ALGEBRAIC K-THEORY OF THE FIRST MORAVA K-THEORY

CHRISTIAN AUSONI AND JOHN ROGNES

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], 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. It is a homotopy commutative ring spectrum for p ≥ 5, with a preferred periodic class v_{2} ∈ V (1)_{∗}. 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 computed the V (1)-homotopy of K(ℓ_{p}) in [AR02], showing that it is essentially a free P (v_{2})-module on (4p + 4) generators. In particular, there are preferred classes λ_{1}, λ_{2} ∈ V (1)_{∗}K(ℓ_{p}) generating an exterior subalgebra E(λ_{1}, λ_{2}). Hence V (1)_{∗}K(ℓ/p) is an E(λ_{1}, λ_{2}) ⊗ P (v_{2})-module. The following is our main result, corresponding to Theorem 7.10 in the body of the paper.

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 E(λ_{1}, λ_{2}) ⊗ 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). Along the way we evaluate V (1)_{∗}T HH(ℓ/p), where T HH denotes topological Hochschild homology, as well as V (1)_{∗}T C(ℓ/p), see Proposition 4.6 and Theorem 7.8.

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) → K(ku) → K(KU) of Blumberg and Mandell [BM08] has the mod p Adams analogue

K(ℤ/p) → K(ℓ/p) → K(L/p) .

Using Quillen’s computation [Qui72] of K(ℤ/p), we obtain the following consequence:

2000 Mathematics Subject Classification. 19D55, 55N15.

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 E(λ_{1}, λ_{2}) ⊗ P (v_{2}^{±1})-modules

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

If the relation λ_{2} = v_{2} dlog v_{1} holds in V (1)_{∗}K(L/p), then there is an isomorphism of E(dlog v_{1}, λ_{1}) ⊗ 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 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}) appears to be a localization of L_{p} away from L/p, less drastic than the algebraic localization L_{p}[p^{−1}] = Lℚ_{p}. So far we do not have a proper definition of this S-algebraic fraction field, but by analogy with the localization sequence above, we expect that its algebraic K-theory appears in a localization cofiber sequence

K(L/p) → K(L_{p}) → K(ff (ℓ_{p})) ,

where the transfer map on the left is a K(L_{p})-module map. Taking this as a preliminary definition of the symbol K(ff (ℓ_{p})), we can use our computations to evaluate its V (1)- homotopy:

Theorem 1.3. Let p ≥ 5 be a prime, and define K(ff (ℓ_{p})) as the homotopy cofiber above. There is an isomorphism of P (v_{2}^{±1})-modules

V (1)_{∗}K(ff (ℓ_{p}))[v_{2}^{−1}] ≅ P (v_{2}^{±1}) ⊗ Λ_{∗}

where

Λ_{∗} ≅ E(∂v_{2}, dlog p, dlog v_{1})
⊕ E(dlog v_{1}) ⊗ 𝔽_{p}{t^{d}λ_{1} | 0 < d < p}
⊕ E(dlog v_{1}) ⊗ 𝔽_{p}{t^{d}v_{2} dlog p | 0 < d < p^{2} − p, p ∤ d}
⊕ E(dlog p) ⊗ 𝔽_{p}{t^{dp}λ_{2} | 0 < d < p} .

Here | dlog p | = 1, and the degrees of the other classes are as in Theorem 1.1. The localization homomorphism

V (1)_{∗}K(ff (ℓ_{p})) → V (1)_{∗}K(ff (ℓ_{p}))[v_{2}^{−1}]

is an isomorphism in degrees ∗ ≥ 2p.

In particular, the homotopy cofiber K(ff (ℓ_{p})) cannot be equivalent to the K(ℚ_{p})- module K(Lℚ_{p}), since V (1)_{∗}K(ℚ_{p}) is a torsion P (v_{2})-module.

We may now conjecturally interpret V (1)_{∗}K(ff (ℓ_{p}))[v_{2}^{−1}] in terms of Galois descent. Indeed, the second author conjectured that if Ω_{1} is an S-algebraic “separable closure” of ff (ℓ_{p}), then there is a homotopy equivalence

L_{K(2)}K(Ω_{1}) ≃ E_{2} .

Here E_{2} is Morava’s second E-theory [GH04], with coefficients (E_{2})_{∗} = 𝕎(𝔽_{p^{2}})[[u_{1}]][u^{±1}], and L_{K(2)} denotes Bousfield localization with respect to the second Morava K-theory K(2), with coefficients K(2)_{∗} = 𝔽_{p}[v_{2}^{±1}]. The v_{2}-periodic V (1)-homotopy groups of K(Ω_{1}) will then be given by

V (1)_{∗}K(Ω_{1})[v_{2}^{−1}] ≅ 𝔽_{p^{2}}[u^{±1}].

We would expect to have a corresponding Galois descent spectral sequence

E_{s,t}^{2} = H_{Gal}^{−s}(ff (ℓ_{p}); 𝔽_{p^{2}}(t/2)) ⟹ V (1)_{s+t}K(ff (ℓ_{p}))[v_{2}^{−1}] .

If this spectral sequence collapses at E^{2} when p ≥ 5, as is the case for p-adic number fields when p ≥ 3, we get a conjectural description of the Galois cohomology of ff (ℓ_{p}) with coefficients in 𝔽_{p^{2}}(t/2), for all even t. Promisingly, this fits very well with the example of the Galois cohomology of ℚ_{p} with coefficients in 𝔽_{p}(t/2), with the difference that the absolute Galois group of ff (ℓ_{p}) has p-cohomological dimension 3 instead of 2. Also, by analogy with Tate–Poitou duality [Tat63] in the Galois cohomology of local number fields, there appears to be a perfect arithmetic duality pairing in the conjectural Galois cohomology of ff (ℓ_{p}), with fundamental class dual to ∂v_{2} · dlog p · dlog v_{1} in H_{Gal}^{3}(ff (ℓ_{p}); 𝔽_{p^{2}}(2)). This indicates that ff (ℓ_{p}) ought to be a form of S-algebraic two-dimensional local field, mixing three different residue characteristics. We elaborate more on this 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 T HH(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 T HH(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) and K(ℓ/p) in V (1)-homotopy.

2. Base change squares of S-algebras

We fix some notations. Let p be a prime, even or odd for now. Write X_{(p)} and X_{p} for the p-localization and the p-completion, respectively, of any spectrum or abelian group X. 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] 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.

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) ≅ T HH_{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}]
⟹ T HH_{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-localization in the category of ku-modules [EKMM97, VIII.4], which makes it an associative KUalgebra. 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 ℓalgebra, 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

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

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.

3. Topological Hochschild homology

We shall compute the V (1)-homotopy of the topological Hochschild homology T HH(−) and topological cyclic homology T C(−) of the S-algebras in diagram (2.2), 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 T HH 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 T HH(HR) to T HH(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 T HH-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.

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 H_{∗}(−) for homology with mod p coefficients. It takes values in A_{∗}-comodules, where A_{∗} is the dual Steenrod algebra [Mil58]. 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]. Then the zeroth Postnikov sections 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_{∗}(T HH(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}

They 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], [Hun96] or [Aus05, 4.3], 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 → T HH(B) is split for commutative B by the augmentation ϵ: T HH(B) → B. Thus there are unique representatives in Bökstedt filtration 1, with zero augmentation, for each of the classes σx. They correspond to 1 ⊗ x − x ⊗ 1 in the Hochschild complex, or just 1 ⊗ x in the normalized Hochschild complex. There are multiplicative extensions (στ̄_{k})^{p} = στ̄_{k+1} for k ≥ 0, see [AR05, 5.9], so

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

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

(3.2)  \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}

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

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

as an A_{∗}-comodule module over E_{∗∗}^{∞}(ℓ). In order to obtain H_{∗}(T HH(ℓ/p)), we need to resolve the A_{∗}-comodule and H_{∗}(T HH(ℓ))-module extensions. This is achieved in Lemma 3.6 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_{∗}(T HH(ℓ/p)), mapping to classes with the same names in H_{∗}(T HH(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_{∗}(T HH(ℓ/p)) as in H_{∗}(T HH(ℤ/p)), cf. (3.2).

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

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

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

to

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

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

In degree 2p, the map π_{∗} goes from

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

to

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

Again the latter extension is canonically split.

Lemma 3.4. There is a unique class y in H_{2p−1}(T HH(ℓ/p)) that is represented by τ̄_{0}(στ̄_{0})^{p−1} in E_{p−1,p}^{∞}(ℓ/p) and maps by π_{∗} to τ̄_{0}(στ̄_{0})^{p−1} − τ̄_{1} in H_{∗}(T HH(ℤ/p)).

Proof. This follows from naturality of the suspension operator σ and the multiplicative relation (στ̄_{0})^{p} = στ̄_{1} in H_{∗}(T HH(ℤ/p)). A class y in H_{2p−1}(T HH(ℓ/p)) represented by τ̄_{0}(στ̄_{0})^{p−1} is determined modulo ξ̄_{1}τ̄_{0}. Its image in H_{2p−1}(T HH(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}(T HH(ℓ/p)) = 𝔽_{p}{ξ̄_{1}στ̄_{0}}, so its image in H_{2p}(T HH(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. □

Remark 3.5. For p = 2 this can alternatively be read off from the explicit form [Wür91] of the commutator for the product μ in ℓ/p. The coequalizer C of the two maps

\xymatrix{ℓ/p ∧ ℓ/p \ar[r]^{μ} \ar[r]_{μτ} & ℓ/p}

maps to (the 1-skeleton of) T HH(ℓ/p). The commutator μ − μτ factors as

ℓ/p ∧ ℓ/p ⟶^{β∧β} Σℓ/p ∧ Σℓ/p ⟶^{μ} Σ^{2}ℓ/p ⟶^{v_{1}} ℓ/p

where β is the mod p Bockstein associated to the cofiber sequence ℓ ⟶^{p} ℓ ⟶^{i} ℓ/p and the cofiber of v_{1} is Hℤ/p. We get a map of cofiber sequences

[figure: ℓ/p ∧ ℓ/p; μ−μτ; μ(β∧β); ℓ/p; C; Σ^{2}ℓ/p; v_{1}; ℓ/p; Hℤ/p ,]

so there is a class in H_{3}(C) that maps to ξ̄_{1} ⊗ ξ̄_{1} in H_{2}(ℓ/p ∧ ℓ/p) and to ξ̄_{1}σξ̄_{1} in H_{3}(T HH(ℓ/p)), which also maps to ξ̄_{2} in the cofiber of v_{1}, i.e., whose A_{∗}-coaction contains the term ξ̄_{2} ⊗ 1. (The classes τ̄_{0} and τ̄_{1} go by the names ξ̄_{1} and ξ̄_{2} at p = 2.)

For odd primes there is a similar interpretation of how the non-commutativity of the product on ℓ/p provides an obstruction to splitting off the 0-simplices from the (p − 1)- skeleton of T HH(ℓ/p), where the cyclic permutation of the p factors in the (p−1)-simplex τ̄_{0}(στ̄_{0})^{p−1}, represented by the Hochschild cycle τ̄_{0} ⊗ ⋯ ⊗ τ̄_{0}, plays a similar role to the twist map τ above.

Let

H_{∗}(T HH(ℓ))/(σξ̄_{1}) ≅ H_{∗}(ℓ) ⊗ E(σξ̄_{2}) ⊗ P (στ̄_{2})

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

Lemma 3.6. There is an isomorphism of H_{∗}(T HH(ℓ))-modules

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

Hence H_{∗}(T HH(ℓ/p)) is a free module of rank 2p over H_{∗}(T HH(ℓ))/(σξ̄_{1}), generated by classes

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

in degrees 0 through 2p − 1. These generators are represented in E_{∗∗}^{∞}(ℓ/p) by the classes

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

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

τ̄_{0}(στ̄_{0})^{p−1} − τ̄_{1} .

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_{∗}(T HH(ℓ)) induce multiplication by the same symbols in E_{∗∗}^{∞}(ℓ/p), as given in (3.3). 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_{∗}(T HH(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_{∗}(T HH(ℤ/p)) is injective. The action of σξ̄_{1} is trivial on H_{∗}(T HH(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, the thing to check is that

\begin{aligned}(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 \end{aligned}

equals

(1 ⊗ π_{∗})(1 ⊗ y + τ̄_{0} ⊗ (στ̄_{0})^{p−1} − τ̄_{0} ⊗ ξ̄_{1} − τ̄_{1} ⊗ 1) .
□

We note that these results do not visibly depend on the particular choice of ℓ-algebra structure on ℓ/p.

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], [Oka84]. 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) .

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)_{∗}T HH(ℓ/p) as a module over V (1)_{∗}T HH(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_{∗}. 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}

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

H_{∗}(V (1) ∧ T HH(B)) ≅ H_{∗}(V (1)) ⊗ H_{∗}(T HH(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_{∗}(T HH(B)). Let

(4.1)  \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}

These are all A_{∗}-comodule primitive, where defined. By a dimension count,

(4.2)  \begin{aligned}V (1)_{∗}T HH(ℤ/p) & = E(ϵ_{0}, ϵ_{1}) ⊗ P (μ_{0}) \\ V (1)_{∗}T HH(ℤ_{(p)}) & = E(ϵ_{1}) ⊗ E(λ_{1}) ⊗ P (μ_{1}) \\ V (1)_{∗}T HH(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}}T HH(ℓ) was simply denoted μ in [AR02].

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

(4.3)  μ_{0}^{i} := 1 ∧ (στ̄_{0})^{i}

for 0 ≤ i ≤ p − 1 and

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

for 0 ≤ i ≤ p − 2 are A_{∗}-comodule primitive, hence lift uniquely to V (1)_{∗}T HH(ℓ/p). These map to the classes ϵ_{0}^{δ}μ_{0}^{i} in V (1)_{∗}T HH(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) ∧ T HH(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

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

is A_{∗}-comodule primitive. Its image under π_{∗} in H_{∗}(V (1) ∧ T HH(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)_{∗}T HH(ℓ)/(λ_{1}) ≅ E(λ_{2}) ⊗ P (μ_{2})

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

Proposition 4.6. There is an isomorphism of V (1)_{∗}T HH(ℓ)-modules

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

where the classes μ_{0}^{i}, ϵ_{0}μ_{0}^{i} and ϵ̄_{1} are defined in (4.3), (4.4) and (4.5) above. Multiplication by λ_{1} is 0, so this is a free module on the 2p generators

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

over V (1)_{∗}T HH(ℓ)/(λ_{1}). The map π_{∗} to V (1)_{∗}T HH(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. The multiplication by λ_{1} is 0 for degree and filtration reasons: λ_{1} has Bökstedt filtration 1 and cannot map to ϵ̄_{1} in Bökstedt filtration (p − 1). Similarly in higher degrees. □

5. The C_{p}-Tate construction

Let C = C_{p^{n}} denote the cyclic group of order p^{n}, considered as a closed subgroup of the circle group S^{1}. For each spectrum X with C-action, X_{hC} = EC_{+} ∧_{C} X and X^{hC} = F (EC_{+}, X)^{C} denote its homotopy orbit and homotopy fixed point spectra, as usual. We now write X^{tC} = [ẼC ∧ F (EC_{+}, X)]^{C} for the C-Tate construction on X, which was denoted t_{C}(X)^{C} in [GM95] and ℍ̂(C, X) in [AR02]. There are C-homotopy fixed point and C-Tate spectral sequences in V (1)-homotopy for X, with

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

and

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

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. 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.

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

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

of horizontal cofiber sequences. We abbreviate Ê_{∗∗}^{2}(C, T HH(B)) to Ê_{∗∗}^{2}(C, 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^{∗}(C, B) is a module spectral sequence over Ê^{∗}(C, A). The map R^{h} corresponds to the inclusion E_{∗∗}^{2}(C, B) → Ê_{∗∗}^{2}(C, B) from the second quadrant to the upper half-plane, for connective B.

In this section we compute V (1)_{∗}T HH(ℓ/p)^{tC_{p}} by means of the C_{p}-Tate spectral sequence in V (1)-homotopy for T HH(ℓ/p). In Propositions 5.8 and 5.9 we show that the comparison map Γ̂_{1} : V (1)_{∗}T HH(ℓ/p) → V (1)_{∗}T HH(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 T HH(ℤ/p), with V (0)- and V (1)-coefficients. We have V (0)_{∗}T HH(ℤ/p) = E(ϵ_{0}) ⊗ P (μ_{0}), and with an obvious notation 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 C-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 [HM97, Prop. 5.3] that the comparison map

Γ̂_{1} : V (0)_{∗}T HH(ℤ/p) → V (0)_{∗}T HH(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_{∗}T HH(ℤ/p) → π_{∗}T HH(ℤ/p)^{tC_{p}} is (−2)-coconnected, meaning that it induces an injection in degree (−2) and an isomorphism in all higher degrees. From this we can deduce the following behavior of the comparison map Γ̂_{1} in V (1)-homotopy.

Lemma 5.1. The map

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

takes the classes ϵ_{0}^{δ}μ_{0}^{i} from V (0)_{∗}T HH(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 T HH(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

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

The Bökstedt trace map tr: K(B) → T HH(B) admits a preferred lift tr_{n} through each fixed point spectrum T HH(B)^{C_{p^{n}}}, which homotopy equalizes the iterated restriction and Frobenius maps R^{n} and F^{n} to T HH(B), see [BHM93, 2.5]. In particular, the circle action and the σ-operator act trivially 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)_{∗}T HH(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_{∗}T HH(ℤ/p) → π_{∗}T HH(ℤ/p)^{tC_{p}} is (−2)-coconnected, 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)_{∗}T HH(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 of the diagram, we reach a class (Γ_{1} ∘ tr_{1})(ϵ̄_{1}), represented in total degree (2p − 1) of 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)_{∗}T HH(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}). □

We can lift the algebraic K-theory class ϵ̄_{1} to ℓ/p.

Definition 5.3. 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, 10.9]. We can therefore choose a class

ϵ̄_{1}^{K} ∈ V (1)_{2p−1}K(ℓ/p)

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

the trace image tr(ϵ̄_{1}^{K}) in V (1)_{∗}T HH(ℓ/p) must map under π_{∗} to tr(ϵ̄_{1}) = ϵ_{0}μ_{0}^{p−1} − ϵ_{1} in V (1)_{∗}T HH(ℤ/p), which by Proposition 4.6 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 T HH(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

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

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

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

(up to units in 𝔽_{p}, which we will always suppress) 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)_{∗}T HH(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}}, respectively, inducing the algebraic localization map and identification

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

Lemma 5.6. 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 nonzero 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 is nonzero. Considering the E^{3}-term in (5.5), the only possible source of a previous differential hitting u_{1}t^{p^{2}−p}λ_{2} is ϵ̄_{1}. 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 Frobenius (group restriction) map from the S^{1}-Tate spectral sequence to the C_{p}-Tate spectral sequence, which takes Ê_{∗∗}^{2}(S^{1}, B) isomorphically to the even columns of Ê_{∗∗}^{2}(C_{p}, B), any such differential from an even to an odd column must be zero. □

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

Lemma 5.7. The classes 1, ϵ_{0}, μ_{0}, ϵ_{0}μ_{0}, …, μ_{0}^{p−1} and ϵ̄_{1} in V (1)_{∗}T HH(ℓ/p) map under Γ̂_{1} to classes in V (1)_{∗}T HH(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)_{∗}T HH(ℓ/p) \ar[r]^{Γ̂_{1}} \ar[d]^{π_{∗}} & V (1)_{∗}T HH(ℓ/p)^{tC_{p}} \ar[d]^{π_{∗}} \\ V (1)_{∗}T HH(ℤ/p) \ar[r]^{Γ̂_{1}} & V (1)_{∗}T HH(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 4.6 and Lemma 5.1.

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 5.6 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 5.8. The C_{p}-Tate spectral sequence in V (1)-homotopy for T HH(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 following 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 (5.5) 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)_{∗}T HH(ℓ/p), by Proposition 4.6, 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 5.7. 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)_{∗}T HH(ℓ) and V (1)_{∗}T HH(ℓ)^{tC_{p}} for degree reasons, and therefore also on V (1)_{∗}T HH(ℓ/p) and V (1)_{∗}T HH(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 (5.5) 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 5.9. The comparison map Γ̂_{1} takes the classes ϵ_{0}^{δ}μ_{0}^{i}, ϵ̄_{1}, λ_{2} and μ_{2} in V (1)_{∗}T HH(ℓ/p) to classes in V (1)_{∗}T HH(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), respectively. Thus

V (1)_{∗}T HH(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} factors as the algebraic localization map and identification

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

In particular, this map is (2p − 2)-coconnected.

Proof. The action of the map Γ̂_{1} on the classes 1, ϵ_{0}, μ_{0}, ϵ_{0}μ_{0}, …, μ_{0}^{p−1} and ϵ̄_{1} was given in Lemma 5.7, and the action on the classes λ_{2} and μ_{2} was already recalled from [AR02]. The structure of V (1)_{∗}T HH(ℓ/p)^{tC_{p}} is then immediate from the E^{∞}-term in Proposition 5.8. 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) = holim_{n,F} T HH(B)^{C_{p^{n}}}
T R(B) = holim_{n,R} T HH(B)^{C_{p^{n}}}

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

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

respectively.

Corollary 5.10. The comparison maps

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

for n ≥ 1, and

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

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

Proof. This follows from a theorem of Tsalidis [Tsa98] and Proposition 5.9 above, just like in [AR02, 5.7]. See also [BBLNR]. □

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, 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 (5.2), by [AR02, 2.8] and Lemma 5.4. For tμ_{2} it follows from [AR02, 4.8], by naturality.

Theorem 6.1. The C_{p^{n}}-Tate spectral sequence in V (1)-homotopy for T HH(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)_{∗}T HH(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)_{∗}T HH(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 is covered by Proposition 5.8. We can therefore assume that Theorem 6.1 holds for some fixed 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)-homotopy for T HH(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}

ALGEBRAIC K-THEORY OF THE FIRST MORAVA K-THEORY 19

(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} | 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} | 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} | 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} | 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} | 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} | 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 5.8. 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} | 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} | 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 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 T HH(ℓ/p) with its C_{p}-Tate construction.

In general, there is a commutative diagram

(6.3)  \xymatrix{T HH(B)^{C_{p^{n}}} \ar[r]^{R} \ar[d]^{Γ_{n}} & T HH(B)^{C_{p^{n−1}}} \ar[r]^{Γ_{n−1}} \ar[d]^{Γ̂_{n}} & T HH(B)^{hC_{p^{n−1}}} \ar[d]^{Γ̂_{1}^{hC_{p^{n−1}}}} \\ T HH(B)^{hC_{p^{n}}} \ar[r]^{R^{h}} & T HH(B)^{tC_{p^{n}}} \ar[r]^{G_{n−1}} & (T HH(B)^{tC_{p}})^{hC_{p^{n−1}}}}

where 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 T HH(B)^{tC_{p}}, in view of the identification

(T HH(B)^{tC_{p}})^{C_{p^{n−1}}} = T HH(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 5.10, so G_{n−1} is at least (2p − 1)-coconnected. (We shall see in Lemma 6.11 that V (1)_{∗}G_{n−1} is an isomorphism in all degrees.) By Proposition 5.9 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 T HH(ℓ/p)^{tC_{p}} is obtained by inverting μ_{2} in E_{∗∗}^{2}(C_{p^{n}}, ℓ/p). We denote it by μ_{2}^{−1}E^{∗}(C_{p^{n}}, ℓ/p), even though in later terms only a power of μ_{2} is present.

Theorem 6.4. The C_{p^{n}}-homotopy fixed point spectral sequence μ_{2}^{−1}E^{∗}(C_{p^{n}}, ℓ/p) in V (1)- homotopy for T HH(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)_{∗}(T HH(ℓ/p)^{tC_{p}})^{hC_{p^{n}}}, which receives a (2p − 2)-coconnected map (Γ̂_{1})^{hC_{p^{n}}} from V (1)_{∗}T HH(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 6.5. The initial differential in the C_{p^{n}}-homotopy fixed point spectral sequence in V (1)-homotopy for T HH(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} | 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} | 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} | 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} | 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} | 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} | 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} | 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} | 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, 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

(6.6) \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} | 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} | 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}

Next we use the following commutative diagram, where we abbreviate T HH(B) to T (B):

(6.7)  [figure: (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; T (B)^{tC_{p}}; T (B); Γ̂_{1}; T (B); Γ̂_{1}; T (B)^{tC_{p}}]

The horizontal maps all induce (2p − 2)-coconnected maps in V (1)-homotopy for B = ℓ/p. Here F is the Frobenius map, forgetting part of the equivariance. 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)_{∗}T HH(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)_{∗}T HH(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 5.9 it takes ϵ̄_{1}, λ_{2} and μ_{2} to ϵ̄_{1}, λ_{2} and t^{−p^{2}}, respectively, and similarly for monomials in these classes.

We focus on the summand

E(u_{n}, λ_{2}) ⊗ 𝔽_{p}{μ_{2}^{j} | 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)_{∗}T HH(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 6.8 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)_{∗}T HH(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 (6.6), 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 6.9 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, 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 6.8. For v_{p}(j) = 2n − 2, 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 6.5, the classes of lower filtration than y all lie in the terms

E(u_{n}, ϵ̄_{1}) ⊗ 𝔽_{p}{λ_{2}μ_{2}^{i} | 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 6.9. 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 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 6.10. V (1)_{∗}T HH(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 6.11. The map G_{n} induces an isomorphism

V (1)_{∗}T HH(ℓ/p)^{tC_{p^{n+1}}} ⟶^{≅} V (1)_{∗}(T HH(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)_{∗}T HH(ℓ/p)^{tS^{1}} ⟶^{≅} V (1)_{∗}(T HH(ℓ/p)^{tC_{p}})^{hS^{1}} .

Proof. As remarked after diagram (6.3), 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 6.12. (a) The associated graded of V (1)_{∗}T HH(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} | v_{p}(j) = 2k − 2} ⊗ P_{ρ(2k−3)}(tμ_{2})

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

⊕ ⨁_{k≥1} E(ϵ̄_{1}) ⊗ 𝔽_{p}{λ_{2}μ_{2}^{j} | 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. Furthermore, it takes multiples by ϵ̄_{1}, λ_{2} or tμ_{2} in the source to the same multiples in the target.

Proof. Claims (a) and (b) follow by passage to the limit over n from Corollaries 6.2 and 6.5. 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 (6.7). □

7. Topological cyclic homology

By definition, there is a fiber sequence

T C(B) ⟶^{π} T F (B) ⟶^{R−1} T F (B)

inducing a long exact sequence

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

in V (1)-homotopy. By Corollary 5.10, there are (2p − 2)-coconnected maps Γ and Γ̂ from V (1)_{∗}T F (ℓ/p) to V (1)_{∗}T HH(ℓ/p)^{hS^{1}} and V (1)_{∗}T HH(ℓ/p)^{tS^{1}}, respectively. We model V (1)_{∗}T F (ℓ/p) in degrees > (2p − 2) by the map Γ̂ to the S^{1}-Tate construction. Then, by diagram (6.3), R is modeled in the same range of degrees by the chain of maps below.

\xymatrix{V (1)_{∗}T HH(B)^{tS^{1}} \ar@{-->}[dr]^{G} & V (1)_{∗}T HH(B)^{hS^{1}} \ar[d]^{(Γ̂_{1})^{hS^{1}}} \ar[r]^{R^{h}} & V (1)_{∗}T HH(B)^{tS^{1}} \\  & V (1)_{∗}(T HH(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) ,

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 6.11, and the middle (wrong-way) map is (2p − 2)-coconnected.

Proposition 7.2. 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.

Proof. Consider the summands of E^{∞}(S^{1}, ℓ/p) and Ê^{∞}(S^{1}, ℓ/p), as given in Theorem 6.12. 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 7.3. 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 7.4. 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), represented by A, B_{k}, C_{k} and D in Ê^{∞}(S^{1}, ℓ/p), respectively, such that the homomorphism 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) ≅ Ã⊕B̃⊕C̃⊕D̃, where B̃ = ∏_{k≥2} B̃_{k} and C̃ = ∏_{k≥2} C̃_{k}.

Proof. In terms of the model T HH(ℓ/p)^{tS^{1}} for T F (ℓ/p), the restriction map R is given in these degrees as the composite of the isomorphism G, computed in Theorem 6.12(c), and the map Ê^{∞}(R^{h}), computed in Proposition 7.2. This gives the desired formulas at the level of E^{∞}-terms. The rest of the argument is the same as that for Theorem 7.7 of [AR02], using Corollary 6.10 to control the topologies, and will be omitted. □ Remark 7.5. Here we have followed the basic computational strategy of [BM94], [BM95] and [AR02]. It would be interesting to have a more concrete construction of the lifts B̃_{k}, C̃_{k} and D̃, in terms of de Rham–Witt operators R, F, V and d = σ, like in the algebraic case of [HM97] and [HM03]. Proposition 7.6. 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) ≅ 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 7.4, 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) ⟶^{π} ker(R − 1) → 0 ,

from which the formula for the middle term follows. □

Remark 7.7. 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. Under the canonical map T F (ℓ/p) → T HH(ℓ/p)^{C_{p}}, modeled here by T HH(ℓ/p)^{tS^{1}} → (T HH(ℓ/p)^{tC_{p}})^{hC_{p}}, these map to the classes μ_{2}^{−d}. Since we are only concerned with degrees > (2p − 2) we may equally well use their v_{2}-power multiplies (tμ_{2})^{d} · μ_{2}^{−d} = t^{d} as generators, with the advantage that these are in the image of the localization map T HH(ℓ/p)^{hC_{p}} → (T HH(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) → T HH(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) → V (1)_{∗}T C(ℤ/p), by [BM94, 10.9] and [Dun97]. Here T C(ℤ/p) ≃ Hℤ_{p} ∨ Σ^{−1}Hℤ_{p} and V (1)_{∗}T C(ℤ/p) ≅ E(∂, ϵ̄_{1}), so we can recover V (1)_{∗}T C(ℓ/p) in degrees ≤ (2p − 2) from this map.

Theorem 7.8. There is an isomorphism of E(λ_{1}, λ_{2}) ⊗ P (v_{2})-modules

V (1)_{∗}T C(ℓ/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 v_{2} · dlog v_{1} = λ_{2}. The degrees are |∂| = −1, |ϵ̄_{1}| = |λ_{1}| = 2p − 1, |λ_{2}| = 2p^{2} − 1 and |v_{2}| = 2p^{2} − 2. The formal multipliers have degrees |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).

Proof. Only the additive generators t^{d} for 0 < d < p^{2} − p, p ∤ d from Proposition 7.6 do not appear in V (1)_{∗}T C(ℓ/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] the cyclotomic trace map of [BHM93] induces cofiber sequences

(7.9)  K(B_{p})_{p} ⟶^{trc} T C(B)_{p} ⟶^{g} Σ^{−1}Hℤ_{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) ⟶^{g} Σ^{−1}E(ϵ̄_{1}) → ….

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

Theorem 7.10. There is an isomorphism of E(λ_{1}, λ_{2}) ⊗ 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. □

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Mathematical Institute, University of Münster, DE–48149 Münster, Germany

E-mail address: ausoni@uni-muenster.de

Department of Mathematics, University of Oslo, NO–0316 Oslo, Norway

E-mail address: rognes@math.uio.no