2p^{2} − 2)e mod 2p^{2n} we must have e ≡ −p/(1 − p^{2}) ≡ p^{2n} − p^{2n−1} − ⋯ − p mod p^{2n}, which does not happen for e in the allowable range.

The total degree of λ_{2} ·t^{j}(tμ_{2})^{e} is (2p^{2} −1)−2j +(2p^{2} −2)e ≡ (2p^{2} −1)+(2p^{2} −2)e mod 2p^{2n}. To have (2p − 1) ≡ (2p^{2} − 1) + (2p^{2} − 2)e mod 2p^{2n} we must have e ≡ −p/(1 + p) ≡ ρ(2n − 1) − 1 mod p^{2n}, which does not happen.

The total degree of u_{n+1}ϵ̄_{1}λ_{2} ·t^{j}(tμ_{2})^{e} is −1+(2p−1)+(2p^{2} −1)−2j +(2p^{2} −2)e ≡ (2p − 1) + (2p^{2} − 2)(e + 1) mod 2p^{2n}. To have (2p − 1) ≡ (2p − 1) + (2p^{2} − 2)(e + 1) mod 2p^{2n} we must have (e + 1) ≡ 0 mod p^{2n}, so e ≡ p^{2n} − 1 mod p^{2n}, which does not happen.

The total degree of ϵ̄_{1} · t^{j}(tμ_{2})^{e} is (2p − 1) − 2j + (2p^{2} − 2)e ≡ (2p − 1) + (2p^{2} − 2)e mod 2p^{2n}. To have (2p − 1) ≡ (2p − 1) + (2p^{2} − 2)e mod 2p^{2n}, we must have e ≡ 0 mod p^{2n}, so e = 0 is the only possibility in the allowable range. In that case, a check of total degrees shows that we must have j = p^{2n+1} − p^{2n+2} + i.

Corollary 6.7. V (1)_{∗}THH(ℓ/p)^{C_{p^{n}}} is finite in each degree.

Proof. This is clear by inspection of the E^{∞}-term in Corollary 6.2.

Lemma 6.8. The map G_{n} induces an isomorphism

V (1)_{∗}THH(ℓ/p)^{tC_{p^{n+1}}} ⟶^{≅} V (1)_{∗}(ρ_{p}^{∗}THH(ℓ/p)^{tC_{p}})^{hC_{p^{n}}}

in all degrees. In the limit over the Frobenius maps F, there is a map G inducing an isomorphism

V (1)_{∗}THH(ℓ/p)^{tS^{1}} ⟶^{≅} V (1)_{∗}(ρ_{p}^{∗}THH(ℓ/p)^{tC_{p}})^{hS^{1}} .  (6.4)

Proof. As remarked after diagram (6.1), G_{n} induces an isomorphism in V (1)-homotopy above degree (2p − 2). The permanent cycle t^{−p^{2n+2}} in Ê^{∞}(C_{p^{n+1}}, ℓ) acts invertibly on Ê^{∞}(C_{p^{n+1}}, ℓ/p), and its image G_{n}(t^{−p^{2n+2}}) = μ_{2}^{p^{2n}} in μ_{2}^{−1}E^{∞}(C_{p^{n}}, ℓ) acts invertibly on μ_{2}^{−1}E^{∞}(C_{p^{n}}, ℓ/p). Therefore the module action derived from the ℓ-algebra structure on ℓ/p ensures that G_{n} induces isomorphisms in V (1)-homotopy in all degrees.

Theorem 6.9. The isomorphism (6.4) admits the following description at the associated graded level:

(a) The associated graded of V (1)_{∗}THH(ℓ/p)^{tS^{1}} for the S^{1}-Tate spectral sequence is

Ê^{∞}(S^{1}, ℓ/p) = E(λ_{2}) ⊗ 𝔽_{p}{t^{−i} | 0 < i < p} ⊗ P (t^{±p^{2}})

⊕ ⨁_{k≥2} E(λ_{2}) ⊗ 𝔽_{p}{t^{j} | j ∈ ℤ, v_{p}(j) = 2k − 2} ⊗ P_{ρ(2k−3)}(tμ_{2})

⊕ ⨁_{k≥2} E(ϵ̄_{1}) ⊗ 𝔽_{p}{t^{j}λ_{2} | j ∈ ℤ, v_{p}(j) = 2k − 1} ⊗ P_{ρ(2k−2)}(tμ_{2})

⊕ E(ϵ̄_{1}, λ_{2}) ⊗ P (tμ_{2}) .

(b) The associated graded of V (1)_{∗}THH(ℓ/p)^{hS^{1}} for the S^{1}-homotopy fixed point spectral sequence maps by a (2p − 2)-coconnected map to

μ_{2}^{−1}E^{∞}(S^{1}, ℓ/p) = E(λ_{2}) ⊗ 𝔽_{p}{μ_{0}^{i} | 0 < i < p} ⊗ P (μ_{2}^{±1})

⊕ ⨁_{k≥1} E(λ_{2}) ⊗ 𝔽_{p}{μ_{2}^{j} | j ∈ ℤ, v_{p}(j) = 2k − 2} ⊗ P_{ρ(2k−1)}(tμ_{2})

⊕ ⨁_{k≥1} E(ϵ̄_{1}) ⊗ 𝔽_{p}{λ_{2}μ_{2}^{j} | j ∈ ℤ, v_{p}(j) = 2k − 1} ⊗ P_{ρ(2k)}(tμ_{2})

⊕ E(ϵ̄_{1}, λ_{2}) ⊗ P (tμ_{2}) .

(c) The isomorphism from (a) to (b) induced by G takes t^{−i} to μ_{0}^{i} for 0 < i < p and t^{i} to μ_{2}^{j} for i+p^{2}j = 0, up to a unit factor in 𝔽_{p}. Furthermore, it takes multiples by ϵ̄_{1}, λ_{2} or tμ_{2} in the source to the same multiples in the target, up to a unit factor in 𝔽_{p}.

Proof. Claims (a) and (b) follow by passage to the limit over n from Corollaries 6.2 and 6.4. Claim (c) follows by passage to the same limit from the formulas for the isomorphism induced by Γ̂_{n+1}, which were given below diagram (6.3).

7 Topological cyclic homology

By definition, there is a fiber sequence

T C(B; p) ⟶^{π} T F (B; p) ⟶^{R−1} T F (B; p) → ΣT C(B; p)

inducing a long exact sequence

… ⟶^{∂} V (1)_{∗}T C(B; p) ⟶^{π} V (1)_{∗}T F (B; p) ⟶^{R_{∗}−1} V (1)_{∗}T F (B; p) ⟶^{∂} …  (7.1)

in V (1)-homotopy. By Corollary 5.9, there are (2p − 2)-coconnected maps Γ and Γ̂ from V (1)_{∗}T F (ℓ/p; p) to V (1)_{∗}THH(ℓ/p)^{hS^{1}} and V (1)_{∗}THH(ℓ/p)^{tS^{1}}, respectively. We model V (1)_{∗}T F (ℓ/p; p) in degrees > (2p − 2) by the map Γ̂ to the S^{1}-Tate construction. Then, by diagram (6.1), R_{∗} is modeled in the same range of degrees by the chain of maps below :

\xymatrix{V (1)_{∗}THH(B)^{tS^{1}} \ar@{-->}[dr]^{G} & V (1)_{∗}THH(B)^{hS^{1}} \ar[d]^{(Γ̂_{1})^{hS^{1}}} \ar[r]^{R_{∗}^{h}} & V (1)_{∗}THH(B)^{tS^{1}} \\  & V (1)_{∗}(ρ_{p}^{∗}THH(B)^{tC_{p}})^{hS^{1}} .}

Here R^{h} induces a map of spectral sequences

E^{∗}(R^{h}): E^{∗}(S^{1}, B) → Ê^{∗}(S^{1}, B)

(abutting to R_{∗}^{h}), which at the E^{2}-term equals the inclusion that algebraically inverts t. When B = ℓ/p, the left hand map G is an isomorphism by Lemma 6.8, and the middle (wrong-way) map is (2p − 2)-coconnected.

Proposition 7.1. In degrees > (2p − 2), the homomorphism

E^{∞}(R^{h}): E^{∞}(S^{1}, ℓ/p) → Ê^{∞}(S^{1}, ℓ/p)

maps

(a) E(ϵ̄_{1}, λ_{2}) ⊗ P (tμ_{2}) identically to the same expression;

(b) E(λ_{2})⊗𝔽_{p}{μ_{2}^{−j}}⊗P_{ρ(2k−1)}(tμ_{2}) surjectively onto E(λ_{2})⊗𝔽_{p}{t^{j}}⊗P_{ρ(2k−3)}(tμ_{2}) for each k ≥ 2, j = dp^{2k−2}, 0 < d < p^{2} − p and p ∤ d;

(c) E(ϵ̄_{1})⊗𝔽_{p}{λ_{2}μ_{2}^{−j}}⊗P_{ρ(2k)}(tμ_{2}) surjectively onto E(ϵ̄_{1})⊗𝔽_{p}{t^{j}λ_{2}}⊗P_{ρ(2k−2)}(tμ_{2}) for each k ≥ 2, j = dp^{2k−1} and 0 < d < p;

(d) the remaining terms to zero.

Notice that in the statements (b) and (c) above, we abuse notation and indentify the components of degree > 2p − 2 of E^{∞}(S^{1}, ℓ/p) and μ_{2}^{−1}E^{∞}(S^{1}, ℓ/p), using Theorem 6.9(b).

Proof. Consider the summands of E^{∞}(S^{1}, ℓ/p) and Ê^{∞}(S^{1}, ℓ/p) given in Theorem 6.9. Clearly, the first term E(λ_{2}) ⊗ 𝔽_{p}{μ_{0}^{i} | 0 < i < p} ⊗ P (μ_{2}) goes to zero (these classes are hit by d^{2}-differentials), and the last term E(ϵ̄_{1}, λ_{2}) ⊗ P (tμ_{2}) maps identically to the same term. This proves (a) and part of (d).

For each k ≥ 1 and j = dp^{2k−2} with p ∤ d, the term E(λ_{2}) ⊗ 𝔽_{p}{μ_{2}^{−j}} ⊗ P_{ρ(2k−1)}(tμ_{2}) maps to the term E(λ_{2}) ⊗ 𝔽_{p}{t^{j}} ⊗ P_{ρ(2k−3)}(tμ_{2}), except that the target is zero for k = 1. In symbols, the element λ_{2}^{δ}μ_{2}^{−j}(tμ_{2})^{i} maps to the element λ_{2}^{δ}t^{j}(tμ_{2})^{i−j}. If d < 0, then the t-exponent in the target is bounded above by dp^{2k−2} + ρ(2k − 3) < 0, so the target lives in the right half-plane and is essentially not hit by the source, which lives in the left half-plane. If d > p^{2} − p, then the total degree in the source is bounded above by (2p^{2} − 1) − 2dp^{2k} + ρ(2k − 1)(2p^{2} − 2) < 2p − 2, so the source lives in total degree < (2p−2) and will be disregarded. If 0 < d < p^{2}−p, then ρ(2k−1)−dp^{2k−2} > ρ(2k−3) and −dp^{2k−2} < 0, so the source surjects onto the target. This proves (b) and part of (d).

Lastly, for each k ≥ 1 and j = dp^{2k−1} with p ∤ d, the term E(ϵ̄_{1}) ⊗ 𝔽_{p}{λ_{2}μ_{2}^{−j}} ⊗ P_{ρ(2k)}(tμ_{2}) maps to the term E(ϵ̄_{1}) ⊗ 𝔽_{p}{t^{j}λ_{2}} ⊗ P_{ρ(2k−2)}(tμ_{2}). The target is zero for k = 1. If d < 0, then dp^{2k−1} + ρ(2k − 2) < 0 so the target lives in the right half-plane. If d > p, then (2p − 1) + (2p^{2} − 1) − 2dp^{2k+1} + ρ(2k)(2p^{2} − 2) < 2p − 2, so the source lives in total degree < (2p − 2). If 0 < d < p, then ρ(2k) − dp^{2k−1} > ρ(2k − 2) and −dp^{2k−1} < 0, so the source surjects onto the target. This proves (c) and the remaining part of (d).

Definition 7.2. Let

\begin{aligned}A & = E(ϵ̄_{1}, λ_{2}) ⊗ P (tμ_{2}) \\ B_{k} & = E(λ_{2}) ⊗ 𝔽_{p}\{t^{dp^{2k−2}} | 0 < d < p^{2} − p, p ∤ d\} ⊗ P_{ρ(2k−3)}(tμ_{2}) \\ C_{k} & = E(ϵ̄_{1}) ⊗ 𝔽_{p}\{t^{dp^{2k−1}}λ_{2} | 0 < d < p\} ⊗ P_{ρ(2k−2)}(tμ_{2}) \end{aligned}

for k ≥ 2 and let D be the span of the remaining monomials in Ê^{∞}(S^{1}, ℓ/p). Let B = ⨁_{k≥2} B_{k} and C = ⨁_{k≥2} C_{k}. Then Ê^{∞}(S^{1}, ℓ/p) = A ⊕ B ⊕ C ⊕ D.

Proposition 7.3. In degrees > (2p − 2), there are closed subgroups Ã = E(ϵ̄_{1}, λ_{2}) ⊗ P (v_{2}), B̃_{k}, C̃_{k} and D̃ in V (1)_{∗}T F (ℓ/p; p), represented by the subgroups A, B_{k}, C_{k} and D of Ê^{∞}(S^{1}, ℓ/p), respectively, such that the homomorphism R_{∗} = V (1)_{∗}R induced by the restriction map R

(a) is the identity on Ã;

(b) maps B̃_{k+1} surjectively onto B̃_{k} for all k ≥ 2;

(c) maps C̃_{k+1} surjectively onto C̃_{k} for all k ≥ 2;

(d) is zero on B̃_{2}, C̃_{2} and D̃.

In these degrees, V (1)_{∗}T F (ℓ/p; p) ≅ Ã ⊕ B̃ ⊕ C̃ ⊕ D̃, where B̃ = ∏_{k≥2} B̃_{k} and C̃ = ∏_{k≥2} C̃_{k}.

Proof. The proof is the same as the proof of [AR02, Thm. 7.7], except that in the present paper we work with the Tate model THH(ℓ/p)^{tS^{1}} for T F (ℓ/p; p), in place of the homotopy fixed point model THH(ℓ/p)^{hS^{1}}. The computations are made in V (1)-homotopy, and we disregard all classes in total degrees ≤ (2p − 2). For example with this convention we write μ_{2}^{−1}E^{∞}(S^{1}, ℓ/p) ≅ E^{∞}(S^{1}, ℓ/p), using the same abuse of notation as in Proposition 7.1.

In these terms, the restriction homomorphism R_{∗} is given at the level of E^{∞}-terms as the composite of the isomorphism

G_{∗}: Ê^{∞}(S^{1}, ℓ/p) → μ_{2}^{−1}E^{∞}(S^{1}, ℓ/p) ≅ E^{∞}(S^{1}, ℓ/p)

and the map

E^{∞}(R^{h}): E^{∞}(S^{1}, ℓ/p) → Ê^{∞}(S^{1}, ℓ/p) .

As an endomorphism of Ê^{∞}(S^{1}, ℓ/p), this composite E^{∞}(R^{h})G_{∗} is the identity on A, maps B_{k+1} onto B_{k} and C_{k+1} onto C_{k} for all k ≥ 2, and is zero on B_{2}, C_{2} and D, by Theorem 6.9(c) and Proposition 7.1. The task is to find closed lifts of these groups to V (1)_{∗}T F (ℓ/p; p) such that R_{∗} has similar properties.

Let Ã = E(ϵ̄_{1}, λ_{2}) ⊗ P (v_{2}) ⊂ V (1)_{∗}T F (ℓ/p; p) be the (degreewise finite, hence closed) subalgebra generated by the images of the classes ϵ̄_{1}^{K}, λ_{2} and v_{2} in V (1)_{∗}K(ℓ/p). Then Ã lifts A and consists of classes in the image of the trace map from V (1)_{∗}K(ℓ/p). Hence R_{∗} is the identity on Ã.

We fix k ≥ 2 and choose, for all n ≥ 0, a subgroup B_{k}^{n} ⊂ B_{k+n}, as follows. We take

B_{k}^{0} = B_{k} ∩ ker(E^{∞}(R^{h})G_{∗})

= \begin{cases}B_{2} & for k = 2, \\ E(λ_{2}) ⊗ ⨁_{0<d<p^{2}−p, p∤d}𝔽_{p}\{t^{dp^{2k−2}}\} ⊗ P_{dp^{2k−4}+ρ(2k−5)}^{ρ(2k−3)−1}(tμ_{2}) & for k ≥ 3, \end{cases}

where P_{a}^{b}(tμ_{2}) = 𝔽_{p}{(tμ_{2})^{c} | a ≤ c ≤ b}. We proceed by induction on n for n ≥ 1, choosing a subgroup B_{k}^{n} of B_{k+n} mapping isomorphically onto B_{k}^{n−1} under E^{∞}(R^{h})G_{∗} (such a group exists by Theorem 6.9(c) and Proposition 7.1(b)). We then have

B_{k} = ⨁_{n=0}^{k−2} B_{k−n}^{n} .

By the argument given on top of page 31 of [AR02], we can choose a lift B̃_{k}^{0} of B_{k}^{0} with

B̃_{k}^{0} ⊂ im(R_{∗}) ∩ ker(R_{∗})

in V (1)_{∗}T F (ℓ/p; p). By induction on n ≥ 1, we choose a lift B̃_{k}^{n} ⊂ im(R_{∗}) of B_{k}^{n} mapping isomorphically onto B̃_{k}^{n−1} under R_{∗}. Such a choice is possible since the image of R_{∗} on V (1)_{∗}T F (ℓ/p; p) equals the image of its restriction to im(R_{∗}), see [AR02, p. 30]. Now

B̃_{k} = ⨁_{n=0}^{k−2} B̃_{k−n}^{n}

is a (degreewise finite, hence closed) lift of B_{k} with R_{∗}(B̃_{2}) = 0 and R_{∗}(B̃_{k}) = B̃_{k−1} for k ≥ 3.

To construct C̃_{k} we proceed as for B̃_{k} above, starting with C_{2}^{0} = C_{2} and

\begin{aligned}C_{k}^{0} & = C_{k} ∩ ker(E^{∞}(R^{h})G_{∗}) \\ & = E(ϵ̄_{1}) ⊗ ⨁_{0<d<p} 𝔽_{p}\{t^{dp^{2k−1}}λ_{2}\} ⊗ P_{dp^{2k−3}+ρ(2k−4)}^{ρ(2k−2)−1}(tμ_{2}) \end{aligned}

for k ≥ 3, and using Theorem 6.9(c) and Proposition 7.1(c) to choose C_{k}^{n} for n ≥ 1.

It remains to construct D̃. By Proposition 7.1(d), the isomorphism G_{∗} maps D into ker(E^{∞}(R^{h})). By [AR02, Lem. 7.3] the representatives in E^{∞}(S^{1}, ℓ/p) of the kernel of R_{∗}^{h} equal the kernel of E^{∞}(R^{h}). It follows that the representatives in Ê^{∞}(S^{1}, ℓ/p) of the kernel of R_{∗} are mapped isomorphically by G_{∗} to ker(E^{∞}(R^{h})). Hence we can pick a vector space basis for D, choose a representative in ker(R_{∗}) ⊂ V (1)_{∗}T F (ℓ/p; p) of each basis element, and let D̃ ⊂ V (1)_{∗}T F (ℓ/p; p) be the closure of the vector space spanned by these chosen representatives. This closure is contained in ker(R_{∗}) since R_{∗} is continuous. Hence R_{∗} is zero on D̃. Proposition 7.4. In degrees > (2p − 2) there are isomorphisms

ker(R_{∗} − 1) ≅ Ã ⊕ lim_{k} B̃_{k} ⊕ lim_{k} C̃_{k}
≅ E(ϵ̄_{1}, λ_{2}) ⊗ P (v_{2})
⊕ E(λ_{2}) ⊗ 𝔽_{p}{t^{d} | 0 < d < p^{2} − p, p ∤ d} ⊗ P (v_{2})
⊕ E(ϵ̄_{1}) ⊗ 𝔽_{p}{t^{dp}λ_{2} | 0 < d < p} ⊗ P (v_{2})

and cok(R_{∗} − 1) ≅ Ã = E(ϵ̄_{1}, λ_{2}) ⊗ P (v_{2}). Hence there is an isomorphism

V (1)_{∗}T C(ℓ/p; p) ≅ E(∂, ϵ̄_{1}, λ_{2}) ⊗ P (v_{2})
⊕ E(λ_{2}) ⊗ 𝔽_{p}{t^{d} | 0 < d < p^{2} − p, p ∤ d} ⊗ P (v_{2})
⊕ E(ϵ̄_{1}) ⊗ 𝔽_{p}{t^{dp}λ_{2} | 0 < d < p} ⊗ P (v_{2})

in these degrees, where ∂ has degree −1 and represents the image of 1 under the connecting map ∂ in (7.1). Proof. By Proposition 7.3, the homomorphism R_{∗} − 1 is zero on Ã and an isomorphism on D̃. Furthermore, there is an exact sequence

0 → lim_{k} B̃_{k} → ∏_{k≥2} B̃_{k} ⟶^{R_{∗}−1} ∏_{k≥2} B̃_{k} → lim^{1}_{k} B̃_{k} → 0

and similarly for the C’s. The derived limit on the right vanishes since each B̃_{k+1} surjects onto B̃_{k}.

Multiplication by tμ_{2} in each B_{k} is realized by multiplication by v_{2} in B̃_{k}. Each B̃_{k} is a sum of 2(p − 1)^{2} cyclic P (v_{2})-modules, and since ρ(2k − 3) grows to infinity with k their limit is a free P (v_{2})-module of the same rank, with the indicated generators t^{d} and t^{d}λ_{2} for 0 < d < p^{2} − p, p ∤ d. The argument for the C’s is practically the same.

The long exact sequence (7.1) yields the short exact sequence

0 → Σ^{−1} cok(R_{∗} − 1) ⟶^{∂} V (1)_{∗}T C(ℓ/p; p) ⟶^{π} ker(R_{∗} − 1) → 0 ,

from which the formula for the middle term follows. Remark 7.5. A more obvious set of E(λ_{2})⊗P (v_{2})-module generators for lim_{k} B̃_{k} would be the classes t^{dp^{2}} in B_{2} ≅ B̃_{2}, for 0 < d < p^{2} −p, p ∤ d. We have a commutative diagram

\xymatrix{T F (ℓ/p; p) \ar[r]^{Γ̂} \ar[d] & THH(ℓ/p)^{tS^{1}} \ar[d]^{F G} \\ THH(ℓ/p)^{C_{p}} \ar[r]^{G_{1}Γ̂_{2}} & (ρ_{p}^{∗}THH(ℓ/p)^{tC_{p}})^{hC_{p}} .}

Under the left-hand canonical map T F (ℓ/p; p) → THH(ℓ/p)^{C_{p}}, modeled here by

F G: THH(ℓ/p)^{tS^{1}} → (ρ_{p}^{∗}THH(ℓ/p)^{tC_{p}})^{hS^{1}} → (ρ_{p}^{∗}THH(ℓ/p)^{tC_{p}})^{hC_{p}} ,

the class t^{dp^{2}} maps to the class μ_{2}^{−d}. Since we are only concerned with degrees > (2p − 2) we may equally well use its v_{2}-power multiple (tμ_{2})^{d} · μ_{2}^{−d} = t^{d} as generator, with the advantage that it is in the image of the localization map

THH(ℓ/p)^{hC_{p}} → (ρ_{p}^{∗}THH(ℓ/p)^{tC_{p}})^{hC_{p}} .

Hence the class denoted t^{d} in lim_{k} B̃_{k} is chosen so as to map under T F (ℓ/p; p) → THH(ℓ/p)^{hC_{p}} to t^{d} in E^{∞}(C_{p}, ℓ/p). Similarly, the class denoted t^{dp}λ_{2} in lim_{k} C̃_{k} is chosen so as to map to t^{dp}λ_{2} in E^{∞}(C_{p}, ℓ/p).

The map π: ℓ/p → ℤ/p is (2p−2)-connected, hence induces (2p−1)-connected maps π_{∗}: K(ℓ/p) → K(ℤ/p) and π_{∗}: V (1)_{∗}T C(ℓ/p; p) → V (1)_{∗}T C(ℤ/p; p), by [BM94, Prop. 10.9] and [Dun97, p. 224]. Here T C(ℤ/p; p) ≃ Hℤ_{p} ∨ Σ^{−1}Hℤ_{p} and we have an isomorphism V (1)_{∗}T C(ℤ/p; p) ≅ E(∂, ϵ̄_{1}), so we can recover V (1)_{∗}T C(ℓ/p; p) in degrees ≤ (2p − 2) from this map.

Theorem 7.6. There is an isomorphism of P (v_{2})-modules

V (1)_{∗}T C(ℓ/p; p) ≅ P (v_{2}) ⊗ E(∂, ϵ̄_{1}, λ_{2})

⊕ P (v_{2}) ⊗ E(dlog v_{1}) ⊗ 𝔽_{p}{t^{d}v_{2} | 0 < d < p^{2} − p, p ∤ d}

⊕ P (v_{2}) ⊗ E(ϵ̄_{1}) ⊗ 𝔽_{p}{t^{dp}λ_{2} | 0 < d < p}

where dlog v_{1}·t^{d}v_{2} = t^{d}λ_{2}. The degrees are |∂| = −1, |ϵ̄_{1}| = |λ_{1}| = 2p−1, |λ_{2}| = 2p^{2}−1, |v_{2}| = 2p^{2} − 2, |t| = −2 and | dlog v_{1}| = 1.

The notation dlog v_{1} for the multiplier v_{2}^{−1}λ_{2} is suggested by the relation v_{1} · dlog p = λ_{1} in V (0)_{∗}T C(ℤ_{(p)}|ℚ; p).

Proof. Only the additive generators t^{d} for 0 < d < p^{2} − p, p ∤ d from Proposition 7.4 do not appear in V (1)_{∗}T C(ℓ/p; p), but their multiples by λ_{2} and positive powers of v_{2} do. This leads to the given formula, where dlog v_{1} · t^{d}v_{2} must be read as t^{d}λ_{2}.

By [HM97, Th. C] the cyclotomic trace map of [BHM93] induces cofiber sequences

K(B_{p})_{p} ⟶^{trc} T C(B; p)_{p} ⟶^{g} Σ^{−1}Hℤ_{p} → ΣK(B_{p})_{p}

for each connective S-algebra B with π_{0}(B_{p}) = ℤ_{p} or ℤ/p, and thus long exact sequences

⋯ → V (1)_{∗}K(B_{p}) ⟶^{trc} V (1)_{∗}T C(B; p) ⟶^{g} Σ^{−1}E(ϵ̄_{1}) → ….

This uses the identifications W (ℤ_{p})_{F} ≅ W (ℤ/p)_{F} ≅ ℤ_{p} of Frobenius coinvariants of rings of Witt vectors, and applies in particular for B = Hℤ_{(p)}, Hℤ/p, ℓ and ℓ/p.

36 Christian Ausoni, John Rognes

Theorem 7.7. There is an isomorphism of P (v_{2})-modules

V (1)_{∗}K(ℓ/p) ≅ P (v_{2}) ⊗ E(ϵ̄_{1}) ⊗ 𝔽_{p}{1, ∂λ_{2}, λ_{2}, ∂v_{2}}
⊕ P (v_{2}) ⊗ E(dlog v_{1}) ⊗ 𝔽_{p}{t^{d}v_{2} | 0 < d < p^{2} − p, p ∤ d}
⊕ P (v_{2}) ⊗ E(ϵ̄_{1}) ⊗ 𝔽_{p}{t^{dp}λ_{2} | 0 < d < p} .

This is a free P (v_{2})-module of rank (2p^{2} − 2p + 8) and of zero Euler characteristic.

Proof. In the case B = ℤ/p, K(ℤ/p)_{p} ≃ Hℤ_{p} and the map g is split surjective up to homotopy. So the induced homomorphism to V (1)_{∗}Σ^{−1}Hℤ_{p} = Σ^{−1}E(ϵ̄_{1}) is surjective. Since π: ℓ/p → ℤ/p induces a (2p − 1)-connected map in topological cyclic homology, and Σ^{−1}E(ϵ̄_{1}) is concentrated in degrees ≤ (2p − 2), it follows by naturality that also in the case B = ℓ/p the map g induces a surjection in V (1)-homotopy. The kernel of the surjection P (v_{2}) ⊗ E(∂, ϵ̄_{1}, λ_{2}) → Σ^{−1}E(ϵ̄_{1}) gives the first row in the asserted formula.

Acknowledgments. We would like to thank the referee for his many useful suggestions. Both authors were partially supported by the Max Planck Institute for Mathematics, Bonn.

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