ALGEBRAIC K-THEORY OF THE FRACTION

FIELD OF TOPOLOGICAL K-THEORY

Christian Ausoni and John Rognes

Abstract. We compute the algebraic K-theory modulo p and v_{1} of the S-algebra ℓ/p = k(1), using topological cyclic homology. We use this to compute the homotopy cofiber of a transfer map K(L/p) → K(L_{p}), which we interpret as the algebraic K-theory of the “fraction field” of the p-complete Adams summand of topological K-theory. The results suggest that there is an arithmetic duality theorem for this fraction field, much like Tate–Poitou duality for p-adic fields.

1. Introduction

In this paper we continue the investigation of the algebraic K-theory of topological K-theory initiated in [AuR02]. There we computed the mod p and v_{1} homotopy groups of K(ℓ_{p}) at primes p ≥ 5, denoted by V (1)_{∗}K(ℓ_{p}) or K/(p, v_{1})_{∗}(ℓ_{p}), where ℓ_{p} is the Adams summand of the p-complete connective complex K-theory spectrum ku_{p}, with coefficients π_{∗}(ℓ_{p}) = ℓ_{p∗} = ℤ_{p}[v_{1}], |v_{1}| = 2p − 2. Denoting by P (v_{2}) the polynomial algebra over 𝔽_{p} generated by v_{2}, |v_{2}| = 2p^{2} − 2, we showed that V (1)_{∗}K(ℓ_{p}) is essentially a free P (v_{2})-module on 4p + 4 generators. This is reminiscent, up to a chromatic shift of one in the sense of stable homotopy theory, of the structure of the mod p algebraic K-theory V (0)_{∗}K(ℤ_{p}) = K/(p)_{∗}(ℤ_{p}) of the p-adic integers, which is a free P (v_{1})-module on p + 3 generators.

The structure of the algebraic K-theory of ℤ_{p} can be conceptually explained in terms of localization, Galois descent and motivic truncation. First, there is a localization sequence

K(ℤ/p) → K(ℤ_{p}) → K(ℚ_{p})

relating the algebraic K-theory of ℤ_{p} to that of its fraction field ℚ_{p} = ff (ℤ_{p}) and its residue field ℤ/p. Second, the v_{1}-periodic mod p algebraic K-theory of ℚ_{p} is the target of a Galois descent spectral sequence [Th85]

E_{s,t}^{2} = H_{Gal}^{−s}(ℚ_{p}; 𝔽_{p}(t/2)) ⟹ V (0)_{s+t}K(ℚ_{p})[v_{1}^{−1}] .

The coefficient module 𝔽_{p}(t/2) can be interpreted as V (0)_{t}K(ℚ̄_{p})[v_{1}^{−1}] by Suslin’s Theorem [Su84], and this spectral sequence can be thought of as the continuous homotopy fixed-point spectral sequence in mod p homotopy associated to the action of the absolute Galois group of ℚ_{p} on K(ℚ̄_{p}). Third, the unlocalized mod p algebraic K-theory of ℚ_{p} is the target of a motivic spectral sequence [BL:ss], [FS02]

E_{s,t}^{2} = H_{mot}^{−s}(ℚ_{p}; 𝔽_{p}(t/2)) ⟹ V (0)_{s+t}K(ℚ_{p}) ,

Typeset by AMS-TEX

where

H_{mot}^{r}(ℚ_{p}; 𝔽_{p}(i)) ≅ \{ \begin{matrix}H_{Gal}^{r}(ℚ_{p}; 𝔽_{p}(i)) & for r ≤ i , \\ 0 & for r > i . \end{matrix}

In other words, the mod p motivic cohomology of ℚ_{p} is a specific truncation of its Galois cohomology. In terms of Bloch’s higher Chow groups [Bl86], the vanishing of H_{mot}^{r}(ℚ_{p}; 𝔽_{p}(i)) for r > i follows from the fact that there are no varieties of negative dimension relative to Spec(ℚ_{p}). The identification with Galois cohomology for r ≤ i is part of the Beilinson conjectures [BMS87].

Our aim here is to conceptually understand V (1)_{∗}K(ℓ_{p}) in the same way as we understand V (0)_{∗}K(ℤ_{p}), using the Galois theory for commutative S-algebras developed by the second author [Rog08]. The first task is therefore to identify the fraction field ff (ℓ_{p}) of ℓ_{p}. There is a localization sequence

K(ℤ_{p}) → K(ℓ_{p}) → K(L_{p}) ,

the existence of which was conjectured by the second author and established by Blumberg and Mandell [BM08], relating the algebraic K-theory of ℓ_{p} to that of the periodic p-complete Adams summand L_{p}. The fact that p is not invertible in π_{∗}(L_{p}) = L_{p∗} = ℤ_{p}[v_{1}^{±1}], and the presence of the nontrivial residue S-algebra L/p = K(1) of L_{p}, give some indications as to why L_{p} should not qualify as the fraction field we are after.

A naive guess would be to take the Bousfield localization L_{p}[p^{−1}] of L_{p} away from p, inverting p in π_{∗}(L_{p}). However, L_{p}[p^{−1}] is an Hℚ_{p}-algebra, and we cannot possibly recover the v_{2}-periodic mod p and v_{1} homotopy of K(ℓ_{p}) or K(L_{p}) from that of K(L_{p}[p^{−1}]), which is v_{2}-torsion. See Remark 3.12. Therefore L_{p}[p^{−1}] does not qualify as the desired fraction field of ℓ_{p}. Instead, we have reasons to expect that if ff (ℓ_{p}) is the “correct” fraction field, by which we mean that it will suit the purpose of understanding algebraic K-theory by Galois descent, then its algebraic K-theory K(ff (ℓ_{p})) ought to fit in the following 3×3-diagram of localization cofiber sequences, see Definition 3.9.

(1.1)  \xymatrix{K(ℤ/p) \ar[r]^{i_{∗}} \ar[d]^{π_{∗}} & K(ℤ_{p}) \ar[r]^{j^{∗}} \ar[d]^{π_{∗}} & K(ℚ_{p}) \ar[d]^{π_{∗}} \\ K(ℓ/p) \ar[r]^{i_{∗}} \ar[d]^{ρ^{∗}} & K(ℓ_{p}) \ar[r]^{j^{∗}} \ar[d]^{ρ^{∗}} & K(p^{−1}ℓ_{p}) \ar[d]^{ρ^{∗}} \\ K(L/p) \ar[r]^{i_{∗}} & K(L_{p}) \ar[r]^{j^{∗}} & K(ff (ℓ_{p})) .}

We emphasize that for now p^{−1}ℓ_{p} and ff (ℓ_{p}) are just formal symbols, not the same as the Bousfield localizations ℓ_{p}[p^{−1}] and L_{p}[p^{−1}]. The V (1)-homotopy groups of K(ℤ/p), K(ℤ_{p}) and K(ℓ_{p}) are known by [Qu72], [HM97] and [AuR02]. It therefore remains to compute V (1)_{∗}K(ℓ/p) and the transfer maps i_{∗} and π_{∗} in order to be in a position to evaluate the V (1)-homotopy of the iterated cofiber K(ff (ℓ_{p})). This is our main result, corresponding to Theorem 8.10 in the body of the paper.

K-THEORY OF THE FRACTION FIELD OF K-THEORY 3

Theorem 1.2. 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, where p ≥ 5 is assumed.

As explained in Section 2, the spectrum ℓ/p admits infinitely many structures of an associative ℓ_{p}-algebra, none of which are commutative. However, all of these ℓ_{p}-algebras have equivalent underlying S-algebra structures [An:un], and indeed it turns out that the additive structure of V (1)_{∗}K(ℓ/p) is independent of the choice of ℓ_{p}-algebra structure. The classes λ_{1}, λ_{2} and v_{2} are present in V (1)_{∗}K(ℓ_{p}), and in the statement above we refer to the V (1)_{∗}K(ℓ_{p})-module structure of V (1)_{∗}K(ℓ/p). We prove this theorem by means of the cyclotomic trace map to topological cyclic homology T C(ℓ/p). On 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 5.4 and Theorem 8.8.

Defining K(ff (ℓ_{p})) as the iterated cofiber in diagram (1.1), we can then evaluate its V (1)-homotopy. The formulation given in Theorem 9.4 readily implies the following statement.

Theorem 1.3. Let p ≥ 5 be a prime. 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}

is displayed in Figure 10.3. Here | dlog p | = 1, and the degrees of the other classes are as in Theorem 1.2. The localization homomorphism

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

is an isomorphism in degrees ∗ ≥ 2p.

This proves that the iterated cofiber K(ff (ℓ_{p})) cannot be a K(ℚ_{p})-module, since V (1)_{∗}K(ℚ_{p}) is a torsion P (v_{2})-module.

In the final part of this paper, we conjecturally interpret the computation of Theorem 1.3 in terms of Galois descent. Indeed, the second author conjectured that if Ω_{1} is a 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 E-theory [GH04] with coefficients π_{∗}E_{2} = 𝕎(𝔽_{p^{2}})[[u_{1}]][u^{±1}], and L_{K(2)} denotes Bousfield localization with respect to the Morava K-theory K(2) with coefficients π_{∗}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 therefore conjecture 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}] ,

see Conjecture 10.5. Starting with V (1)_{∗}K(ff (ℓ_{p}))[v_{2}^{−1}] and working backwards, we give a conjectural description of the Galois cohomology of ff (ℓ_{p}) with coefficients in V (1)_{∗}K(Ω_{1})[v_{2}^{−1}], see Figure 10.3. This fits very well with the example of the Galois cohomology of ℚ_{p} with coefficients in V (0)_{∗}K(ℚ̄_{p})[v_{1}^{−1}], with the difference that the absolute Galois group of ff (ℓ_{p}) has cohomological dimension 3 instead of 2. Also, there appears to be a perfect arithmetic duality pairing in the conjectural Galois cohomology of ff (ℓ_{p}), analogous to Tate–Poitou duality for local number fields, which is not present in the analogous cohomology rings for ℓ_{p} or L_{p}. This indicates that ff (ℓ_{p}) ought to be a form of S-algebraic two-local field, mixing three different residue characteristics. As argued in section 3 of this paper, we expect very similar results to hold for ℓ_{p} replaced by ku_{p}.

The paper is organized as follows. In section 2 we fix our notations, show that ℓ/p admits the structure of an associative ℓ-algebra, and give a similar discussion for ku/p and the periodic versions L/p and KU/p. We give derived algebro-geometric and log-geometric interpretations of these constructions. In section 3, we discuss localization sequences in algebraic K-theory. Section 4 contains the computation of the mod p homology of T HH(ℓ/p), and in section 5 we evaluate its V (1)-homotopy. Sections 6, 7 and 8 deal with the computation of T C(ℓ/p) and K(ℓ/p) in V (1)- homotopy. In section 9 the various computations are assembled in a diagram of localization sequences, to evaluate V (1)_{∗}K(ff (ℓ_{p})). In section 10, the structure of V (1)_{∗}K(ff (ℓ_{p})) is interpreted in terms of Galois descent, yielding a conjectural description of the Galois cohomology of ff (ℓ_{p}).

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 Ktheory 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 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 ku-algebra. 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 [La01, §§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 KU-algebra, 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 [An08, §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 the sequence of choices α ∈ 𝔽_{p}.

The periodic spectrum KU/p can be obtained from ku/p by Bousfield KUlocalization 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. Similarly, for each ν ≥ 1 we can realize the ku-module spectrum ku/p^{ν}, with (ku/p^{ν})_{∗} = ℤ/p^{ν}[u], as an associative ku-algebra, and with a little care the ku-algebra structures may be so chosen that the p-adic tower

(2.2)  ⋯ → ku/p^{ν+1} → ku/p^{ν} → ⋯ → ku/p

is one of associative ku-algebras. Uncountably many choices of structures remain, though. Again, the same remarks apply over ℓ, where the p-adic tower {ℓ/p^{ν}}_{ν} can be chosen to be one of associative ℓ-algebras.

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

(2.3)  \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-algebras

(2.4)  \xymatrix{ℓ \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 → Hℤ_{(p)}, and similarly in the p-complete case. By omission of structure, these squares are also diagrams of S-algebras and S-algebra maps.

Remark 2.5. We might think algebro-geometrically about these diagrams. In other words, we would like to view the S-algebras in diagram (2.3) as the rings of functions on a diagram

\xymatrix{Spec(ℤ/p) \ar[r]^{i} \ar[d]^{π} & Spec(ℤ) \ar[d]^{π} \\ Spnc(ku/p) \ar[r]^{i} & Spec(ku)}

of affine derived schemes, in the sense of Jacob Lurie, or affine brave new schemes in the sense of Toën and Vezzosi. Here Spec(ku) has the same underlying prime ideal space as Spec(π_{0}(ku)) = Spec(ℤ), but it is ringed in commutative S-algebras, rather than in ordinary commutative rings. Over the open subset Spec(ℤ[f^{−1}]), where f is a finite product of primes, the ring of functions is the commutative S-algebra ku[f^{−1}] with coefficient ring ku_{∗}[f^{−1}]. However, since ku/p is not a commutative S-algebra, we do not know how to make good sense of its structure space. At best, i: Spnc(ku/p) → Spec(ku) can be though of as a map from a non-commutative derived scheme to the derived scheme Spec(ku). The base change assertion above then says that Spec(ℤ/p) is the fiber of Spnc(ku/p) over the closed subspace Spec(ℤ) in Spec(ku), which in turn happens to be an honest (commutative) scheme.

Remark 2.6. At this point we recall how Spec of the fraction field F = ff (R) of an integral domain R is obtained geometrically by cutting away all the proper closed subspaces of Spec(R). The remaining open subset represents the generic point of Spec(R). For any localization T of R, with R ⊂ T ⊂ F, we also have F = ff (T), so F is simultaneously the fraction field of R and of T. In the case of Spec(ku), with R = ku, the analogous (sub-)spaces appear to be Spec(ℤ) (included by π) and the Spnc(ku/p) (mapped by i) for all primes p, so Spec of the fraction field ff (ku) should consist of the complement in Spec(ku) of all of these spaces. In the p-complete case, this means cutting out the two spaces Spec(ℤ_{p}) and Spnc(ku/p) from Spec(ku_{p}), meeting precisely along their intersection Spec(ℤ/p). The remainder should be Spec(ff (ku_{p})). We expect that the complement of Spec(ℤ_{p}) in Spec(ku_{p}) will play the role of Spec(KU_{p}), and that the complement of Spec(ℤ/p) in Spnc(ku/p) will play the role of Spnc(KU/p), so the S-algebraic fraction field of ku_{p} is also realized as the S-algebraic fraction field of KU_{p}, denoted ff (ku_{p}) = ff (KU_{p}) = p^{−1}KU_{p}. Similarly, Spec of the fraction field

ff (ℓ_{p}) = ff (L_{p}) = p^{−1}L_{p}

of ℓ_{p} or L_{p} would be the complement of Spec(ℤ_{p}) and Spnc(ℓ/p) in Spec(ℓ_{p}), meeting at Spec(ℤ/p).

Based on the algebraic K-theory computations to follow, we have strong reasons to believe that the complement Spec(p^{−1}ku) of Spnc(ku/p) in Spec(ku) is not equal to the derived algebraic scheme associated to the commutative ku-algebra ku[p^{−1}] that is obtained by (algebraic) Bousfield localization away from p. See Remark 3.12 for the calculational justification of our expectation. Similarly, in the p-complete case the complement Spec(p^{−1}ku_{p}) of Spnc(ku/p) in Spec(ku_{p}) is not equal to Spec(ku_{p}[p^{−1}]), and in the periodic case the complement Spec(p^{−1}KU_{p}) = Spec(ff (ku_{p})) is different from Spec(KU_{p}[p^{−1}]). In fact, we do not expect that Spec(p^{−1}ku_{p}) and Spec(ff (ku_{p})) are affine derived schemes in the sense that they are equal to Spec(B) for a commutative S-algebra B. Still, they may be derived stacks, in a sense approximated by the following remark.

Remark 2.7. Since Spnc(ku/p) is not actually a closed subscheme of Spec(ku_{p}), but only maps to the complement of the open subscheme Spec(ku_{p}[p^{−1}]), we expect that a derived scheme Spec(p^{−1}ku_{p}) that is complementary to Spnc(ku/p), is a derived algebro-geometric object which is intermediate between Spec(ku_{p}[p^{−1}]) and Spec(ku_{p}). In forthcoming work, by the second author together with Clark Barwick and Steffen Sagave, we hope to give good sense to this intermediate object as a logarithmic S-algebra (ku_{p}, M), where M is a pointed E_{∞} space. See also Remark 3.3. This makes (ku_{p}, M) a logarithmic S-algebra and Spec(ku_{p}, M) an affine derived logarithmic scheme. Following Martin Olsson’s work in the classical algebraic case [Ol03], this derived logarithmic scheme can be faithfully represented by the derived stack representing the ∞-category of derived logarithmic schemes over it. Hence we expect that this moduli stack of Spec(ku_{p}, M) will provide the desired derived stack interpretation of Spec(p^{−1}ku_{p}). Related topological logarithmic structures on ku_{p} will then provide useful interpretations for Spec(KU_{p}) and Spec(ff (ku_{p})). See [Rog09] for a set of foundations of topological logarithmic geometry, and [Rog:ltc] for the construction of topological cyclic homology of a logarithmic S-algebra.

3. Localization squares in algebraic K-theory

We are interested in the algebraic K-theory spectra of the S-algebras considered in the previous section. For an S-algebra B, we adapt [EKMM97, VI] and let C_{B} be the category of finite cell B-modules and B-module maps. This is a category with cofibrations and weak equivalences in the sense of [Wa85, §1], with cofibrations the maps that are isomorphic to the inclusion of a subcomplex, and weak equivalences the homotopy equivalences. The algebraic K-theory of B, denoted K(B), is defined to be the algebraic K-theory of this category with cofibrations and weak equivalences. When B = HR is an Eilenberg–Mac Lane spectrum, it is known that K(HR) ≃ K(R) recovers Quillen’s algebraic K-theory of the ring R.

If φ: A → B is a map of S-algebras, the exact functor B∧_{A}(−): C_{A} → C_{B} induces a map φ^{∗}: K(A) → K(B), making K(−) a covariant functor in the S-algebra. We shall follow the variance conventions of algebraic geometry, thinking of K(−) as a contravariant functor in a (not yet generally defined) geometric object Spec(−) associated to the S-algebra, so that φ^{∗} is derived from the inverse image functor on sheaves. This is done to avoid confusion with the transfer maps φ_{∗}: K(B) → K(A), derived from the direct image functor on sheaves.

Suppose that φ: A → B makes B a finite cell A-module. Then each finite cell B-module M can be viewed as an A-module along φ, and each B-cell attachment needed to build M can be achieved by attaching finitely many A-cells, one for each A-cell in B. The resulting exact functor C_{B} → C_{A} induces the transfer map φ_{∗}: K(B) → K(A).

When A is a commutative S-algebra, the smash product (−) ∧_{A} (−) induces a pairing on C_{A} that makes K(A) a commutative S-algebra. When A is central in B, so that B is an A-algebra, the smash product makes K(B) a K(A)-module, and the maps φ^{∗}: K(A) → K(B) and φ_{∗}: K(B) → K(A) (when defined) are K(A)- module maps. In other words, the transfer is a module map over its target. When B is commutative, then K(B) is likewise a commutative S-algebra, and φ^{∗} is a commutative S-algebra map. The K(A)-linearity of φ_{∗} can then be expressed by the projection/reciprocity formula φ_{∗}(φ^{∗}(x) ∧_{B} y) = x ∧_{A} φ_{∗}(y).

We apply this to the maps i: ku → ku/p and π: ku → Hℤ. Here ku/p = ku∪_{p}e^{1} and Hℤ = ku ∪_{u} e^{3} in the category of ku-modules, meaning that ku/p is obtained from ku by attaching a 1-cell along p, and that Hℤ is obtained from ku by attaching a 3-cell along u. The transfer maps i_{∗} and π_{∗} and their variants then define the upper left hand square in the following commutative diagram, where all rows and columns are cofiber sequences.

(3.1)  \xymatrix{K(ℤ/p) \ar[r]^{i_{∗}} \ar[d]^{π_{∗}} & K(ℤ) \ar[r]^{j^{∗}} \ar[d]^{π_{∗}} & K(ℤ[p^{−1}]) \ar[d]^{π_{∗}} \\ K(ku/p) \ar[r]^{i_{∗}} \ar[d]^{ρ^{∗}} & K(ku) \ar[r]^{j^{∗}} \ar[d]^{ρ^{∗}} & K(p^{−1}ku) \ar[d]^{ρ^{∗}} \\ K(KU/p) \ar[r]^{i_{∗}} & K(KU) \ar[r]^{j^{∗}} & K(p^{−1}KU)}

The upper row is a cofiber sequence by Quillen’s localization theorem for Dedekind domains [Qu73]. Andrew Blumberg and Mike Mandell [BM08] have proved that the middle column is a cofiber sequence, as conjectured by the second author. The result is analogous to that of Quillen, in that ku/u ≃ Hℤ and ku[u^{−1}] ≃ KU. The same argument shows that the left hand column is a cofiber sequence.

We complete the 3 × 3 diagram of cofiber sequences by making the following ad hoc definition.

Definition 3.2. Let K(p^{−1}ku) and K(p^{−1}KU) denote the (homotopy) cofibers of the transfer maps i_{∗}: K(ku/p) → K(ku) and i_{∗}: K(KU/p) → K(KU), respectively.

Remark 3.3. The theory of logarithmic structures on S-algebras, see [Rog09], is designed to model intermediate objects between a commutative S-algebra B, like ku, and its localizations, like ku[p^{−1}]. A logarithmic structure on B is given by a commutative S-algebra map S[M] → B from the spherical monoid ring on a “commutative monoid” M, where the latter term should be interpreted in a sufficiently flexible category to also encompass E_{∞} spaces. In joint work with Steffen Sagave, the second author has defined a category of logarithmic modules over a logarithmic S-algebra (B, M), and one may define the logarithmic K-theory K(B, M) as the algebraic K-theory of the category of suitably finite objects in that module category. It still remains to be seen if one can realize K(p^{−1}ku) as K(ku, M) for a suitable logarithmic S-algebra (ku, M).

Remark 3.4. There is a similar 3×3 diagram to (3.1), in which the left hand column is replaced by the sum of corresponding columns for all primes p, and i_{∗} is replaced by the sum of the transfer maps. By Quillen’s localization sequence the upper right hand term can be written as K(ℚ). We expect that the lower right hand term then plays the role of the algebraic K-theory of the S-algebraic fraction field of ku, denoted K(ff (ku)). The hypothetical fraction field has a valuation with uniformizer u, and the middle right hand term K(O) in the 3 × 3 diagram plays the role of the algebraic K-theory of an S-algebraic valuation ring O of ff (ku). However, we do not expect that this fraction field ff (ku) is the rationalization L_{0}KU = KUℚ of the S-algebra KU. This expectation is again supported by the algebraic K-theory computations that follow.

We shall use the topological cyclic homology T C(−) and the cyclotomic trace map trc: K(B) → T C(B) of [BHM93] to compute the algebraic K-theory spectra and maps in the diagram (3.1) above, after making two modifications. Firstly, we will replace ku by its p-completion ku_{p}, and secondly we will pass to the Adams summand ℓ_{p} of ku_{p}. We will now motivate these two changes, and discuss their effect on the algebraic K-theory diagram.

First, the cyclotomic trace map is very close to a p-complete equivalence when applied to connective p-complete S-algebras B with π_{0}(B) = ℤ/p or ℤ_{p}, or a finite extension of these. More precisely, K(B)_{p} is the connective cover of T C(B)_{p} in these cases [HM97]. Therefore we shall pass to the p-complete situation, replacing ku and Hℤ by ku_{p} and Hℤ_{p}, respectively. In view of [Du97], the iterated (homotopy) fiber of the square

\xymatrix{K(ku)_{p} \ar[r]^{κ^{∗}} \ar[d]^{π^{∗}} & K(ku_{p})_{p} \ar[d]^{π^{∗}} \\ K(ℤ)_{p} \ar[r]^{κ^{∗}} & K(ℤ_{p})_{p} ,}

where κ: ku → ku_{p} and κ: Hℤ → Hℤ_{p} denote the completion maps, is homotopy equivalent to the iterated fiber of the corresponding square of topological cyclic homology spectra, and the latter is trivial, since T C(B)_{p} ≃ T C(B_{p})_{p}. Therefore the fiber of K(ku)_{p} → K(ku_{p})_{p} is homotopy equivalent to the fiber of K(ℤ)_{p} → K(ℤ_{p})_{p}, so the change created by passing to the p-complete case is as well understood for ku and ku_{p} as in the number ring case.

In fact, our calculations shall concentrate on the v_{2}-periodic behavior of the algebraic K-theory of the S-algebras related to topological K-theory, rather than the subsequent issues of divisibility by v_{0} = p and v_{1} = u^{p−1}. We achieve this by working with homotopy with mod p and v_{1} coefficients, i.e., with the V (1)-homotopy functor X ↦ V (1)_{∗}X = π_{∗}(V (1) ∧ X), where V (1) = S/(p, v_{1}) is the Smith–Toda complex [Sm70] defined for p odd 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 = 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 Smith–Toda complex V (1) is a homotopy commutative and associative ring spectrum for p ≥ 5 [Ok84], and its BP -homology satisfies BP_{∗}V (1) ≅ BP_{∗}/(p, v_{1}). In this sense V (1) generalizes the mod p Moore spectrum representing mod p homotopy, and represents mod p and v_{1} homotopy. There is an element v_{2} ∈ π_{2p^{2}−2}V (1), with image v_{2} in BP_{∗}V (1). So V (1) is a finite complex of type 2 in the sense of [HoSm98], with telescope V (1)[v_{2}^{−1}] the spectrum that represents v_{2}-periodic homotopy, V (1)_{∗}(X)[v_{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.

It is known by [BM94] and [BM95] that V (1)_{∗}K(ℤ_{p}) is a finite ℤ/p-module. Furthermore, the announced proof of the Bloch–Kato conjecture by Vladimir Voevodsky and Markus Rost also implies the Lichtenbaum–Quillen conjecture on the algebraic K-theory of rings of integers in number fields, which in turn implies that V (1)_{∗}K(ℤ) is a finite ℤ/p-module. So the common fiber of the maps K(ℤ)_{p} → K(ℤ_{p})_{p} and K(ku)_{p} → K(ku_{p})_{p} has (totally) finite V (1)-homotopy. Therefore, replacing K(ℤ) by K(ℤ_{p}) and K(ku) by K(ku_{p}) in diagram (3.1), and still defining the lower row and the right hand column so as to obtain a 3×3 square of cofibrations, only changes the V (1)-homotopy in the diagram by a finite ℤ/pmodule. In this case the upper right hand corner is K(ℚ_{p}), since ℤ_{p}[p^{−1}] = ℚ_{p}, and the lower right hand corner p^{−1}KU_{p} plays the role of the S-algebraic fraction field of ku_{p}, denoted ff (ku_{p}). It follows that the v_{2}-periodic homotopy, given by V (1)_{∗}K(ku)[v_{2}^{−1}], etc., does not change at all:

(3.5)  V (1)_{∗}K(p^{−1}KU)[v_{2}^{−1}] ≅ V (1)_{∗}K(ff (ku_{p}))[v_{2}^{−1}] .

Along the same lines, we might have started with the version of diagram (3.1) discussed in Remark 3.4, inverting all primes in ℤ rather than just p. This adds a copy of the cofiber sequence K(ℤ/r) ⟶^{π_{∗}} K(ku/r) ⟶ K(KU/r) to the left hand column, for each prime r ≠ p. Here π^{∗}: K(ku/r)_{p} → K(ℤ/r)_{p} is easily seen to be an equivalence, using the BGL(−)^{+}-model for algebraic K-theory, and from the F ψ^{r}-model for K(ℤ/r) from [Qu72] it is clear that V (1)_{∗}K(ℤ/r) is finite and concentrated in degrees 0 ≤ ∗ < 2p−2. Therefore, replacing the right hand column by the cofiber sequence K(ℚ) ⟶^{π_{∗}} K(O) → K(ff (ku)), where O = O_{ff (ku)} is the valuation ring of the fraction field of ku, only changes the V (1)-homotopy in a finite range of degrees, and the v_{2}-periodic homotopy is unchanged:

(3.6)  V (1)_{∗}K(p^{−1}KU)[v_{2}^{−1}] ≅ V (1)_{∗}K(ff (ku))[v_{2}^{−1}] .

Second, the Hurewicz image of the Bott element u is nilpotent in the mod p homology of the spectrum ku, since u^{p−1} = v_{1} has Adams filtration 1, and this significantly complicates the algebra involved in computing the topological Hochschild homology T HH(−) and the topological cyclic homology T C(−) of ku, compared to the computations for its Adams summand ℓ. Compare [MS93], [AuR02] and [Au05]. Therefore, for the detailed calculations in this paper we shall restrict attention to the Adams summands ℓ_{p} and ℓ/p of ku_{p} and ku/p, respectively. The map L_{p} → KU_{p} is a Δ-Galois extension of commutative S-algebras, in the sense of [Rog08, §4.1], where Δ = (ℤ/p)^{×} ≅ ℤ/(p − 1) acts on KU_{p} through the p-adic Adams operations fixing L_{p}. Generalizing the Lichtenbaum–Quillen conjecture, it is therefore to be expected that the map

Φ: K(L_{p}) → K(KU_{p})^{hΔ}

is (close to) an equivalence after p-adic completion, and similarly for the fraction fields. Indeed, the first expectation is known to be correct: Consider the map of horizontal cofiber sequences

(3.7)  \xymatrix{K(ℤ_{p})_{p} \ar[r]^{π_{∗}} \ar[d]^{p−1} & K(ℓ_{p})_{p} \ar[r] \ar[d]^{φ} & K(L_{p})_{p} \ar[d]^{Φ} \\ K(ℤ_{p})_{p}^{hΔ} \ar[r]^{π_{∗}^{hΔ}} & K(ku_{p})_{p}^{hΔ} \ar[r] & K(KU_{p})_{p}^{hΔ} ,}

where the lower row is obtained from the Blumberg–Mandell localization sequence by taking Δ-homotopy fixed points (Δ acts trivially on ℤ_{p} = π_{0}(ku_{p})), and the upper row is its analogue for the Adams summand. The left hand vertical map multiplies by the ramification index (p − 1), since the uniformizer v_{1} in L_{p} maps to that power of the uniformizer u in KU_{p}. More directly, the upper map π_{∗} takes the unit class of Hℤ_{p} to that of the complex v_{1} : Σ^{2p−2}ℓ_{p} → ℓ_{p}, which extends along ℓ_{p} → ku_{p} to the class of v_{1} = u^{p−1} : Σ^{2p−2}ku_{p} → ku_{p}. This is equivalent to (p − 1) times the class of u: Σ^{2}ku_{p} → ku_{p}, which is the image of the unit class of Hℤ_{p} under the lower map π_{∗}^{hΔ}. Either way, the left hand vertical map is a p-adic equivalence. Thus the middle vertical map φ is an equivalence if and only if the right hand vertical map Φ is one. These equivalent conclusions are indeed precisely verified by the calculations of the first author in [Au05], showing that V (1)_{∗}K(ℓ_{p}) ≅ V (1)_{∗}K(ku_{p})^{Δ}, from which the corresponding assertion for p-adic integral homotopy groups follows by a Bockstein argument.

Conversely, the later calculations of [Au:tcku] show that there is an element b ∈ V (1)_{2p+2}K(ku) with b^{p−1} = −v_{2}, related to a_{(0,1)} ∈ K(2)_{2p+2}K(ℤ/p, 2) [RaWi80, §9] via the composite map

K(ℤ/p, 2) → K(ℤ, 3) → BGL_{1}(ku) → K(ku)

and the unit map V (1) → V (1) ∧ E(2) = K(2). Furthermore, there is an isomorphism

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

where P (x^{±1}) denotes the Laurent polynomial algebra over 𝔽_{p} generated by x. The left hand side can also be written as the algebraic extension

V (1)_{∗}K(ℓ_{p})[v_{2}^{−1}, b]/(b^{p−1} + v_{2}) .

Here Δ acts faithfully on b, fixing b^{p−1} = −v_{2}, so this both shows how the v_{2}- periodic algebraic K-theory of ku_{p} is generated by that of ℓ_{p} and the element b, and conversely how the v_{2}-periodic algebraic K-theory of ℓ_{p} can be recovered from that of ku_{p} as the fixed points of the Galois action. In view of the cofiber sequences (3.7) above, the corresponding formulas also hold with L_{p} and KU_{p} in place of ℓ_{p} and ku_{p}.

We are therefore confident that ku-based calculations of K(ku/p), K(ku_{p}) and K(ff (ku_{p})), like the ℓ-based calculations of K(ℓ/p), K(ℓ_{p}) and K(ff (ℓ_{p})) to be presented here, will verify the, for now conjectural, formula

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

and show that V (1)_{∗}K(ff (ℓ_{p}))[v_{2}^{−1}] can be recovered as the Δ-invariants of the right hand side. Thus we expect these more complicated calculations to give qualitatively the same answers. One reason to perform such more complicated calculations, other than to confirm our expectations, would be to precisely identify the image of V (1)_{∗}K(ff (ku_{p})) in V (1)_{∗}K(ff (ku_{p}))[v_{2}^{−1}], and to check that the map inverting v_{2} is very close to being injective, but we will not address this matter in the present paper.

With this we are done with our justifications for concentrating on the following framework. We collect the relevant definitions in one place, for the reader’s convenience.

Definition 3.9. Let ℓ_{p} be the p-complete connective Adams summand, which is a well-defined commutative S-algebra, with π_{∗}ℓ_{p} = ℤ_{p}[v_{1}]. For each choice of associative ℓ_{p}-algebra structure on ℓ/p, the p-complete form

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

of diagram (2.4) is a base change square of associative ℓ_{p}-algebras. Each map i: ℓ_{p} → ℓ/p, i: Hℤ_{p} → Hℤ/p, π: ℓ_{p} → Hℤ_{p} and π: ℓ/p → Hℤ/p makes the target a finite 2-cell complex over the source. The respective transfer maps in algebraic K-theory define the upper left hand square in the diagram (3.10) below.

The upper row, and the left hand and middle columns, are cofiber sequences by the algebraic K-theory localization sequences of Quillen and Blumberg–Mandell, respectively. We define K(p^{−1}ℓ_{p}) and K(p^{−1}L_{p}) to be the cofibers of the transfer maps i_{∗} in the middle and lower rows, respectively. This makes K(p^{−1}L_{p}) the iterated cofiber of the upper left hand square. Furthermore, we write ff (ℓ_{p}) = ff (L_{p}) for the symbol p^{−1}L_{p}, and consider it as an S-algebraic fraction field of ℓ_{p}. Likewise, we consider the symbol p^{−1}ℓ_{p} as an S-algebraic valuation ring for a valuation on ff (ℓ_{p}) with uniformizer v_{1}. Then the following diagram is a 3 × 3 square of cofiber sequences.

(3.10)  \xymatrix{K(ℤ/p) \ar[r]^{i_{∗}} \ar[d]^{π_{∗}} & K(ℤ_{p}) \ar[r]^{j^{∗}} \ar[d]^{π_{∗}} & K(ℚ_{p}) \ar[d]^{π_{∗}} \\ K(ℓ/p) \ar[r]^{i_{∗}} \ar[d]^{ρ^{∗}} & K(ℓ_{p}) \ar[r]^{j^{∗}} \ar[d]^{ρ^{∗}} & K(p^{−1}ℓ_{p}) \ar[d]^{ρ^{∗}} \\ K(L/p) \ar[r]^{i_{∗}} & K(L_{p}) \ar[r]^{j^{∗}} & K(ff (ℓ_{p}))}

Remark 3.11. We gave the existence of a non-trivial S-algebra homomorphism L_{p} → L/p as one argument for why L_{p} is not an S-algebraic field. However, this is in an essential way a non-commutative phenomenon. It follows from [BMMS86, III.4.1] that any commutative S-algebra with p = 0 in π_{0} is an Hℤ/p-module, hence any commutative L_{p}-algebra with p = 0 in π_{0} is trivial, since Hℤ/p∧L_{p} ≃ ∗. Algebraic K-theory is a functor of associative S-algebras, not just of commutative S-algebras, so this should not be perceived as an insurmountable obstacle, but it still illustrates some of the subtleties involved in thinking algebro-geometrically about these S-algebras.

Remark 3.12. It is time to explain why we do not think of K(p^{−1}ku) as the algebraic K-theory of the S-algebra ku[p^{−1}], etc. For primes p ≥ 5 we computed V (1)_{∗}K(ℓ_{p}) in [AuR02], and we shall compute V (1)_{∗}K(ℓ/p) in Theorem 8.10 below. The results are different, also after inverting v_{2}, so V (1)_{∗}K(p^{−1}ℓ_{p})[v_{2}^{−1}] is certainly nonzero. This is a direct summand of V (1)_{∗}K(p^{−1}ku_{p})[v_{2}^{−1}], which is isomorphic to V (1)_{∗}K(p^{−1}ku)[v_{2}^{−1}], V (1)_{∗}K(ff (ku_{p}))[v_{2}^{−1}] and V (1)_{∗}K(ff (ku))[v_{2}^{−1}], so all of these are also nonzero. On the other hand, ku_{p} is an S_{p}-algebra, so the localization ku_{p}[p^{−1}] is an S_{p}[p^{−1}] ≃ Hℚ_{p}-algebra. Therefore V (1)_{∗}K(ku_{p}[p^{−1}]) is a module over V (1)_{∗}K(ℚ_{p}), which is finite, so V (1)_{∗}K(ku_{p}[p^{−1}])[v_{2}^{−1}] is a module over V (1)_{∗}K(ℚ_{p})[v_{2}^{−1}], which is zero. Hence V (1)_{∗}K(ku_{p}[p^{−1}])[v_{2}^{−1}] = 0. Therefore p^{−1}ku_{p} ≄ ku_{p}[p^{−1}] and ff (ku_{p}) ≄ KU_{p}[p^{−1}].

4. Topological Hochschild homology

In this and the following sections, 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 the upper left hand square in diagram (3.10), 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, which by the projection formula for the transfer maps allows us to compute the V (1)-homotopy of the remaining entries in the diagram, including K(p^{−1}L_{p}) = K(ff (ℓ_{p})). 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 [Bo:zzp], [BM94], [BM95] and [HM97] for Hℤ/p and Hℤ, and of [MS93] and [AuR02] for ℓ. See also [AnR05, §§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 [Mi58]. 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

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)

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

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 .

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

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) .

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 [Bo:zzp], [Hu96] or [Au05, 4.3], leaving

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) .

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, so

(4.1)  \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

(4.2)  \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 map π^{∗}: T HH(ℓ) → T HH(ℤ_{(p)}) takes σξ̄_{2} to 0 and στ̄_{2} to (στ̄_{1})^{p}. The map i^{∗}: T HH(ℤ_{(p)}) → T HH(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 Salgebras. 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

(4.3)  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). We need to resolve the A_{∗}-comodule and H_{∗}(T HH(ℓ))-module extensions in order to obtain H_{∗}(T HH(ℓ/p)). This is achieved in Lemma 4.6 below.

The 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. (4.2).

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) there is a map π^{∗} 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 4.4. 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 that maps to τ̄_{0}(στ̄_{0})^{p−1} − τ̄_{1} in H_{∗}(T HH(ℤ/p)).

Proof. This follows by 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 4.5. For p = 2 this can alternatively be read off from the explicit form [Wu91] 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 4.6. 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 (4.3). 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, from which the claim follows.

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

(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

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.

5. Passage to V (1)-homotopy

For p ≥ 5 the Smith–Toda complex V (1) is a homotopy commutative ring spectrum. 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

ν(τ_{0}) = 1 ⊗ τ_{0} + τ_{0} ⊗ 1
ν(τ_{1}) = 1 ⊗ τ_{1} + ξ_{1} ⊗ τ_{0} + τ_{1} ⊗ 1 .

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 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

(5.1)  \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,

(5.2)  \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 [AuR02].

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

(5.3.a)  μ_{0}^{i} := 1 ∧ (στ̄_{0})^{i}

for 0 ≤ i ≤ p − 1 and

(5.3.b)  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. 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

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

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

(5.3.c)  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 5.4. 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 (5.3.abc) 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. □

We end this section with a suggestive conjectural calculation of the topological Hochschild homology of the fraction field ff (ℓ) = p^{−1}L, which may play the role of the de Rham complex over Spec(ff (ℓ)) in derived algebraic geometry. The calculation is not needed for the rest of the paper. We work with the p-local ℓ, but could equally well have worked with the p-complete ℓ_{p}.

Thus consider a 3 × 3 square of cofiber sequences

(5.5)  \xymatrix{T HH(ℤ/p) \ar[r]^{i_{∗}} \ar[d]^{π_{∗}} & T HH(ℤ_{(p)}) \ar[r]^{j^{∗}} \ar[d]^{π_{∗}} & T HH(ℤ_{(p)}|ℚ) \ar[d]^{π_{∗}} \\ T HH(ℓ/p) \ar[r]^{i_{∗}} \ar[d]^{ρ^{∗}} & T HH(ℓ) \ar[r]^{j^{∗}} \ar[d]^{ρ^{∗}} & T HH(ℓ|p^{−1}ℓ) \ar[d]^{ρ^{∗}} \\ T HH(ℓ/p|L/p) \ar[r]^{i_{∗}} & T HH(ℓ|L) \ar[r]^{j^{∗}} & T HH(ff (ℓ))}

generated by the upper left hand square. The transfer map i_{∗}: T HH(ℤ/p) → T HH(ℤ_{(p)}) was properly defined in [HM03], and T HH(ℤ_{(p)}|ℚ) is its cofiber. To construct the remaining transfer maps one may use the definition in [BM:loc] of T HH in terms of spectral categories. By analogy with the algebraic case, we write T HH(ℓ|p^{−1}ℓ), T HH(ℓ/p|L/p) and T HH(ℓ|L) for three of the remaining cofibers in the diagram. We might have denoted the term in the lower right hand corner by something like T HH(ℓ|p^{−1}ℓ, L|p^{−1}L), but for simplicity we abbreviate this to T HH(ff (ℓ)).

In the top row we get [HM03, 2.4.1] a long exact sequence in V (1)-homotopy

… ⟶^{∂} E(ϵ_{0}, ϵ_{1}) ⊗ P (μ_{0}) ⟶^{i_{∗}} E(ϵ_{1}, λ_{1}) ⊗ P (μ_{1}) ⟶^{j^{∗}} E(dlog p, ϵ_{1}) ⊗ P (κ_{0}) ⟶^{∂} …

where i_{∗}(ϵ_{0}μ_{0}^{p−1}) = λ_{1}, j^{∗}(μ_{1}) = κ_{0}^{p}, ∂(dlog p) = 1 and ∂(κ_{0}) = ϵ_{0}, tensored with the identity on E(ϵ_{1}).

In the middle row we expect a long exact sequence

… ⟶^{∂} 𝔽_{p}{1, ϵ_{0}, μ_{0}, ϵ_{0}μ_{0}, …, μ_{0}^{p−1}, ϵ̄_{1}} ⊗ E(λ_{2}) ⊗ P (μ_{2}) ⟶^{i_{∗}}

⟶^{i_{∗}} E(λ_{1}, λ_{2}) ⊗ P (μ_{2}) ⟶^{j^{∗}} E(dlog p, λ_{2}) ⊗ P_{p}(κ_{0}) ⊗ P (μ_{2}) ⟶^{∂} …

where i_{∗}(ϵ̄_{1}) = λ_{1}, ∂(dlog p) = 1 and ∂(κ_{0}) = ϵ_{0}, tensored with the identity on E(λ_{2}).

In the middle column we expect [Au05, §10] a long exact sequence

… ⟶^{∂} E(ϵ_{1}, λ_{1}) ⊗ P (μ_{1}) ⟶^{π_{∗}} E(λ_{1}, λ_{2}) ⊗ P (μ_{2}) ⟶^{ρ^{∗}} E(dlog v_{1}, λ_{1}) ⊗ P (κ_{1}) ⟶^{∂} …

where π_{∗}(ϵ_{1}μ_{1}^{p−1}) = λ_{2}, ρ^{∗}(μ_{2}) = κ_{1}^{p}, ∂(dlog v_{1}) = 1 and ∂(κ_{1}) = ϵ_{1}, tensored with the identity on E(λ_{1}).

In the right hand column we expect a long exact sequence

… ⟶^{∂} E(dlog p, ϵ_{1}) ⊗ P (κ_{0}) ⟶^{π_{∗}}

⟶^{π_{∗}} E(dlog p, λ_{2}) ⊗ P_{p}(κ_{0}) ⊗ P (μ_{2}) ⟶^{ρ^{∗}} E(dlog p, dlog v_{1}) ⊗ P (κ_{0}) ⟶^{∂} …

where π_{∗}(ϵ_{1}κ_{0}^{p^{2}−p}) = λ_{2}, ρ^{∗}(μ_{2}) = κ_{0}^{p^{2}}, ∂(dlog v_{1}) = 1 and ∂(κ_{0}^{p}) = ϵ_{1}, tensored with the identity on E(dlog p). Note that this ρ^{∗} is not multiplicative, since it takes the truncated polynomial algebra P_{p}(κ_{0}) into P (κ_{0}).

This leads to the following formula. When compared with [HM03, 2.4.1], it strongly suggests that E(dlog p, dlog v_{1}) is the V (1)-homotopy of a de Rham complex for ℓ with logarithmic poles along (p) and (v_{1}).

Conjecture 5.6. There is an isomorphism

V (1)_{∗}T HH(ff (ℓ)) ≅ E(dlog p, dlog v_{1}) ⊗ P (κ_{0})

with | dlog p | = | dlog v_{1}| = 1 and |κ_{0}| = 2. Here T HH(ff (ℓ)) is defined as the iterated cofiber of the upper left hand square in diagram (5.5), dlog p is in the image from π_{1}T HH(ℓ|p^{−1}ℓ), with ∂(dlog p) = 1 in π_{0}T HH(ℓ/p), dlog v_{1} is in the image from π_{1}T HH(ℓ|L), with ∂(dlog v_{1}) = 1 in π_{0}T HH(ℤ_{(p)}), and κ_{0} satisfies

κ_{0}^{p^{2}} = μ_{2}, with μ_{2} in the image from V (1)_{2p^{2}}T HH(ℓ).

6. 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, denoted t_{C}(X)^{C} in [GM95] and ℍ̂(C, X) in [AuR02]. 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 6.8 and 6.9 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

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}) .

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 (5.1), so

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}) .

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) = Ê_{∗∗}^{∞}(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 6.1. 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

(6.2)  \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 equalizes the iterated restriction and Frobenius maps R^{n} and F^{n} to T HH(B) [BHM93]. 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} after p-adic completion, 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 6.3. The map π: ℓ/p → Hℤ/p is (2p − 2)-connected, so it 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)_{∗}K(ℤ/p) ≅ ℤ/p.

Lemma 6.4. 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 5.4 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{matrix}μ_{0}^{i} & for δ = 1, 0 < i < p, \\ 0 & otherwise \end{matrix}

is readily deduced from (5.1), and σ(ϵ̄_{1}) = 0 since ϵ̄_{1} is in the image of tr. Thus

(6.5)  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 [AuR02, 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 [AuR02, 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

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}

up to units in 𝔽_{p}, which we will always suppress, leave the terms

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}})

with Ê^{2p^{2}+2} = Ê^{∞}, converging to V (1)_{∗}T HH(ℓ)^{tC_{p}}. The comparison map Γ̂_{1} maps 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 6.6. 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, up to a unit in 𝔽_{p}, which we have already declared that we will suppress. 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 (6.5), 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 6.7. 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 5.4 and Lemma 6.1.

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 6.6 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 6.8. 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 (6.5) 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 Lemma 5.4, 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 6.7. 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 (6.5) that

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}) .

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

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}) .

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

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}) .

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

Proposition 6.9. 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 6.7, and the action on the classes λ_{2} and μ_{2} was already recalled from [AuR02]. The structure of V (1)_{∗}T HH(ℓ/p)^{tC_{p}} is then immediate from the E^{∞}-term in Proposition 6.8. 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 6.10. The comparison maps

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}}}

for n ≥ 1, and

Γ: T F (ℓ/p) → T HH(ℓ/p)^{hS^{1}}
Γ̂: T F (ℓ/p) → T HH(ℓ/p)^{tS^{1}}

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

Proof. This follows from a theorem of Tsalidis [Ts98] and Proposition 6.9 above, just like in [AuR02, 5.7]. See also [BBLR:cf]. □

7. Higher fixed points

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

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}

for k ≥ 0, so ρ(−1) = 1 and ρ(0) = 0. For even arguments, ρ(2k) = r(2k) as defined in [AuR02, 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 (6.2), by [AuR02, 2.8] and Lemma 6.4. For tμ_{2} it follows from [AuR02, 4.8], by naturality.

Theorem 7.1. The C_{p^{n}}-Tate spectral sequence Ê^{∗}(C_{p^{n}}, ℓ/p) 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 7.1 by induction on n. The base case n = 1 is covered by Proposition 6.8. We can therefore assume that Theorem 7.1 holds for some fixed n ≥ 1. First we make the following deduction.

Corollary 7.2. 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

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})

(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 6.8. 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

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}) .

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

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}) .

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

(7.3)  \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 6.10, so G_{n−1} is at least (2p − 1)-coconnected. (We shall see in Lemma 7.11 that V (1)_{∗}G_{n−1} is an isomorphism in all degrees.) By Proposition 6.9 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 7.4. 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 7.1 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 7.5. 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

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}) .

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 7.2. 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

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}) ,

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

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}) .

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 7.1. 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 [AuR02, 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 7.1 for n + 1 that concerns the differentials leading up to the term

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}) (7.6)

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

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

(7.7)  [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 6.9 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 7.8 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 (7.6), 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 7.9 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 [AuR02, 6.1], hence follow in the present case by the module structure. With this we have established the complete differential pattern asserted by Theorem 7.1. □

Lemma 7.8. 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 7.5, 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 7.9. 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 7.10. 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 7.2. □

Lemma 7.11. 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 (7.3), 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 7.12. (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 7.2 and 7.5. 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 (7.7). □

8. 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

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

in V (1)-homotopy. By Corollary 6.10, 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 (7.3), 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 7.11, and the middle (wrongway) map is (2p − 2)-coconnected.

Proposition 8.2. 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 7.12. 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 halfplane. 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 8.3. Let

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})

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 8.4. 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 7.12(c), and the map Ê^{∞}(R^{h}), computed in Proposition 8.2. 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 [AuR02], using Corollary 7.10 to control the topologies, and will be omitted. □

Remark 8.5. Here we have followed the basic computational strategy of [BM94],

[BM95] and [AuR02]. It would be desirable 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 8.6. 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 (8.1).

Proof. By Proposition 8.4, 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 (8.1) 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 8.7. 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 [Du97]. 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 8.8. 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 8.6 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

(8.9)  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 8.10. 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, where p ≥ 5 is assumed.

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. □

9. The fraction field

We wish to determine the effect on algebraic K-theory of excising the closed subspaces Spec(ℤ_{p}) and Spnc(ℓ/p) from Spec(ℓ_{p}), meeting at the closed point Spec(ℤ/p). It will be a little cleaner to express the computation in topological cyclic homology. Therefore we consider another 3 × 3 diagram of cofiber sequences

(9.1)  \xymatrix{T C(ℤ/p) \ar[r]^{i_{∗}} \ar[d]^{π_{∗}} & T C(ℤ_{(p)}) \ar[r]^{j^{∗}} \ar[d]^{π_{∗}} & T C(ℤ_{(p)}|ℚ) \ar[d]^{π_{∗}} \\ T C(ℓ/p) \ar[r]^{i_{∗}} \ar[d]^{ρ^{∗}} & T C(ℓ) \ar[r]^{j^{∗}} \ar[d]^{ρ^{∗}} & T C(ℓ|p^{−1}ℓ) \ar[d]^{ρ^{∗}} \\ T C(ℓ/p|L/p) \ar[r]^{i_{∗}} & T C(ℓ|L) \ar[r]^{j^{∗}} & T C(ff (ℓ)) ,}

generated as usual by the upper left hand square. Like in the T HH-case displayed in diagram (5.5), the upper horizontal transfer map i_{∗} was defined in [HM03], and the remaining transfer maps can be defined using [BM:loc].

In the top row, the transfer map i_{∗} from

V (1)_{∗}T C(ℤ/p) = E(∂, ϵ̄_{1})

to

V (1)_{∗}T C(ℤ_{(p)}) = E(∂, λ_{1}) ⊕ 𝔽_{p}{t^{d}λ_{1} | 0 < d < p}

is a module map over the target. We claim that i_{∗}(1) = 0 and i_{∗}(ϵ̄_{1}) = λ_{1}, up to a unit in 𝔽_{p}, so

V (1)_{∗}T C(ℤ_{(p)}|ℚ) = E(∂, dlog p) ⊕ 𝔽_{p}{t^{d}λ_{1} | 0 < d < p} ,

where the connecting map from (9.1) takes the degree 1 class dlog p to 1. To prove the claim, recall from [HM03] that v_{1} · dlog p = λ_{1} in the V (0)-homotopy of T C(ℤ_{(p)}|ℚ). Hence λ_{1} in V (1)_{∗}T C(ℤ_{(p)}) maps to zero under j^{∗}, and must be in the image of i_{∗}. The only class that can hit it is ϵ̄_{1}, up to a unit.

In the middle row, the transfer map i_{∗} from

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}

(computed in Theorem 8.8), to

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

(computed in [AuR02, 8.4]), is also a module map over the target. By naturality in the diagram

\xymatrix{T C(ℓ/p) \ar[r]^{i_{∗}} \ar[d]^{π^{∗}} & T C(ℓ) \ar[d]^{π^{∗}} \ar[r]^{j^{∗}} & T C(ℓ|p^{−1}ℓ) \ar[d]^{π^{∗}} \\ T C(ℤ/p) \ar[r]^{i_{∗}} & T C(ℤ_{(p)}) \ar[r]^{j^{∗}} & T C(ℤ_{(p)}|ℚ)}

we find that i_{∗}(1) = 0 and i_{∗}(ϵ̄_{1}) = λ_{1} in V (1)_{∗}T C(ℓ), since the vertical maps π^{∗} are all (2p − 1)-connected.

To be precise, i_{∗}(ϵ̄_{1}) will be a unit times λ_{1} plus a multiple of t^{p^{2}−p}λ_{2}, but λ_{1} was only defined modulo such indeterminacy in [AuR02, 1.3], and this is a good occasion to make a more definite choice. In p-adic homotopy, π^{∗}: π_{2p−1}T C(ℓ)_{p} → π_{2p−1}T C(ℤ_{(p)})_{p} ≅ ℤ_{p}{λ_{1}} is surjective with kernel isomorphic to π_{2p−2}(ℓ)_{p} ≅ ℤ_{p}, so

π_{2p−1}T C(ℓ)_{p} → V (1)_{2p−1}T C(ℓ) = 𝔽_{p}{λ_{1}, t^{(p−1)p}λ_{2}}

is surjective, and this ensures that we may, indeed, choose λ_{1}^{K} ∈ K_{2p−1}(ℓ_{p})_{p} so that its cyclotomic trace image λ_{1} ∈ π_{2p−1}T C(ℓ)_{p} reduces mod p and v_{1} to a unit times i_{∗}(ϵ̄_{1}).

Furthermore, i_{∗}(t^{d}v_{2}) = 0 for 0 < d < p^{2} − p, p ∤ d, since the target is zero in these degrees, namely the even degrees strictly between 2p − 2 and 2p^{2} − 2 that are not congruent to −2 mod 2p. Similarly, i_{∗}(t^{d}v_{2} dlog v_{1}) = 0 for the same d. Therefore the cofiber of i_{∗} is

V (1)_{∗}T C(ℓ|p^{−1}ℓ) = P (v_{2}) ⊗ E(∂, dlog p, λ_{2})
⊕ P (v_{2}) ⊗ E(λ_{2}) ⊗ 𝔽_{p}{t^{d}λ_{1} | 0 < d < p}

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

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

where the connecting map from (9.1) takes dlog p to 1.

In the middle column the transfer map π_{∗}: V (1)_{∗}T C(ℤ_{(p)}) → V (1)_{∗}T C(ℓ) is zero. For the natural map π^{∗} is (2p − 1)-connected and thus surjective, and the composite π_{∗}∘π^{∗} is multiplication by the cyclotomic trace class of Hℤ_{(p)} in π_{0}T C(ℓ), realized as the mapping cone of v_{1} : Σ^{2p−2}ℓ → ℓ, which represents zero already in π_{0}K(ℓ).

We introduce a degree 1 class dlog v_{1} in the cofiber V (1)_{∗}T C(ℓ|L), and suppose that this notation is compatible with that of Theorem 8.8, in the sense that v_{2} · dlog v_{1} = λ_{2} as elements of V (1)_{∗}T C(ℓ|L), not just formally as in V (1)_{∗}T C(ℓ/p). This is again to be expected by analogy with the case of V (0)_{∗}T C(ℤ_{(p)}|ℚ). Then the terms E(∂, λ_{1}) ⊗ 𝔽_{p}{λ_{2}} and 𝔽_{p}{t^{d}λ_{1}λ_{2} | 0 < d < p} in V (1)_{∗}T C(ℓ) become once divisible by v_{2} in the cofiber, which we can write as

V (1)_{∗}T C(ℓ|L) = P (v_{2}) ⊗ E(∂, λ_{1}, dlog v_{1})
⊕ P (v_{2}) ⊗ E(dlog v_{1}) ⊗ 𝔽_{p}{t^{d}λ_{1} | 0 < d < p}
⊕ P (v_{2}) ⊗ E(λ_{1}) ⊗ 𝔽_{p}{t^{dp}v_{2} dlog v_{1} | 0 < d < p} .

In the left hand column, a very similar argument gives π_{∗} = 0 and

V (1)_{∗}T C(ℓ/p|L/p) = P (v_{2}) ⊗ E(∂, ϵ̄_{1}, dlog v_{1})

⊕ 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} .

We are most interested in the right hand column, where again π^{∗} is surjective and π_{∗} ∘ π^{∗} = 0, so π_{∗} = 0.

Proposition 9.2. There is an exact sequence

0 → V (1)_{∗}T C(ℓ|p^{−1}ℓ) ⟶^{ρ^{∗}} V (1)_{∗}T C(ff (ℓ)) → V (1)_{∗−1}T C(ℤ_{(p)}|ℚ) → 0 .

If v_{2} · dlog v_{1} = λ_{2}, where the (right hand) connecting map takes dlog v_{1} to 1, then there is an isomorphism

V (1)_{∗}T C(ff (ℓ)) = P (v_{2}) ⊗ E(∂, dlog p, dlog v_{1})
⊕ P (v_{2}) ⊗ E(dlog v_{1}) ⊗ 𝔽_{p}{t^{d}λ_{1} | 0 < d < p}

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

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

This is a free module over P (v_{2}) of rank (2p^{2} + 6) and of zero Euler characteristic.

Proof. The assumption in the second clause is that the terms E(∂, dlog p)⊗𝔽_{p}{λ_{2}} and 𝔽_{p}{t^{d}λ_{1}λ_{2} | 0 < d < p} in V (1)_{∗}T C(ℓ|p^{−1}ℓ) become once divisible by v_{2} in V (1)_{∗}T C(ff (ℓ)), and the formula then follows from those for V (1)_{∗}T C(ℤ_{(p)}|ℚ) and V (1)_{∗}T C(ℓ|p^{−1}ℓ). □

Remark 9.3. We expect to verify the relation v_{2} · dlog v_{1} = λ_{2} in V (1)_{∗}T C(ℓ|L) and V (1)_{∗}T C(ff (ℓ)) by means of a logarithmic geometric model for T C(ℓ|L), to be discussed in [Rog:ltc].

Theorem 9.4. There is an exact sequence

0 → Σ^{−2}E(dlog p, dlog v_{1}) ⊗ 𝔽_{p}{ϵ̄_{1}} → V (1)_{∗}K(ff (ℓ_{p})) ⟶^{trc}

⟶^{trc} V (1)_{∗}T C(ff (ℓ)) → Σ^{−1}E(dlog p, dlog v_{1}) → 0 .

Thus, if v_{2} · dlog v_{1} = λ_{2}, then there is an isomorphism of P (v_{2})-modules

V (1)_{∗}K(ff (ℓ_{p})) ≡ P (v_{2}) ⊗ Λ_{∗}

modulo the kernel Σ^{−2}E(dlog p, dlog v_{1}) ⊗ 𝔽_{p}{ϵ̄_{1}} of trc, 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}v_{2} dlog v_{1} | 0 < d < p} .

See Figure 10.3 below for a chart of Λ_{∗}.

Proof. To pass to algebraic K-theory, we consider the map from diagram (3.10) to diagram (9.1) induced by the cyclotomic trace map in (8.9). The cofiber is the 3×3 diagram of cofiber sequences with a copy of Σ^{−1}Hℤ_{p} in each corner of the upper left hand square, and V (1)_{∗}Σ^{−1}Hℤ_{p} = Σ^{−1}E(ϵ̄_{1}). The transfer maps i_{∗} and π_{∗} in this diagram are all trivial, since the natural maps i^{∗} induce the isomorphisms W (ℤ_{p})_{F} ≅ W (ℤ/p)_{F} and the maps π^{∗} induce identity maps. So there is a long exact sequence

⋯ → V (1)_{∗}K(ff (ℓ_{p})) ⟶^{trc} V (1)_{∗}T C(ff (ℓ)) ⟶^{g} Σ^{−1}E(ϵ̄_{1}, dlog p, dlog v_{1}) → ….

We claim that the image of g equals Σ^{−1}E(dlog p, dlog v_{1}). It is clear that the term E(dlog p, dlog v_{1}) ⊗ 𝔽_{p}{∂} in V (1)_{∗}T C(ff (ℓ)) has this image. We must argue that Σ^{−1}E(dlog p, dlog v_{1}) ⊗ 𝔽_{p}{ϵ̄_{1}} is not in the image of g.

In degree (2p−2) the class Σ^{−1}ϵ̄_{1} could only be hit by tλ_{1} ·dlog v_{1}, but tλ_{1} is the image of α_{1} in π_{2p−3}(S) and g takes dlog v_{1} to zero, so g also takes tλ_{1} · dlog v_{1} to zero, since it is a map of S-modules. In degree (2p−1) the class t^{p^{2}−p}v_{2} dlog v_{1} comes from t^{p^{2}−p}λ_{2} in V (1)_{∗}T C(ℓ), so by naturality it cannot map to Σ^{−1}ϵ̄_{1} dlog p or Σ^{−1}ϵ̄_{1} dlog v_{1}. In degree 2p the class t^{p^{2}−p}v_{2} dlog p dlog v_{1} comes from t^{p^{2}−p}λ_{2} dlog p in V (1)_{∗}T C(ℓ|p^{−1}ℓ), so it cannot hit Σ^{−1}ϵ̄_{1} dlog p dlog v_{1}. This exhausts all possibilities, and concludes the proof. □

The low-degree v_{2}-divisibility assumptions needed for Proposition 9.2 and Theorem 9.4 become irrelevant upon inverting v_{2}, since the V (1)-homotopy of both T C(ℤ_{(p)}|ℚ) and Σ^{−1}Hℤ_{p} is v_{2}-torsion. Hence we have the following unconditional result, for primes p ≥ 5.

Theorem 9.5. In v_{2}-periodic homotopy there are isomorphisms of P (v_{2}^{±1})-modules

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

where Λ_{∗} is as in Theorem 9.4.

10. Galois cohomology

For motivation, let F → E be a G-Galois extension of local or global number fields, i.e., finite extensions of ℚ or ℚ_{p} for some prime p. There is an induced Gaction on K(E) and a natural map K(F) → K(E)^{hG}. In their simplest form, the Lichtenbaum–Quillen conjectures predict that this map induces an isomorphism in mod p homotopy, in sufficiently high degrees. In other words, the abutment of the homotopy fixed point spectral sequence

E_{s,t}^{2} = H_{gp}^{−s}(G; V (0)_{t}K(E)) ⟹ V (0)_{s+t}K(E)^{hG}

is conjectured to agree with V (0)_{∗}K(F) in sufficiently high degrees. Thomason [Th85, 0.1, 4.1] proved these conjectures for algebraic K-theory with the Bott element inverted (in much greater generality than the present one), or equivalently, for the v_{1}-periodic algebraic K-theory functor V (0)_{∗}K(−)[v_{1}^{−1}]. Hence there is a spectral sequence

E_{s,t}^{2} = H_{gp}^{−s}(G; V (0)_{t}K(E)[v_{1}^{−1}]) ⟹ V (0)_{s+t}K(F)[v_{1}^{−1}]

for each Galois extension E of F, with G = Gal(E/F). Passing to the colimit over all such Galois extensions contained in a separable closure F̄ of F still gives a spectral sequence

E_{s,t}^{2} = H_{cont}^{−s}(G_{F}; V (0)_{t}K(F̄)[v_{1}^{−1}]) ⟹ V (0)_{s+t}K(F)[v_{1}^{−1}] .

Here H_{cont}^{r}(G_{F}; M) equals the continuous group cohomology of the absolute Galois group G_{F} = Gal(F̄/F), for any discrete G_{F} -module M, which is also denoted H_{Gal}^{r}(F; M) and known as Galois cohomology. By Suslin’s theorem [Su84], K(F̄)_{p} ≃ ku_{p}, so V (0)_{t}K(F̄)[v_{1}^{−1}] ≅ V (0)_{t}KU = 𝔽_{p}(t/2) equals the G_{F} -module μ_{p}^{⊗(t/2)} for t even, and 0 for t odd. Here 𝔽_{p}(1) = V (0)_{2}K(F̄) ≅ μ_{p}(F̄) = μ_{p} is the group of p-th roots of unity in F̄, considered as the p-torsion in K_{1}(F̄) = F̄^{×}. Thus the spectral sequence above can be rewritten as the Galois descent spectral sequence

(10.1)  E_{s,t}^{2} = H_{Gal}^{−s}(F; 𝔽_{p}(t/2)) ⟹ V (0)_{s+t}K(F)[v_{1}^{−1}] .

We will conjecture below that there are similar results for G-Galois extensions ff (A) → ff (B) of the fraction fields of a class of K(1)-local commutative S-algebras, containing L_{p} and KU_{p}, when considering the v_{2}-periodic algebraic K-theory functor V (1)_{∗}K(−)[v_{2}^{−1}]. Recall from (3.5) and (3.6) that this functor does not distinguish between the fraction fields of L and L_{p}, or between the fraction fields of KU, KU_{(p)} or KU_{p}, since it vanishes on finite, global and local fields in the ordinary sense. Note that we here shift our focus from connective spectra like ℓ and ℓ_{p}, which are convenient for topological cyclic homology calculations, to K(1)-local spectra like L_{p}, which are convenient for Galois theory. As discussed in Remark 2.6, ff (ℓ_{p}) = ff (L_{p}) and ff (ku_{p}) = ff (KU_{p}), so the shift is only one of perspective.

More precisely, we might consider a K(1)-local pro-Galois extension L_{K(1)}S → C of commutative S-algebras in the sense of [Rog08, 4.1], with profinite Galois group H, and a pair K ⊂ N of closed subgroups of H, with K normal in N. Letting A = C^{hN}, B = C^{hK} and G = N/K we obtain a diagram

J_{p} = L_{K(1)}S → A ⟶^{G} B → C .

Here J_{p} is the p-complete image-of-J spectrum, which plays an analogous initial role in the K(1)-local category as Hℚ = L_{0}S does in the rational category. There is a K(1)-local pro-Γ-Galois extension J_{p} → L_{p}, where Γ = 1 + pℤ_{p} ⊂ ℤ_{p}^{×} acts on L_{p} through p-adic Adams operations, and Spec(J_{p}) = [Spec(L_{p})/Γ] is essentially the derived orbit stack for the corresponding Γ-action on the derived scheme Spec(L_{p}). Similar remarks apply in a non-commutative sense to J/p → L/p, so Spec(ff (J_{p})) may be interpreted as the derived orbit stack for Γ acting on Spec(ff (L_{p})). In both cases, some care in interpretation is needed, since Γ is a profinite group.

For such a K(1)-local G-Galois extension A → B as above, we expect that the map K(A) → K(B)^{hG} induces an isomorphism in V (1)-homotopy, in sufficiently high degrees, so that there is a weak equivalence

V (1)_{∗}K(A)[v_{2}^{−1}] ⟶^{≃} V (1)_{∗}K(B)^{hG}[v_{2}^{−1}]

and a homotopy fixed point spectral sequence

E_{s,t}^{2} = H_{gp}^{−s}(G; V (1)_{t}K(B)[v_{2}^{−1}]) ⟹ V (1)_{s+t}K(A)[v_{2}^{−1}] .

In the special case of the Δ-Galois extension L_{p} → KU_{p}, this conjecture is somewhat trivially compatible with the predicted formula (3.8), since H_{gp}^{r}(Δ; P (b^{±1})) equals P (v_{2}^{±1}) for r = 0 and is 0 otherwise.

Considering A and B as valuation rings in their fraction fields, when defined, these extensions correspond to unramified Galois extensions of the fraction fields. It is difficult to use the obstruction theories of [GH04] or [Rob03] to construct ramified extensions in commutative S-algebras, since the ramification creates highly nontrivial obstruction groups of André–Quillen type. In fact, there are only a few unramified Galois extensions of L_{p} and KU_{p}. By [BR08, 1.1, 7.9], the extended Lubin–Tate spectrum KU_{p}^{nr}, with π_{∗}KU_{p}^{nr} = 𝕎(𝔽̄_{p})[u^{±1}], is the maximal pro-Galois extension of KU_{p}, both as commutative S-algebras and as K(1)-local commutative S-algebras.

More generally, we suppose that there is a notion of K(1)-local Galois extensions in an extended framework that contains the fraction fields of commutative S-algebras discussed in Section 3, but which also allows for ramified Galois extensions. Then let Ω_{1} be a maximal connected K(1)-local pro-Galois extension of ff (L_{p}), or equivalently of ff (J_{p}). In other words, Ω_{1} is to be a separable closure of the fraction field of L_{p} (or J_{p}). Then for each intermediate such K(1)-local G-Galois extension

ff (J_{p}) = ff (L_{K(1)}S) → ff (A) ⟶^{G} ff (B) → Ω_{1}

(where ff (B) or K(ff (B)) might only exist in the extended framework), we expect to have a weak equivalence

V (1)_{∗}K(ff (A))[v_{2}^{−1}] ⟶^{≃} V (1)_{∗}K(ff (B))^{hG}[v_{2}^{−1}]

and a homotopy fixed point spectral sequence

E_{s,t}^{2} = H_{gp}^{−s}(G; V (1)_{t}K(ff (B))[v_{2}^{−1}]) ⟹ V (1)_{s+t}K(ff (A))[v_{2}^{−1}] ,

as above. Fixing ff (A) and passing to the colimit over all such Galois extensions contained in Ω_{1}, we are then led to a spectral sequence

E_{s,t}^{2} = H_{cont}^{−s}(G_{ff (A)}; V (1)_{t}K(Ω_{1})[v_{2}^{−1}]) ⟹ V (1)_{s+t}K(ff (A))[v_{2}^{−1}] ,

where G_{ff (A)} = Gal(Ω_{1}/ff (A)) is the absolute Galois group, defined as the limit of the Galois groups G above.

In the case A = L_{p} for p ≥ 5, we achieved a concrete calculation of the abutment of this spectral sequence in Theorem 9.5, by methods completely independent of the conjectural Galois descent properties. We are therefore entitled to make an educated guess at the structure of the E^{2}-term of this spectral sequence, which in turn makes strong suggestions about the form of the Galois module V (1)_{∗}K(Ω_{1})[v_{2}^{−1}], and about the absolute Galois group G_{ff (L_{p})}.

We therefore study the spectral sequence

(10.2) E_{s,t}^{2} = H_{cont}^{−s}(G_{ff (L_{p})}; V (1)_{t}K(Ω_{1})[v_{2}^{−1}]) ⟹ V (1)_{s+t}K(ff (L_{p}))[v_{2}^{−1}] ,

which we assume to be an algebra spectral sequence, and one that collapses at the E^{2}-term for p ≥ 5, as is the case for the Galois descent spectral sequence (10.1) for any local number field, as well as for any global number field when p ≥ 3. The abutment V (1)_{∗}K(ff (L_{p}))[v_{2}^{−1}] is isomorphic to P (v_{2}^{±1}) tensored with Λ_{∗}, as given in Theorem 9.4. Hence the E^{2} = E^{∞}-term of (10.2) is the associated graded for a “Galois” filtration of this P (v_{2}^{±1})-module. Since v_{2} is invertible, it must be represented in Galois filtration s = 0. By consideration of localization sequences in motivic cohomology, and the relation of motivic cohomology to Galois cohomology, we can partially justify why the other algebra generators ∂v_{2}, dlog p, dlog v_{1}, t^{d}λ_{1}, t^{d}v_{2} dlog p and t^{dp}v_{2} dlog v_{1} must all be represented in Galois filtration s = −1. It follows that the entire spectral sequence is concentrated in the four columns −3 ≤ s ≤ 0, with the product ∂v_{2} · dlog p · dlog v_{1} as the only generator in filtration s = −3.

We display the placement in the E^{2}-term of the P (v_{2}^{±1})-module generators for V (1)_{∗}K(ff (L_{p}))[v_{2}^{−1}], or equivalently the 𝔽_{p}-basis for Λ_{∗}, in Figure 10.3 below. To

make room on the page, we write \frac{dp}{p} and \frac{dv_{1}}{v_{1}} for dlog p and dlog v_{1}, respectively. Multiple generators in the same bidegree are separated by colons. The symbol (v_{2}) indicates the placement of this v_{2}-multiple of 1. Also to save space, the chart is drawn for p = 3, even if we are really assuming p ≥ 5.

∂v_{2} \frac{dp}{p} \frac{dv_{1}}{v_{1}}  .  .  .  2p^{2} + 2 .  ∂v_{2} \frac{dp}{p} : ∂v_{2} \frac{dv_{1}}{v_{1}} : tv_{2} \frac{dp}{p} \frac{dv_{1}}{v_{1}}  .  .  2p^{2} .  t^{2}v_{2} \frac{dp}{p} \frac{dv_{1}}{v_{1}}  ∂v_{2} : tv_{2} \frac{dp}{p}  (v_{2})  2p^{2} − 2 .  t^{p}v_{2} \frac{dp}{p} \frac{dv_{1}}{v_{1}}  t^{2}v_{2} \frac{dp}{p}  . .  t^{p+1}v_{2} \frac{dp}{p} \frac{dv_{1}}{v_{1}}  t^{p}v_{2} \frac{dv_{1}}{v_{1}}  . .  t^{p+2}v_{2} \frac{dp}{p} \frac{dv_{1}}{v_{1}}  t^{p+1}v_{2} \frac{dp}{p}  . .  t^{2p}v_{2} \frac{dp}{p} \frac{dv_{1}}{v_{1}}  t^{p+2}v_{2} \frac{dp}{p}  . .  tλ_{1} \frac{dv_{1}}{v_{1}}  t^{2p}v_{2} \frac{dv_{1}}{v_{1}}  .  2p .  \frac{dp}{p} \frac{dv_{1}}{v_{1}} : t^{2}λ_{1} \frac{dv_{1}}{v_{1}}  tλ_{1}  .  2p − 2 .  .  \frac{dp}{p} : \frac{dv_{1}}{v_{1}} : t^{2}λ_{1}  .  2 .  .  .  1  0 −3  −2  −1  0  s\t

Figure 10.3: P (v_{2}^{±1})-basis for E_{s,t}^{2} ⟹ V (1)_{s+t}K(ff (L_{p}))[v_{2}^{−1}]

From these considerations, it is apparent that V (1)_{t}K(Ω_{1})[v_{2}^{−1}] will be nontrivial for each even t, and zero for each odd t. Let us briefly write M_{t} for this Galois module. The presence of nonzero products by dlog p or dlog v_{1} from E_{s,t}^{2} to E_{s−1,t+2}^{2}, for each even t and some s, implies that the pairing

M_{t} ⊗ M_{2} → M_{t+2}

(tensor product over M_{0}, here and below) is also nontrivial for each even t, since the product in the spectral sequence should be induced by the group cohomology cup product and this pairing of the coefficients. This can most simply be the case if these pairings induce an isomorphism

M_{2}^{⊗(t/2)} ⟶^{≅} M_{t}

for all even t, which we now assume. Since E_{0,t}^{2} = 0 for 0 < t < 2p^{2} − 2, the Galois

action on M_{2}^{⊗(t/2)} should have no invariants for these t, but M_{2}^{⊗(p^{2}−1)} should have trivial action. This indicates that the M_{0}-linear automorphism group of M_{2} should have exponent (p^{2} − 1), which excludes 𝔽_{p}, but strongly suggests the minimal example M_{2} = 𝔽_{p^{2}}(1), with some Galois automorphism of Ω_{1} over ff (L_{p}) acting by multiplication by a generator of the group of units 𝔽_{p^{2}}^{×} ≅ ℤ/(p^{2} − 1). Then M_{0} = 𝔽_{p^{2}}(0) with the trivial Galois action, and letting u ∈ M_{2} = V (1)_{2}K(Ω_{1})[v_{2}^{−1}] be a generator for M_{2} as an M_{0}-module, we see that u^{p^{2}−1} generates M_{2p^{2}−2} = M_{0}{v_{2}}, hence equals a unit multiple of v_{2}. Thus we deduce that the coefficient module in (10.2) can most simply be

V (1)_{∗}K(Ω_{1})[v_{2}^{−1}] = M_{∗} = 𝔽_{p^{2}}[u^{±1}] = π_{∗}(K_{2})

where K_{2} is the 2-periodic form of the Morava K-theory spectrum K(2), related to the (elliptic) Lubin–Tate spectrum E_{2} with π_{∗}E_{2} = 𝕎(𝔽_{p^{2}})[[u_{1}]][u^{±1}] by K_{2} ≃ V (1) ∧ E_{2}. Here K(2) → K_{2} takes v_{2} to u^{p^{2}−1}, and v_{1} acts on V (0) ∧ E_{2} by multiplication by u_{1}u^{p−1}. In other words, π_{∗}(K_{2}) ≅ π_{∗}(E_{2})/(p, u_{1}) = π_{∗}(E_{2})/(p, v_{1}). We are therefore led to the formula

V (1)_{∗}K(Ω_{1})[v_{2}^{−1}] ≅ V (1)_{∗}E_{2} .

By [HoSt99, 7.2], the left hand side is the V (1)-homotopy of L_{K(2)}K(Ω_{1}), and E_{2} is K(2)-local.

Conjecture 10.4. K(Ω_{1}) is a connective form of E_{2}, in the sense that there is an equivalence L_{K(2)}K(Ω_{1}) ≃ E_{2}.

This would generalize Suslin’s theorem, which can be formulated as an equivalence L_{K(1)}K(F̄) ≃ KU_{p} = E_{1}. Assuming this conjecture, we write 𝔽_{p^{2}}(t/2) for the Galois module V (1)_{t}K(Ω_{1})[v_{2}^{−1}] = V (1)_{t}E_{2} = π_{t}(K_{2}), which is 𝔽_{p^{2}}{u^{t/2}} for t even, and zero for t odd. Then the Galois descent spectral sequence takes the following form.

Conjecture 10.5. For K(1)-local fraction fields ff (A) contained in the separable closure Ω_{1} of ff (J_{p}) = ff (L_{K(1)}S) there is a natural algebra spectral sequence

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

where H_{Gal}^{r}(ff (A); M) = H_{cont}^{r}(G_{ff (A)}; M) is the continuous group cohomology of the absolute Galois group of ff (A).

The apparent self-duality of the 𝔽_{p}-algebra Λ_{∗} displayed in Figure 10.3 suggests that there is such a self-duality in the Galois cohomology of most or all K(1)-local fraction fields contained in Ω_{1}. Recall [Se97, §II.5] that for each p-local number field F there is a canonical isomorphism H_{Gal}^{2}(F, 𝔽_{p}(1)) ≅ 𝔽_{p}, and that by the Tate–Poitou arithmetic duality theorem the cup product

H_{Gal}^{r}(F; 𝔽_{p}(i)) ⊗ H_{Gal}^{2−r}(F; 𝔽_{p}(1 − i)) ⟶^{∪} H_{Gal}^{2}(F; 𝔽_{p}(1)) ≅ 𝔽_{p}

is a perfect pairing for each r and i.

Conjecture 10.6. For each finite K(1)-local Galois extension ff (A) of ff (L_{p}) there is a canonical isomorphism

H_{Gal}^{3}(ff (A); 𝔽_{p^{2}}(2)) ≅ 𝔽_{p}

and the cup product

H_{Gal}^{r}(ff (A); 𝔽_{p^{2}}(i)) ⊗ H_{Gal}^{3−r}(ff (A); 𝔽_{p^{2}}(2 − i)) ⟶^{∪} H_{Gal}^{3}(ff (A); 𝔽_{p^{2}}(2)) ≅ 𝔽_{p}

is a perfect pairing for each r and i.

Remark 10.7. Assuming these conjectures, it would be interesting to interpret the Galois module 𝔽_{p^{2}}(1) ≅ V (1)_{2}K(Ω_{1})[v_{2}^{−1}] directly in terms of K(Ω_{1}), perhaps as suitable torsion points of a derived algebraic or formal group over a Galois extension ff (A) of ff (L_{p}). In particular, it would be illuminating to find a (minimal) such extension for which

V (1)_{2}K(ff (A))[v_{2}^{−1}] → V (1)_{2}K(Ω_{1})[v_{2}^{−1}]

is surjective. This would play the role of the p-th cyclotomic field ℚ(ζ_{p}) in the case of mod p algebraic K-theory of number fields, since there is a Bott element β ∈ V (0)_{2}K(ℚ(ζ_{p}))[v_{1}^{−1}] mapping to u ∈ V (0)_{2}K(ℚ̄)[v_{1}^{−1}] ≅ 𝔽_{p}(1), up to a unit in 𝔽_{p}. We recall that the choice of a p-th root of unity ζ_{p} determines a group homomorphism C_{p} → GL_{1}(ℚ(ζ_{p})), a map of E_{∞} spaces

BC_{p} → BGL_{1}(ℚ(ζ_{p})) → Ω_{⊗}^{∞}K(ℚ(ζ_{p})) ,

and a map of commutative S-algebras S[BC_{p}] → K(ℚ(ζ_{p})). The Bott class β ∈ V (0)_{2}(BC_{p}) satisfies β^{p} = v_{1}β, as can be seen by mapping along V (0)_{∗}(BC_{p}) → K(1)_{∗}(BC_{p}), and its image in V (0)_{2}K(ℚ(ζ_{p})) satisfies β^{p−1} = v_{1} = u^{p−1}.

The calculations of [Au:tcku, 1.1] show that a generator in the (p + 1)-st tensor power 𝔽_{p^{2}}(p + 1) ≅ V (1)_{2p+2}K(Ω_{1})[v_{2}^{−1}] of the Galois module 𝔽_{p^{2}}(1) is realized by the unramified Δ-Galois extension L_{p} → KU_{p}, and its fraction field analogue ff (L_{p}) → ff (KU_{p}). For there is a higher Bott element b ∈ V (1)_{2p+2}K(ku) that satisfies b^{p−1} = −v_{2}, whose image in V (1)_{2p+2}K(Ω_{1})[v_{2}^{−1}] ≅ 𝔽_{p^{2}}{u^{p+1}} must have the form wu^{p+1}, for some element w ∈ 𝔽_{p^{2}} with w^{p−1} = −1. Hence ku_{p} → Ω_{1} realizes the 𝔽_{p}-subalgebra generated by wu^{p+1} and u^{±(p^{2}−1)} in 𝔽_{p^{2}}[u^{±1}]. The higher Bott element is constructed from the map

K(ℤ, 2) ≃ BU(1) → BU_{⊗} → GL_{1}(ku)

that interprets the left hand abelian group as the v_{1}-torsion points in the homotopy units of ku. More precisely, BU_{⊗} ≃ BU(1) × BSU_{⊗}, and p-locally BSU ≃ BSU_{⊗} [AP76] is v_{1}-torsion free, so K(ℤ, 2) is the v_{1}-torsion in BU_{⊗}. The delooped E_{∞} map K(ℤ, 3) → BGL_{1}(ku) → Ω_{⊗}^{∞}(ku) is adjoint to a map S[K(ℤ, 3)] → K(ku) of commutative S-algebras. The Bott element β ∈ V (1)_{2p+2}(K(ℤ, 3)) satisfies β^{p} = −v_{2}β, as can be detected in K(2)_{∗}(K(ℤ, 3)) using [RaWi80, 9.2, 12.1], and its image b in V (1)_{2p+2}K(ku) satisfies b^{p−1} = −v_{2} = −u^{p^{2}−1}. A generator δ of Δ multiplies b by a generator of 𝔽_{p}^{×} ≅ ℤ/(p − 1), so an extension of δ to a Galois automorphism of Ω_{1} must multiply u by a generator of 𝔽_{p^{2}}^{×} ≅ ℤ/(p^{2} − 1). Hence the absolute Galois group of ff (L_{p}) will act faithfully on 𝔽_{p^{2}}(1).

Now suppose that there is a homotopy-commutative H-space X with

K(2)_{∗}(X) ≅ K(2)_{∗}[x]/(x^{p^{2}} = v_{2}^{j}x) ,

realizing one of the finite irreducible commutative Morava–Hopf algebras [SW98, 1.4]. Here |x| = 2j, and we assume that (j, p + 1) = 1. Let ΩX be the loop group of X, and let S[ΩX] be the spherical monoid ring. As usual, there will be a homotopy commutative S-algebra map

S[X] → K(S[ΩX]) = A(X) ,

where A(X) is as in [Wa85]. The unit map S → E(2) induces a map V (1)_{∗}(X) → K(2)_{∗}(X), and we assume that x lifts to a class in V (1)_{2j}(X), with image ξ ∈ V (1)_{2j}K(S[ΩX]), still satisfying ξ^{p^{2}} = v_{2}^{j}ξ. If one can extend S[ΩX] to a K(1)- local Galois extension ff (A) of ff (J_{p}) or ff (L_{p}), then the image ξ ∈ V (1)_{2j}K(ff (A)) will map to u^{j} ∈ V (1)_{2j}K(Ω_{1})[v_{2}^{−1}] ≅ 𝔽_{p^{2}}(j), up to a unit in 𝔽_{p^{2}}. If we also arrange that ff (A) contains ff (KU_{p}), then ff (A) will realize all of 𝔽_{p^{2}}[u^{±1}], since this is generated as an 𝔽_{p}-algebra by the classes b ↦ wu^{p+1}, ξ ↦ u^{j} and v_{2}^{±1} ↦ u^{±(p^{2}−1)}.

Remark 10.8. The optimistic reader can now extend these conjectures for all n ≥ 1 to Galois extensions in the K(n)-local category, as seen by the v_{n+1}-periodic algebraic K-theory functor V (n)_{∗}K(−)[v_{n+1}^{−1}], where the Smith–Toda complex V (n) might be replaced with any fixed type (n+1) finite complex (to ensure that it exists). The extension J_{p} → L_{p} is best replaced with the K(n)-local pro-Galois extension L_{K(n)}S → E_{n} with Galois group 𝔾_{n} constructed in [DH04], see [Rog08, 5.4] and [AuR08]. We suggest writing Ω_{n} for the K(n)-local separable closure of the fraction field of L_{K(n)}S or E_{n}. Then Conjecture 10.4 may be extended to the prediction

L_{K(n+1)}K(Ω_{n}) ≃ E_{n+1} ,

and similarly for Conjecture 10.5. If correct, then e_{n+1} = K(Ω_{n})_{p} is a good connective form of the p-complete commutative S-algebra E_{n+1}.

References

[AP76] J. F. Adams and S. B. Priddy, Uniqueness of BSO, Math. Proc. Cambridge Philos. Soc. 80 (1976), 475–509.

[An08] Vigleik Angeltveit, Topological Hochschild homology and cohomology of A_{∞} ring spectra, Geometry and Topology 12 (2008), 987–1032.

[An:un] Vigleik Angeltveit, Uniqueness of Morava K-theory, arXiv:0810.5032 preprint.

[AnR05] Vigleik Angeltveit and John Rognes, Hopf algebra structure on topological Hochschild homology, Algebr. Geom. Topol. 5 (2005), 1223–1290.

[Au05] Christian Ausoni, Topological Hochschild homology of connective complex K-Theory, Amer. J. Math. 127 (2005), 1261–1313.

[Au:tcku] Christian Ausoni, On the algebraic K-theory of the complex K-theory spectrum,

arXiv:math.AT/0902.2334 preprint.

[AuR02] Christian Ausoni and John Rognes, Algebraic K-theory of topological K-theory, Acta Math. 188 (2002), 1–39.

[AuR08] Christian Ausoni and John Rognes, The chromatic red-shift in algebraic K-theory, Guido’s Book of Conjectures, Monographie de L’Enseignement Mathématique, vol. 40, 2008, pp. 13–15.

[BJ02] Andrew Baker and Alain Jeanneret, Brave new Hopf algebroids and extensions of MU-algebras, Homology Homotopy Appl. 4 (2002), 163–173.

[BR05] Andrew Baker and Birgit Richter, On the Γ-cohomology of rings of numerical polynomials and E_{∞} structures on K-theory, Comment. Math. Helv. 80 (2005), 691–723.

[BR08] Andrew Baker and Birgit Richter, Galois extensions of Lubin–Tate spectra, Homology, Homotopy Appl. 10 (2008), 27–43.

[BMS87] A. Beilinson, R. MacPherson, and V. Schechtman, Notes on motivic cohomology, Duke Math. J. 54 (1987), 679–710.

[Bl86] Spencer Bloch, Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), 267– 304.

[BL:ss] S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology, preprint

at http://www.math.uiuc.edu/K-theory/0062/ (1985).

[BM08] Andrew Blumberg and Mike Mandell, The localization sequence for the algebraic Ktheory of topological K-theory, Acta Math. 200 (2008), 155–179.

[BM:loc] Andrew Blumberg and Mike Mandell, Localization theorems in topological Hochschild

homology and topological cyclic homology, arXiv:math.KT/0802.3938 preprint.

[Bo:zzp] Marcel Bökstedt, The topological Hochschild homology of ℤ and ℤ/p, University of

Bielefeld preprint (ca. 1986).

[BBLR:cf] Marcel Bökstedt, Bob Bruner, Sverre Lunøe–Nielsen and John Rognes, On cyclic fixed

points of spectra, arXiv:math.AT/0712.3476 preprint.

[BHM93] Marcel Bökstedt, Wu Chung Hsiang and Ib Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 465–539.

[BM94] Marcel Bökstedt and Ib Madsen, Topological cyclic homology of the integers, K-theory (Strasbourg, 1992), Astérisque, vol. 226, 1994, pp. 7–8, 57–143.

[BM95] Marcel Bökstedt and Ib Madsen, Algebraic K-theory of local number fields: the unramified case, Prospects in topology (Princeton, NJ, 1994), Ann. of Math. Stud., vol. 138, Princeton Univ. Press, Princeton, NJ, 1995, pp. 28–57.

[BMMS86] R. R. Bruner, J. P. May, J. E. McClure and M. Steinberger, H_{∞} ring spectra and their applications, Lecture Notes in Mathematics, vol. 1176, Springer–Verlag, Berlin, 1986.

[DH04] Ethan S. Devinatz and Michael J. Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004), 1–47.

[Du97] Bjørn Ian Dundas, Relative K-theory and topological cyclic homology, Acta Math. 179 (1997), 223–242.

[EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell and J. P. May, Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997.

[FS02] Eric M. Friedlander and Andrei Suslin, The spectral sequence relating algebraic Ktheory to motivic cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), 773–875.

[GH04] P. G. Goerss and M. J. Hopkins, Moduli spaces of commutative ring spectra, Structured ring spectra, London Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151–200.

[GM95] J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543.

[HM97] Lars Hesselholt and Ib Madsen, On the K-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), 29–101.

[HM03] Lars Hesselholt and Ib Madsen, On the K-theory of local fields, Ann. of Math. (2) 158 (2003), 1–113.

[HoSm98] Michael J. Hopkins and Jeffrey H. Smith, Nilpotence and stable homotopy theory. II, Ann. of Math. (2) 148 (1998), 1–49.

[HoSt99] Mark Hovey and Neil P. Strickland, Morava K-theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666.

[Hu96] Thomas J. Hunter, On the homology spectral sequence for topological Hochschild homology, Trans. Amer. Math. Soc. 348 (1996), 3941–3953.

[La01] Andrey Lazarev, Homotopy theory of A_{∞} ring spectra and applications to MUmodules, K-Theory 24 (2001), 243–281.

[MS93] J. E. McClure and R. E. Staffeldt, On the topological Hochschild homology of bu. I, Amer. J. Math. 115 (1993), 1–45.

[Mi58] John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171.

[MM65] John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264.

[Ok84] Shichirô Oka, Multiplicative structure of finite ring spectra and stable homotopy of spheres, Algebraic topology (Aarhus, 1982), Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 418–441.

[Ol03] Martin C. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. (4) 36 (2003), 747–791.

[Qu72] Daniel Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (72), 552–586.

[Qu73] Daniel Quillen, Higher algebraic K-theory. I, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., vol. 341, Springer, Berlin, 1973, pp. 85–147.

[RaWi80] Douglas C. Ravenel and W. Stephen Wilson, The Morava K-theories of Eilenberg– Mac Lane spaces and the Conner–Floyd conjecture, Amer. J. Math. 102 (1980), 691– 748.

[Rob03] Alan Robinson, Gamma homology, Lie representations and E_{∞} multiplications, Invent. Math. 152 (2003), 331–348.

[Rog98] John Rognes, Trace maps from the algebraic K-theory of the integers (after Marcel Bökstedt), J. Pure Appl. Algebra 125 (1998), 277–286.

[Rog08] John Rognes, Galois extensions of structured ring spectra, Memoirs of the A.M.S. 192 (2008), no. 898, 1–97.

[Rog09] John Rognes, Topological logarithmic structures, New topological contexts for Galois theory and algebraic geometry (BIRS 2008), Geometry & Topology Monographs, vol. 16, 2009, pp. 401–544.

[Rog:ltc] John Rognes, Logarithmic topological cyclic homology, in preparation.

[SW98] Hal Sadofsky and Stephen W. Wilson, Commutative Morava homology Hopf algebras, Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997), Contemp. Math., vol. 220, Amer. Math. Soc., Providence, RI, 1998, pp. 367– 373.

[Se97] Jean–Pierre Serre, Galois cohomology, Springer-Verlag, Berlin, 1997.

[Sm70] Larry Smith, On realizing complex bordism modules. Applications to the stable homotopy of spheres, Amer. J. Math. 92 (1970), 793–856.

[Su84] Andrei A. Suslin, On the K-theory of local fields, J. Pure Appl. Algebra 34 (1984), 301–318.

[Th85] R. W. Thomason, Algebraic K-theory and étale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), 437–552.

[Ts98] Stavros Tsalidis, Topological Hochschild homology and the homotopy descent problem, Topology 37 (1998), 913–934.

[Wa85] Friedhelm Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419.

[Wu91] Urs Würgler, Morava K-theories: a survey, Algebraic topology (Poznań, 1989), Lecture Notes in Math., vol. 1474, Springer, Berlin, 1991, pp. 111–138.

Institute of Mathematics, University of Bonn, DE-53115 Bonn, Germany

E-mail address: ausoni@math.uni-bonn.de

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

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