ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY

Christian Ausoni and John Rognes

Abstract. Let ℓ_{p} be the p-complete connective Adams summand of topological Ktheory, with coefficient ring (ℓ_{p})_{∗} = ℤ_{p}[v_{1}], and let V (1) be the Smith–Toda complex, with BP_{∗}(V (1)) = BP_{∗}/(p, v_{1}). For p ≥ 5 we explicitly compute the V (1)-homotopy of the algebraic K-theory spectrum of ℓ_{p}, denoted V (1)_{∗}K(ℓ_{p}). In particular we find that it is a free finitely generated module over the polynomial algebra P (v_{2}), except for a sporadic class in degree 2p − 3. Thus also in this case algebraic Ktheory increases chromatic complexity by one. The proof uses the cyclotomic trace map from algebraic K-theory to topological cyclic homology, and the calculation is actually made in the V (1)-homotopy of the topological cyclic homology of ℓ_{p}.

Contents

Introduction

1. Classes in algebraic K-theory

2. Topological Hochschild homology

3. Topological cyclotomy

4. Circle homotopy fixed points

5. The homotopy limit property

6. Higher fixed points

7. The restriction map

8. Topological cyclic homology

9. Algebraic K-theory

Introduction

We are interested in the arithmetic of ring spectra. To make sense of this we must work with structured ring spectra, such as S-

algebras [EKMM], symmetric ring spectra [HSS] or Γ-rings [Ly]. We will refer to these as 𝕊-algebras. The commutative objects are then commutative 𝕊-algebras.

The category of rings is embedded in the category of 𝕊-algebras by the Eilenberg– Mac Lane functor R ↦ HR. We may therefore view an 𝕊-algebra as a generalization of a ring in the algebraic sense. The added flexibility of 𝕊-algebras provides room for new examples and constructions, which may eventually also shed light upon the category of rings itself.

In algebraic number theory the arithmetic of the ring of integers in a number field is largely captured by its Picard group, its unit group and its Brauer group. These are in turn reflected in the algebraic K-theory of the ring of integers. Algebraic

Typeset by AMS-TEX

K-theory is defined also in the generality of 𝕊-algebras. We can thus view the algebraic K-theory of an 𝕊-algebra as a carrier of some of its arithmetic properties.

The algebraic K-theory of (connective) 𝕊-algebras can be closely approximated by diagrams built from the algebraic K-theory of rings [Du]. Hence we expect that global structural properties enjoyed by algebraic K-theory as a functor of rings should also have an analogue for algebraic K-theory as a functor of 𝕊-algebras.

We have in mind, in particular, the étale descent property of algebraic K-theory conjectured by Lichtenbaum [Li] and Quillen [Qu], which has been established for several classes of commutative rings [V], [RW], [HM2]. We are thus led to ask when a map of commutative 𝕊-algebras A → B should be considered as an étale covering with Galois group G. In such a situation we may further ask whether the natural map K(A) → K(B)^{hG} to the homotopy fixed point spectrum for G acting on K(B) induces an isomorphism on homotopy in sufficiently high degrees. These questions will be considered in more detail in [Ro2].

One aim of this line of inquiry is to find a conceptual description of the algebraic K-theory of the sphere spectrum, K(S^{0}) = A(∗), which coincides with Waldhausen’s algebraic K-theory of the one-point space ∗. In [Ro1] the second author computed the mod 2 spectrum cohomology of A(∗) as a module over the Steenrod algebra, providing a very explicit description of this homotopy type. However, this result is achieved by indirect computation and comparison with topological cyclic homology, rather than by a structural property of the algebraic K-theory functor. What we are searching for here is a more memorable intrinsic explanation for the homotopy type appearing as the algebraic K-theory of an 𝕊-algebra.

More generally, for a simplicial group G with classifying space X = BG there is an 𝕊-algebra S^{0}[G], which can be thought of as a group ring over the sphere spectrum, and K(S^{0}[G]) = A(X) is Waldhausen’s algebraic K-theory of the space X. When X has the homotopy type of a manifold, A(X) carries information about the geometric topology of that manifold. Hence an étale descent description of K(S^{0}[G]) will be of significant interest in geometric topology, reaching beyond algebraic K-theory itself.

In the present paper we initiate a computational exploration of this ‘brave new world’ of ring spectra and their arithmetic.

We begin by considering some interesting examples of (pro-)étale coverings in the category of commutative 𝕊-algebras. For convenience we will choose to work locally, with 𝕊-algebras that are complete at a prime p. For the purpose of algebraic K-theory this is less of a restriction than it may seem at first. What we have in mind here is that the square diagram

\xymatrix{K(A) \ar[r] \ar[d] & K(A_{p}) \ar[d] \\ K(π_{0}A) \ar[r] & K(π_{0}A_{p})}

is homotopy cartesian after p-adic completion [Du], when A is a connective 𝕊algebra, A_{p} its p-completion, π_{0}A its ring of path components and π_{0}(A_{p}) ≅ (π_{0}A)_{p}.

This reduces the p-adic comparison of K(A) and K(A_{p}) to the p-adic comparison of K(π_{0}A) and K(π_{0}A_{p}), i.e., to a question about ordinary rings, which we view as a simpler question, or at least as one lying in better explored territory.

This leads us to study p-complete 𝕊-algebras, or algebras over the p-complete sphere spectrum S_{p}^{0}. This spectrum is approximated in the category of commutative 𝕊-algebras (or E_{∞} ring spectra) by a tower of chromatic localizations [Ra1]

S_{p}^{0} → ⋯ → L_{n}S_{p}^{0} → ⋯ → L_{1}S_{p}^{0} → L_{0}S_{p}^{0} = Hℚ_{p} .

Here L_{n} = L_{E(n)} is Bousfield’s localization functor [Bo], [EKMM] with respect to the nth Johnson–Wilson theory with coefficient ring E(n)_{∗} = ℤ_{(p)}[v_{1}, …, v_{n}, v_{n}^{−1}]. By the Hopkins–Ravenel chromatic convergence theorem [Ra3, §8], the natural map S_{p}^{0} → holim_{n} L_{n}S_{p}^{0} is a homotopy equivalence. For each n ≥ 1 there is a further map of commutative 𝕊-algebras L_{n}S_{p}^{0} → L_{K(n)}S_{p}^{0} to the Bousfield localization with respect to the nth Morava K-theory with coefficient ring K(n)_{∗} = 𝔽_{p}[v_{n}, v_{n}^{−1}]. This is an equivalence for n = 0, and L_{K(1)}S_{p}^{0} ≃ J_{p} is the non-connective p-complete image-of-J spectrum.

There is a highly interesting sequence of commutative 𝕊-algebras E_{n} constructed by Morava as spectra [Mo], by Hopkins and Miller [Re] as 𝕊-algebras (or A_{∞} ring spectra) and by Goerss and Hopkins [GH] as commutative 𝕊-algebras (or E_{∞} ring spectra). The coefficient ring of E_{n} is (E_{n})_{∗} ≅ W 𝔽_{p^{n}}[[u_{1}, …, u_{n−1}]][u, u^{−1}]. As a special case E_{1} ≃ KU_{p} is the p-complete complex topological K-theory spectrum.

The cited authors also construct a group action on E_{n} through commutative 𝕊algebra maps, by a semidirect product G_{n} = S_{n}⋊C_{n} where S_{n} is the nth (profinite) Morava stabilizer group [Mo] and C_{n} = Gal(𝔽_{p^{n}}/𝔽_{p}) is the cyclic group of order n. There is a homotopy equivalence L_{K(n)}S_{p}^{0} ≃ E_{n}^{hG_{n}}, where the homotopy fixed point spectrum is formed in a continuous sense [DH], which reflects the Morava change of rings theorem [Mo].

Furthermore, the space of self equivalences of E_{n} in the category of commutative 𝕊-algebras is weakly equivalent to its group of path components, which is precisely G_{n}. In fact the extension L_{K(n)}S_{p}^{0} → E_{n} qualifies as a pro-étale covering in the category of commutative 𝕊-algebras, with Galois group weakly equivalent to G_{n}. The weak contractibility of each path component of the space of self equivalences of E_{n} (over either S_{p}^{0} or L_{K(n)}S_{p}^{0}) serves as the commutative 𝕊-algebra version of the unique lifting property for étale coverings. Also the natural map ζ: E_{n} → T HH(E_{n}) is a K(n)-equivalence, cf. [MS1, 5.1], implying that the space of relative Kähler differentials of E_{n} over L_{K(n)}S_{p}^{0} is contractible. See [Ro2] for further discussion.

There are further étale coverings of E_{n}. For example there is one with coefficient ring W 𝔽_{p^{m}}[[u_{1}, …, u_{n−1}]][u, u^{−1}] for each multiple m of n. Let E_{n}^{nr} be the colimit of these, with E_{n∗}^{nr} = W 𝔽̄_{p}[[u_{1}, …, u_{n−1}]][u, u^{−1}]. Then Gal(E_{n}^{nr}/L_{K(n)}S_{p}^{0}) is weakly equivalent to an extension of S_{n} by the profinite integers ℤ̂ = Gal(𝔽̄_{p}/𝔽_{p}). Let Ē_{n} be a maximal pro-étale covering of E_{n}, and thus of L_{K(n)}S_{p}^{0}. What is the absolute Galois group Gal(Ē_{n}/L_{K(n)}S_{p}^{0}) of L_{K(n)}S_{p}^{0} ?

The tower of commutative 𝕊-algebras induces a tower of algebraic K-theory spectra

K(S_{p}^{0}) → ⋯ → K(L_{n}S_{p}^{0}) → ⋯ → K(J_{p}) → K(ℚ_{p})

studied in the p-local case by Waldhausen [Wa2]. The natural map K(S_{p}^{0}) → holim_{n} K(L_{n}S_{p}^{0}) may well be an equivalence, see [MS2]. We are thus led to study the spectra K(L_{n}S_{p}^{0}), and their relatives K(L_{K(n)}S_{p}^{0}). (More precisely, Waldhausen studied finite localization functors L_{n}^{f} characterized by their behavior on finite CWspectra. However, for n = 1 the localization functors L_{1} and L_{1}^{f} agree, and this is the case that we will explore in the body of this paper. Hence we will suppress this distinction in the present discussion.)

Granting that L_{K(n)}S_{p}^{0} → E_{n} qualifies as an étale covering in the category of commutative 𝕊-algebras, the descent question concerns whether the natural map

(0.1)  K(L_{K(n)}S_{p}^{0}) → K(E_{n})^{hG_{n}}

is a p-adic homotopy equivalence in sufficiently high dimensions. We conjecture that it does.

To analyze K(E_{n}) we expect to use a localization sequence in algebraic K-theory to reduce to the algebraic K-theory of connective commutative 𝕊-algebras, and to use the Bökstedt–Hsiang–Madsen cyclotomic trace map to topological cyclic homology to compute these [BHM]. The ring spectra E_{n} and E(n)_{p} are closely related, and for n ≥ 1 we expect that there is a cofiber sequence of spectra

(0.2)  K(BP ⟨n − 1⟩_{p}) → K(BP ⟨n⟩_{p}) → K(E(n)_{p})

analogous to the localization sequence K(𝔽_{p}) → K(ℤ_{p}) → K(ℚ_{p}) in the case n = 0. Something similar should work for E_{n}.

The cyclotomic trace map

trc: K(BP ⟨n⟩_{p}) → T C(BP ⟨n⟩_{p}; p) ≃ T C(BP ⟨n⟩; p)

induces a p-adic homotopy equivalence upon replacing the target with its connective cover [HM1]. Hence a calculation of T C(BP ⟨n⟩; p) is as good as a calculation of K(BP ⟨n⟩_{p}), after p-adic completion. In this paper we present computational techniques which are well suited for calculating T C(BP ⟨n⟩; p), at least when BP ⟨n⟩_{p} is a commutative 𝕊-algebra and the Smith–Toda complex V (n) exists as a ring spectrum. In the algebraic case n = 0, with BP ⟨0⟩ = Hℤ_{(p)}, these techniques simultaneously provide a simplification of the argument in [BM1], [BM2] computing T C(ℤ; p) and K(ℤ_{p}) for p ≥ 3. Presumably the simplification is related to that appearing in different generality in [HM2].

It is also plausible that variations on these techniques apply when replacing V (n) by another finite type n + 1 ring spectrum, and the desired commutative 𝕊-algebra structure on BP ⟨n⟩_{p} is weakened to the existence of an 𝕊-algebra map from a related commutative 𝕊-algebra, such as MU or BP.

The first non-algebraic case occurs for n = 1. Then E_{1} ≃ KU_{p} has an action by G_{1} = ℤ_{p}^{×} ≅ Γ × Δ. Here ℤ_{p} ≅ Γ = 1 + pℤ_{p} ⊂ ℤ_{p}^{×}, ℤ/(p − 1) ≅ Δ ⊂ ℤ_{p}^{×} and k ∈ ℤ_{p}^{×} acts on E_{1} like the p-adic Adams operation ψ^{k} on KU_{p}.

Let L_{p} = E_{1}^{hΔ} be the p-complete Adams summand with coefficient ring (L_{p})_{∗} = ℤ_{p}[v_{1}, v_{1}^{−1}], so L_{p} ≃ E(1)_{p}. Then Γ acts continuously on L_{p} with J_{p} ≃ L_{p}^{hΓ}. Let ℓ_{p} be the p-complete connective Adams summand with coefficient ring (ℓ_{p})_{∗} = ℤ_{p}[v_{1}], so ℓ_{p} ≃ BP ⟨1⟩_{p}. We expect that there is a cofiber sequence of spectra

K(ℤ_{p}) → K(ℓ_{p}) → K(L_{p}) .

The previous calculation of T C(ℤ; p) [BM1], [BM2], and the calculation of T C(ℓ; p) presented in this paper, identify the p-adic completions of K(ℤ_{p}) and K(ℓ_{p}), respectively. Given an evaluation of the transfer map between them, this identifies K(L_{p}). The homotopy fixed points for the Γ-action on K(L_{p}) induced by the Adams operations ψ^{k} for k ∈ 1 + pℤ_{p} should then model K(J_{p}) = K(L_{K(1)}S_{p}^{0}).

This brings us to the contents of the present paper. In §1 we produce two useful classes λ_{1}^{K} and λ_{2}^{K} in the algebraic K-theory of ℓ_{p}. In §2 we compute the V (1)- homotopy of the topological Hochschild homology of ℓ, simplifying the argument of [MS1]. In §3 we present notation concerning topological cyclic homology and the cyclotomic trace map of [BHM]. In §4 we make preparatory calculations in the spectrum homology of the S^{1}-homotopy fixed points of T HH(ℓ). These are applied in §5 to prove that the canonical map from the C_{p^{n}} fixed points to the C_{p^{n}} homotopy fixed points of T HH(ℓ) induces an equivalence on V (1)-homotopy above dimension (2p − 2), using [Ts]. In §6 we inductively compute the V (1)-homotopy of all these (homotopy) fixed point spectra, and their homotopy limit T F (ℓ; p). The action of the restriction map on this limit is then identified in §7. The pieces of the calculation are brought together in Theorem 8.4 of §8, yielding an explicit computation of the V (1)-homotopy of T C(ℓ; p):

Theorem 0.3. Let p ≥ 5. There is an isomorphism of E(λ_{1}, λ_{2}) ⊗ P (v_{2})-modules

V (1)_{∗}T C(ℓ; p) ≅ E(λ_{1}, λ_{2}, ∂) ⊗ P (v_{2})
⊕ E(λ_{2}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{1}t^{e} | 0 < e < p}
⊕ E(λ_{1}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{2}t^{ep} | 0 < e < p}

with |λ_{1}| = 2p − 1, |λ_{2}| = 2p^{2} − 1, |v_{2}| = 2p^{2} − 2, |∂| = −1 and |t| = −2.

The p-completed cyclotomic trace map K(ℓ_{p})_{p} → T C(ℓ_{p}; p) ≃ T C(ℓ; p) identifies K(ℓ_{p})_{p} with the connective cover of T C(ℓ; p). This yields the following expression for the V (1)-homotopy of K(ℓ_{p}), given in Theorem 9.1 of §9:

Theorem 0.4. Let p ≥ 5. There is an exact sequence of E(λ_{1}, λ_{2})⊗P (v_{2})-modules

0 → Σ^{2p−3}𝔽_{p} ⟶ V (1)_{∗}K(ℓ_{p}) ⟶^{trc} V (1)_{∗}T C(ℓ; p) → Σ^{−1}𝔽_{p} → 0

taking the degree 2p − 3 generator in Σ^{2p−3}H𝔽_{p} to a class a ∈ V (1)_{2p−3}K(ℓ_{p}), and taking the class ∂ in V (1)_{−1}T C(ℓ; p) to the degree −1 generator in Σ^{−1}H𝔽_{p}.

The V (1)-homotopy of any spectrum is a P (v_{2})-module, but we emphasize that V (1)_{∗}T C(ℓ; p) is a free finitely generated P (v_{2})-module, and V (1)_{∗}K(ℓ_{p}) is free and finitely generated except for the summand 𝔽_{p}{a} in degree 2p − 3. Hence both K(ℓ_{p})_{p} and T C(ℓ; p) are fp-spectra in the sense of [MR], with finitely presented mod p cohomology as a module over the Steenrod algebra. They both have fptype 2, because V (1)_{∗}K(ℓ_{p}) is infinite while V (2)_{∗}K(ℓ_{p}) is finite, and similarly for T C(ℓ; p). In particular, K(ℓ_{p}) is closely related to elliptic cohomology.

More generally, at least if BP ⟨n⟩_{p} is a commutative 𝕊-algebra and V (n) exists as a ring spectrum, similar calculations to those presented in this paper show that V (n)_{∗}T C(BP ⟨n⟩; p) is a free P (v_{n+1})-module on 2^{n+2} + 2^{n}(n + 1)(p − 1) generators. So algebraic K-theory takes such fp-type n commutative 𝕊-algebras to fp-type (n + 1) commutative 𝕊-algebras. If our ideas about localization sequences are correct then also K(E_{n})_{p} will be of fp-type (n + 1), and if étale descent holds in algebraic K-theory for L_{K(n)}S_{p}^{0} → E_{n} with cd_{p}(G_{n}) < ∞ then also K(L_{K(n)}S_{p}^{0})_{p} will be of fp-type (n + 1). The moral is that algebraic K-theory in many cases increments chromatic complexity by one, i.e., it produces a constant red-shift in stable homotopy theory.

Notations and conventions. For an 𝔽_{p} vector space V, let E(V), P (V) and Γ(V) be the exterior algebra, polynomial algebra and divided power algebra on V, respectively. When V has a basis {x_{1}, …, x_{n}}, we write E(x_{1}, …, x_{n}), P (x_{1}, …, x_{n}) and Γ(x_{1}, …, x_{n}) for these algebras. So Γ(x) = 𝔽_{p}{γ_{j}(x) | j ≥ 0} with γ_{i}(x) · γ_{j}(x) = (i, j)γ_{i+j}(x). Let P_{h}(x) = P (x)/(x^{h} = 0) be the truncated polynomial algebra of height h. For a ≤ b ≤ ∞ let P_{a}^{b}(x) = 𝔽_{p}{x^{k} | a ≤ k ≤ b} as a P (x)-module.

By an infinite cycle in a spectral sequence we mean a class x such that d^{r}(x) = 0 for all r. By a permanent cycle we mean an infinite cycle which is not a boundary, i.e., a class that survives to represent a nonzero class at E^{∞}. Differentials are often only given up to multiplication by a unit.

1. Classes in algebraic K-theory

1.1. E_{∞} ring spectrum models. Let p be an odd prime. Let ℓ = BP ⟨1⟩ be the Adams summand of p-local connective topological K-theory. Its homotopy groups are ℓ_{∗} ≅ ℤ_{(p)}[v_{1}], with |v_{1}| = q = 2p − 2.

Its p-completion ℓ_{p} with ℓ_{p∗} ≅ ℤ_{p}[v_{1}] admits a model as an E_{∞} ring spectrum, which can be constructed as the algebraic K-theory spectrum of a perfect field k′. Let g be a prime power topologically generating the p-adic units and let k′ = colim_{n≥0} 𝔽_{g^{p^{n}}} ⊂ k̄ be a ℤ_{p}-extension of k = 𝔽_{g}. Then ℓ_{p} = K(k′)_{p} is an E_{∞} ring spectrum model for the p-completed Adams summand.

Likewise j_{p} = K(k)_{p} and ku_{p} = K(k̄)_{p} are E_{∞} ring spectrum models for the p-completed image-of-J spectrum and the p-completed connective topological Ktheory spectrum, respectively. The Frobenius automorphism σ_{g}(x) = x^{g} induces the Adams operation ψ^{g} on both ℓ_{p} and ku_{p}. Then k is the fixed field of σ_{g}, and j_{p} is the connective cover of the homotopy fixed point spectrum for ψ^{g} acting on either one of ℓ_{p} or ku_{p}.

The E_{∞} ring spectrum maps S_{p}^{0} → j_{p} → ℓ_{p} → ku_{p} → Hℤ_{p} induce E_{∞} ring spectrum maps on algebraic K-theory:

K(S_{p}^{0}) ⟶ K(j_{p}) ⟶ K(ℓ_{p}) ⟶ K(ku_{p}) ⟶ K(ℤ_{p}) .

In particular these are H_{∞} ring spectrum maps.

1.2. A first class in algebraic K-theory. The Bökstedt trace map tr: K(ℤ_{p}) → T HH(ℤ_{p}) maps onto the first p-torsion in the image, which is T HH_{2p−1}(ℤ_{p}) ≅ ℤ/p{e} [BM1]. Let e^{K} ∈ K_{2p−1}(ℤ_{p}) be a class with tr(e^{K}) = e.

There is a (2p − 2)-connected linearization map ℓ_{p} → Hℤ_{p} of E_{∞} ring spectra, which induces a (2p − 1)-connected map K(ℓ_{p}) → K(ℤ_{p}).

Definition 1.3. Let λ_{1}^{K} ∈ K_{2p−1}(ℓ_{p}) be a chosen class mapping to e^{K} ∈ K_{2p−1}(ℤ_{p}) under the map induced by linearization ℓ_{p} → Hℤ_{p}.

The image tr(λ_{1}^{K}) ∈ T HH_{2p−1}(ℓ_{p}) of this class under the trace map tr: K(ℓ_{p}) → T HH(ℓ_{p}) will map under linearization to e ∈ T HH_{2p−1}(ℤ_{p}).

Remark 1.4. The class λ_{1}^{K} ∈ K_{2p−1}(ℓ_{p}) does not lift further back to K_{2p−1}(S_{p}^{0}), since e^{K} has a nonzero image in π_{2p−2} of the homotopy fiber of K(S_{p}^{0}) → K(ℤ_{p}) [Wa1]. Thus λ_{1}^{K} does also not lift to K_{2p−1}(j_{p}), because the map S_{p}^{0} → j_{p} is (pq − 2)-connected. It is not clear if the induced action of ψ^{g} on K(ℓ_{p}) leaves λ_{1}^{K} invariant.

1.5. Homotopy and homology operations. For a spectrum X, let D_{p}X = EΣ_{p} ⋉_{Σ_{p}} X^{∧p} be its pth extended power. Part of the structure defining an H_{∞} ring spectrum E is a map ξ: D_{p}E → E. Then a mod p homotopy class θ ∈ π_{m}(D_{p}S^{n}; 𝔽_{p}) determines a mod p homotopy operation

θ^{∗}: π_{n}(E) ⟶ π_{m}(E; 𝔽_{p})

natural for maps of H_{∞} ring spectra E. Its value on the homotopy class represented by a map a: S^{n} → E is the image of θ under the composite map

π_{m}(D_{p}S^{n}; 𝔽_{p}) ⟶^{D_{p}(a)} π_{m}(D_{p}E; 𝔽_{p}) ⟶^{ξ} π_{m}(E; 𝔽_{p}) .

Likewise the Hurewicz image h(θ) ∈ H_{m}(D_{p}S^{n}; 𝔽_{p}) induces a homology operation

h(θ)^{∗}: H_{n}(E; 𝔽_{p}) ⟶ H_{m}(E; 𝔽_{p}) ,

and the two operations are compatible under the Hurewicz homomorphisms.

For S^{n} with n = 2k −1 an odd dimensional sphere the bottom two cells of D_{p}S^{n} are in dimensions pn + (p − 2) and pn + (p − 1), and are connected by a mod p Bockstein. Hence the bottom two mod p homotopy classes of D_{p}S^{n} are in these two dimensions, and are called βP^{k} and P^{k}, respectively. Their Hurewicz images induce the Dyer–Lashof operations denoted βQ^{k} and Q^{k} in homology, cf. [Br].

1.6. A second class in algebraic K-theory. We use the H_{∞} ring spectrum structure on K(ℓ_{p}) to produce a further element in its mod p homotopy.

Definition 1.7. Let λ_{2}^{K} = (P^{p})^{∗}(λ_{1}^{K}) ∈ K_{2p^{2}−1}(ℓ_{p}; 𝔽_{p}) be the image under the mod p homotopy operation

(P^{p})^{∗}: K_{2p−1}(ℓ_{p}) ⟶ K_{2p^{2}−1}(ℓ_{p}; 𝔽_{p})

of λ_{1}^{K} ∈ K_{2p−1}(ℓ_{p}) .

Since the trace map tr: K(ℓ_{p}) → T HH(ℓ_{p}) is an E_{∞} ring spectrum map, it follows that tr(λ_{2}^{K}) ∈ T HH_{2p^{2}−1}(ℓ_{p}; 𝔽_{p}) equals the image of tr(λ_{1}^{K}) ∈ T HH_{2p−1}(ℓ_{p}) under the mod p homotopy operation (P^{p})^{∗}. We shall identify this image in the next section.

Remark 1.8. It is not clear whether λ_{2}^{K} lifts to an integral homotopy class in K_{2p^{2}−1}(ℓ_{p}). The image of e^{K} ∈ K_{2p−1}(ℤ_{p}) in K_{2p−1}(ℚ_{p}; 𝔽_{p}) is v_{1}d log p for a class d log p ∈ K_{1}(ℚ_{p}; 𝔽_{p}) mapping to the generator of K_{0}(𝔽_{p}; 𝔽_{p}) in the K-theory localization sequence for ℤ_{p}, [HM2]. It appears that the image of λ_{2}^{K} in V (1)_{2p^{2}−1}(L_{p}) is v_{2}d log v_{1} for a class d log v_{1} ∈ V (1)_{1}K(L_{p}) mapping to the generator of V (0)_{0}K(ℤ_{p}) in the expected K-theory localization sequence for ℓ_{p}. The classes λ_{1}^{K} and λ_{2}^{K} are therefore related to logarithmic differentials for poles at p and v_{1}, respectively, which partially motivates the choice of the letter ‘λ’.

2. Topological Hochschild homology

The topological Hochschild homology functor T HH(−), as well as its refined versions T HH(−)^{hS^{1}}, T HH(−)^{C_{p^{n}}}, T F (−; p) and T C(−; p), preserve p-adic equivalences. Hence we will tend to write T HH(ℤ) and T HH(ℓ) in place of T HH(ℤ_{p}) and T HH(ℓ_{p}), and similarly for the other related functors.

2.1. Homology of T HH(ℓ). The ring spectrum map ℓ → H𝔽_{p} induces an injection on mod p homology, identifying H_{∗}(ℓ; 𝔽_{p}) with the subalgebra

H_{∗}(ℓ; 𝔽_{p}) = E(τ̄_{k} | k ≥ 2) ⊗ P (ξ̄_{k} | k ≥ 1)

of the dual Steenrod algebra A_{∗}.

There is a Bökstedt spectral sequence

(2.2)  E_{∗∗}^{2} = HH_{∗}(H_{∗}(ℓ; 𝔽_{p})) ⟹ H_{∗}(T HH(ℓ); 𝔽_{p})

with

E_{∗∗}^{2} = H_{∗}(ℓ; 𝔽_{p}) ⊗ Γ(στ̄_{k} | k ≥ 2) ⊗ E(σξ̄_{k} | k ≥ 1) .

Here σx ∈ HH_{1}(−) is represented by 1 ⊗ x in degree 1 of the Hochschild complex. It corresponds to the map σ: Σℓ → T HH(ℓ) induced by the S^{1} action on T HH(ℓ) and the inclusion of 0-simplices ℓ → T HH(ℓ).

By naturality with respect to the map ℓ → H𝔽_{p}, the differentials

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

for j ≥ p, found in the Bökstedt spectral sequence for T HH(𝔽_{p}), lift to the spectral sequence (2.2) above. Hence

E_{∗∗}^{p} = H_{∗}(ℓ; 𝔽_{p}) ⊗ P_{p}(στ̄_{k} | k ≥ 2) ⊗ E(σξ̄_{1}, σξ̄_{2})

and this equals the E^{∞}-term for bidegree reasons.

In H_{∗}(T HH(ℓ); 𝔽_{p}) there are Dyer–Lashof operations acting, and (στ̄_{k})^{p} = Q^{p^{k}}(στ̄_{k}) = σ(Q^{p^{k}}(τ̄_{k})) = στ̄_{k+1} for all k ≥ 2 [St]. Thus as an algebra

(2.3)  H_{∗}(T HH(ℓ); 𝔽_{p}) ≅ H_{∗}(ℓ; 𝔽_{p}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ)

where λ_{1}, λ_{2} and μ are represented by σξ̄_{1}, σξ̄_{2} and στ̄_{2}, respectively, in the Bökstedt spectral sequence. Here |λ_{1}| = 2p − 1, |λ_{2}| = 2p^{2} − 1 and |μ| = 2p^{2}. Furthermore Q^{p}(λ_{1}) = Q^{p}(σξ̄_{1}) = σ(Q^{p}(ξ̄_{1})) = σξ̄_{2}, so we may assume that we have chosen λ_{2} = Q^{p}(λ_{1}).

2.4. V (1)-homotopy of T HH(ℓ). Let V (n) be the nth Smith–Toda complex, with homology H_{∗}(V (n); 𝔽_{p}) ≅ E(τ̄_{0}, …, τ̄_{n}). Thus V (−1) = S^{0}, V (0) is the mod p Moore spectrum, and V (1) is the cofiber of the multiplication by v_{1}-map Σ^{q}V (0) → V (0). There are cofiber sequences

S^{0} ⟶^{p} S^{0} ⟶^{i_{0}} V (0) ⟶^{j_{0}} S^{1}

and

Σ^{q}V (0) ⟶^{v_{1}} V (0) ⟶^{i_{1}} V (1) ⟶^{j_{1}} Σ^{q+1}V (0)

defining the maps labeled i_{0}, j_{0}, i_{1} and j_{1}. When p ≥ 5, V (1) is a commutative ring spectrum [Ok].

For a spectrum X the rth (partially defined) v_{1}-Bockstein homomorphism β_{1,r} is defined on the classes x ∈ V (1)_{∗}(X) with j_{1}(x) ∈ V (0)_{∗}(X) divisible by v_{1}^{r−1}. Then for y ∈ V (0)_{∗}(X) with v_{1}^{r−1} · y = j_{1}(x) let β_{1,r}(x) = i_{1}(y) ∈ V (1)_{∗}(X). So β_{1,r} decreases degrees by rq + 1.

Definition 2.5. Let r(n) = 0 for n ≤ 0, and let r(n) = p^{n} + r(n − 2) for all n ≥ 1. Thus r(2n − 1) = p^{2n−1} + ⋯ + p (n odd powers of p) and r(2n) = p^{2n} + ⋯ + p^{2} (n even powers of p). Note that (p^{2} − 1)r(2n − 1) = p^{2n+1} − p, while (p^{2} − 1)r(2n) = p^{2n+2} − p^{2}.

Proposition 2.6 (McClure–Staffeldt). There is an algebra isomorphism

V (1)_{∗}T HH(ℓ) ≅ E(λ_{1}, λ_{2}) ⊗ P (μ)

with |λ_{1}| = 2p − 1, |λ_{2}| = 2p^{2} − 1 and |μ| = 2p^{2}. There are v_{1}-Bocksteins β_{1,p}(μ) = λ_{1}, β_{1,p^{2}}(μ^{p}) = λ_{2} and generally β_{1,r(n)}(μ^{p^{n−1}}) ≠ 0 for n ≥ 1.

Proof. One proof proceeds as follows, leaving the v_{1}-Bockstein structure to the more detailed work of [MS1].

H_{∗}(T HH(ℓ); 𝔽_{p}) is an A_{∗}-comodule algebra. The coaction

ν: H_{∗}(T HH(ℓ); 𝔽_{p}) → A_{∗} ⊗ H_{∗}(T HH(ℓ); 𝔽_{p})

agrees with the coproduct ψ: A_{∗} → A_{∗} ⊗ A_{∗} when both are restricted to the subalgebra H_{∗}(ℓ; 𝔽_{p}) ⊂ A_{∗}. Furthermore ν(σx) = (1 ⊗ σ)ψ(x), so ν(λ_{1}) = 1 ⊗ λ_{1}, ν(λ_{2}) = 1 ⊗ λ_{2} and ν(μ) = 1 ⊗ μ + τ̄_{0} ⊗ λ_{2}.

There are change-of-rings isomorphisms of A_{∗}-comodule algebras

H_{∗}(V (1) ∧ T HH(ℓ); 𝔽_{p}) ≅ (A_{∗}□_{H_{∗}(ℓ;𝔽_{p})}𝔽_{p}) ⊗ (H_{∗}(ℓ; 𝔽_{p}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ))
≅ A_{∗}□_{H_{∗}(ℓ;𝔽_{p})}(H_{∗}(ℓ; 𝔽_{p}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ))
≅ A_{∗} ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ) .

The homology classes 1 ∧ λ_{1}, 1 ∧ λ_{2} and 1 ∧ μ − τ̄_{0} ∧ λ_{2} in H_{∗}(V (1) ∧ T HH(ℓ); 𝔽_{p}) are primitive, and correspond to the classes λ_{1}, λ_{2} and μ under the isomorphisms above. Hence these three homology classes are in the Hurewicz image from spherical classes λ_{1} ∈ V (1)_{2p−1}T HH(ℓ), λ_{2} ∈ V (1)_{2p^{2}−1}T HH(ℓ) and μ ∈ V (1)_{2p^{2}}T HH(ℓ), respectively. □

Proposition 2.7. The classes λ_{1}^{K} ∈ K_{2p−1}(ℓ_{p}) and λ_{2}^{K} ∈ K_{2p^{2}−1}(ℓ_{p}; 𝔽_{p}) map under the trace map to integral and mod p lifts of λ_{1} ∈ V (1)_{2p−1}T HH(ℓ) and λ_{2} ∈ V (1)_{2p^{2}−1}T HH(ℓ), respectively.

Proof. The linearized image in V (1)_{2p−1}T HH(ℤ) of λ_{1} ∈ V (1)_{2p−1}T HH(ℓ) equals the mod p and v_{1} reduction i_{1}i_{0}(e) of the class e ∈ T HH_{2p−1}(ℤ), since both classes have the same Hurewicz image 1 ∧ σξ̄_{1} in H_{2p−1}(V (1) ∧ T HH(ℤ); 𝔽_{p}). Thus the mod p and v_{1} reduction of the trace image tr(λ_{1}^{K}) equals λ_{1}.

The Hurewicz image in H_{2p^{2}−1}(T HH(ℓ); 𝔽_{p}) of tr(λ_{2}^{K}) = (P^{p})^{∗}(tr(λ_{1}^{K})) equals the image of the homology operation Q^{p} on the Hurewicz image λ_{1} = σξ̄_{1} in H_{2p−1}(T HH(ℓ); 𝔽_{p}) of tr(λ_{1}^{K}). This is Q^{p}(σξ̄_{1}) = σQ^{p}(ξ̄_{1}) = σξ̄_{2} = λ_{2}. So the mod v_{1} image in V (1)_{2p^{2}−1}T HH(ℓ) of tr(λ_{2}^{K}) equals λ_{2}, since both classes have the same Hurewicz image 1 ∧ λ_{2} in H_{2p^{2}−1}(V (1) ∧ T HH(ℓ); 𝔽_{p}). □

3. Topological cyclotomy

We now review some terminology and notation concerning topological cyclic homology and the cyclotomic trace map.

3.1. Frobenius, restriction, Verschiebung. As already indicated, T HH(ℓ) is an S^{1}-equivariant spectrum. Let C_{p^{n}} ⊂ S^{1} be the cyclic group of order p^{n}. The Frobenius maps F: T HH(ℓ)^{C_{p^{n}}} → T HH(ℓ)^{C_{p^{n−1}}} are the usual inclusions of fixed point spectra that forget part of the invariance. Their homotopy limit defines

T F (ℓ; p) = holim_{n,F} T HH(ℓ)^{C_{p^{n}}} .

There are also restriction maps R: T HH(ℓ)^{C_{p^{n}}} → T HH(ℓ)^{C_{p^{n−1}}} , defined using the cyclotomic structure of T HH(ℓ), cf. [HM1]. They commute with the Frobenius maps, and thus induce a self map R: T F (ℓ; p) → T F (ℓ; p). Its homotopy equalizer with the identity map defines the topological cyclic homology of ℓ, which was introduced in [BHM]:

\xymatrix{T C(ℓ; p) \ar[r]^{π} & T F (ℓ; p) \ar[r]^{R} \ar[r]_{1} & T F (ℓ; p) .}

Hence there is a cofiber sequence

Σ^{−1}T F (ℓ; p) ⟶^{∂} T C(ℓ; p) ⟶^{π} T F (ℓ; p) ⟶^{1−R} T F (ℓ; p) ,

which we shall use to compute V (1)_{∗}T C(ℓ; p). There are also Verschiebung maps V : T HH(ℓ)^{C_{p^{n−1}}} → T HH(ℓ)^{C_{p^{n}}}, defined up to homotopy in terms of the S^{1}- equivariant transfer.

3.2. The cyclotomic trace map. The Bökstedt trace map admits lifts

tr_{n}: K(ℓ_{p}) → T HH(ℓ)^{C_{p^{n}}}

for all n ≥ 0, with tr = tr_{0}, which commute with the Frobenius maps and homotopy commute with the restriction maps up to preferred homotopy. Hence the limiting map tr_{F} : K(ℓ_{p}) → T F (ℓ; p) homotopy equalizes R and the identity map, and the resulting lift

trc: K(ℓ_{p}) → T C(ℓ; p)

is the Bökstedt–Hsiang–Madsen cyclotomic trace map [BHM].

3.3. The norm–restriction sequences. For each n ≥ 1 there is a homotopy commutative diagram

(3.4)  [figure: K(ℓ_{p}); tr_{n}; tr_{n−1}; T HH(ℓ)_{hC_{p^{n}}}; N; T HH(ℓ)^{C_{p^{n}}}; R; Γ_{n}; T HH(ℓ)^{C_{p^{n−1}}}; Γ̂_{n}; T HH(ℓ)_{hC_{p^{n}}}; N^{h}; T HH(ℓ)^{hC_{p^{n}}}; R^{h}; ℍ̂(C_{p^{n}}, T HH(ℓ)) .]

The lower part is the map of cofiber sequences that arises by smashing the S^{1}- equivariant cofiber sequence ES_{+}^{1} → S^{0} → ẼS^{1} with the S^{1}-equivariant map T HH(ℓ) → F (ES_{+}^{1}, T HH(ℓ)) and taking C_{p^{n}} fixed point spectra. For closed subgroups G ⊆ S^{1} recall that T HH(ℓ)^{hG} = F (ES_{+}^{1}, T HH(ℓ))^{G} is the G homotopy fixed point spectrum of T HH(ℓ), and ℍ̂(G, T HH(ℓ)) = [ẼS^{1} ∧ F (ES_{+}^{1}, T HH(ℓ)]^{G} is the G Tate construction on T HH(ℓ). The remaining terms of the diagram are then identified by the homotopy equivalences

T HH(ℓ)_{hC_{p^{n}}} ≃ [ES_{+}^{1} ∧ T HH(ℓ)]^{C_{p^{n}}} ≃ [ES_{+}^{1} ∧ F (ES_{+}^{1}, T HH(ℓ))]^{C_{p^{n}}}

and

T HH(ℓ)^{C_{p^{n−1}}} ≃ [ẼS^{1} ∧ T HH(ℓ)]^{C_{p^{n}}} .

We call N, R, N^{h} and R^{h} the norm, restriction, homotopy norm and homotopy restriction maps, respectively. We call Γ_{n} and Γ̂_{n} the canonical maps. The middle and lower cofiber sequences are the norm–restriction and homotopy norm–restriction sequences, respectively.

By passage to homotopy limits over Frobenius maps, there is also a limiting diagram

(3.5)  [figure: K(ℓ_{p}); tr_{F}; tr_{F}; ΣT HH(ℓ)_{hS^{1}}; N; T F (ℓ; p); R; Γ; T F (ℓ; p); Γ̂; ΣT HH(ℓ)_{hS^{1}}; N^{h}; T HH(ℓ)^{hS^{1}}; R^{h}; ℍ̂(S^{1}, T HH(ℓ)) .]

Implicit here are the p-adic homotopy equivalences

ΣT HH(ℓ)_{hS^{1}} ≃ holim_{n,F} T HH(ℓ)_{hC_{p^{n}}}
T HH(ℓ)^{hS^{1}} ≃ holim_{n,F} T HH(ℓ)^{hC_{p^{n}}}
ℍ̂(S^{1}, T HH(ℓ)) ≃ holim_{n,F} ℍ̂(C_{p^{n}}, T HH(ℓ)) .

4. Circle homotopy fixed points

4.1. The circle trace map. The circle trace map

tr_{S^{1}} = Γ ∘ tr_{F} : K(ℓ_{p}) → T HH(ℓ)^{hS^{1}} = F (ES_{+}^{1}, T HH(ℓ))^{S^{1}}

is a preferred lift of the trace map tr: K(ℓ_{p}) → T HH(ℓ). We take S^{∞} as our model for ES^{1}. Let

T^{n} = F (S^{∞}/S^{2n−1}, T HH(ℓ))^{S^{1}}

for n ≥ 0, so that there is a descending filtration {T^{n}}_{n} on T^{0} = T HH(ℓ)^{hS^{1}}, with layers T^{n}/T^{n+1} ≅ F (S^{2n+1}/S^{2n−1}, T HH(ℓ))^{S^{1}} ≅ Σ^{−2n}T HH(ℓ).

4.2. The homology spectral sequence. Placing T^{n} in filtration s = −2n and applying homology, we obtain a (not necessarily convergent) homology spectral sequence

(4.3)  E_{s,t}^{2} = H^{−s}(S^{1}; H_{t}(T HH(ℓ); 𝔽_{p})) ⟹ H_{s+t}(T HH(ℓ)^{hS^{1}}; 𝔽_{p})

with

E_{∗∗}^{2} = P (t) ⊗ H_{∗}(ℓ; 𝔽_{p}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ) .

Here t has bidegree (−2, 0) while the other generators are located on the vertical axis. (No confusion should arise from the double usage of t as a polynomial cohomology class and the vertical degree in this or other spectral sequences.)

Proposition 4.4. There are differentials d^{2}(ξ̄_{1}) = tλ_{1}, d^{2}(ξ̄_{2}) = tλ_{2}, d^{2}(τ̄_{2}) = tμ and d^{2p}(ξ̄_{1}^{p}) = t^{p}λ_{2} in the spectral sequence (4.3).

Proof. The d^{2}-differential

d_{0,t}^{2} : E_{0,t}^{2} ≅ H_{t}(T HH(ℓ); 𝔽_{p}){1} ⟶ E_{−2,t+1}^{2} ≅ H_{t+1}(T HH(ℓ); 𝔽_{p}){t}

is adjoint to the S^{1}-action on T HH(ℓ), hence restricts to σ on H_{t}(ℓ; 𝔽_{p}). Thus d^{2}(ξ̄_{i}) = tσξ̄_{i} = tλ_{i} for i = 1 and 2, and d^{2}(τ̄_{2}) = tστ̄_{2} = tμ.

Write q = 2p−2 and let x ∈ C_{q}(T^{0}, T^{1}; 𝔽_{p}) be a chain representing the differential d^{2}(ξ̄_{1}) = tλ_{1}, i.e., x maps to a cycle representing ξ̄_{1} in H_{q}(T HH(ℓ); 𝔽_{p}) = E_{0,2p−2}^{2}, and has boundary dx ∈ C_{q−1}(T^{1}; 𝔽_{p}) mapping to a cycle representing tλ_{1} in H_{q−1}(Σ^{−2}T HH(ℓ); 𝔽_{p}) = E_{−2,2p−1}^{2}. Let ξ: D_{p}T^{0} → T^{0} be the pth H_{∞} structure map for T^{0} = T HH(ℓ)^{hS^{1}}, and form the chain ξ_{∗}(e_{0} ⊗ x^{⊗p}) ∈ C_{pq}(T^{0}, T^{p}; 𝔽_{p}). It maps to a cycle representing ξ̄_{1}^{p} in H_{pq}(T HH(ℓ); 𝔽_{p}) = E_{0,2p^{2}−2p}^{2}, and has boundary ξ_{∗}(e_{0} ⊗ d(x^{⊗p})) ∈ C_{pq−1}(T^{p}; 𝔽_{p}). By a chain level calculation [Br, 3.4] in the extended pth power of the pair (D^{2p−2}, S^{2p−3}), this boundary is homologous to a unit multiple of ξ_{∗}(e_{p−1} ⊗ (dx)^{⊗p}), representing Q^{p−1}(tλ_{1}) = t^{p}Q^{p}(λ_{1}) = t^{p}λ_{2} in H_{pq−1}(Σ^{−2p}T HH(ℓ); 𝔽_{p}) = E_{−2p,2p^{2}−1}^{2}. Hence there is a nontrivial differential d^{2p}(ξ̄_{1}^{p}) = t^{p}λ_{2} in the spectral sequence above. □

4.5. The V (1)-homotopy spectral sequence. Applying V (1)-homotopy to the filtration {T^{n}}_{n}, in place of homology, we obtain a conditionally convergent V (1)- homotopy spectral sequence

(4.6)  E_{s,t}^{2}(S^{1}) = H^{−s}(S^{1}; V (1)_{t}T HH(ℓ)) ⟹ V (1)_{s+t}T HH(ℓ)^{hS^{1}}

with

E_{∗∗}^{2}(S^{1}) = P (t) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ) .

Again t has bidegree (−2, 0) while the other generators are located on the vertical axis.

Definition 4.7. Let

α_{1} ∈ π_{2p−3}(S^{0}), β′_{1} ∈ π_{2p^{2}−2p−1}V (0) and v_{2} ∈ π_{2p^{2}−2}V (1)

be the classes represented in their respective Adams spectral sequences by the cobar cycles h_{10} = [ξ̄_{1}], h_{11} = [ξ̄_{1}^{p}] and [τ̄_{2}]. So j_{1}(v_{2}) = β′_{1} and j_{0}(β′_{1}) = β_{1} ∈ π_{2p^{2}−2p−2}(S^{0}).

Consider the unit map S^{0} → K(ℓ_{p}) → T HH(ℓ)^{hS^{1}}, which is well defined after p-adic completion.

Proposition 4.8. The classes i_{1}i_{0}(α_{1}) ∈ π_{2p−3}V (1), i_{1}(β′_{1}) ∈ π_{2p^{2}−2p−1}V (1) and v_{2} ∈ π_{2p^{2}−2}V (1) map under the unit map V (1)_{∗}S^{0} → V (1)_{∗}T HH(ℓ)^{hS^{1}} to classes represented in E^{∞}(S^{1}) by tλ_{1}, t^{p}λ_{2} and tμ, respectively.

Proof. Consider first the filtration subquotient T^{0}/T^{2} = F (S_{+}^{3}, T HH(ℓ))^{S^{1}}. The unit map V (1) → V (1) ∧ (T^{0}/T^{2}) induces a map of Adams spectral sequences, taking the permanent cycles [ξ̄_{1}] and [τ̄_{2}] in the source Adams spectral sequence to infinite cycles with the same cobar names in the target Adams spectral sequence. These are not boundaries in the cobar complex for the A_{∗}-comodule H_{∗}(T^{0}/T^{2}; 𝔽_{p}), because of the differentials d^{2}(ξ̄_{1}) = tλ_{1} and d^{2}(τ̄_{2}) = tμ that are present in the two-column spectral sequence converging to H_{∗}(T^{0}/T^{2}; 𝔽_{p}). In detail, H_{2p−2}(T^{0}/T^{2}; 𝔽_{p}) = 0 and H_{2p^{2}−1}(T^{0}/T^{2}; 𝔽_{p}) is spanned by the primitives λ_{2} and λ_{1}ξ̄_{1}^{p}.

Thus [ξ̄_{1}] and [τ̄_{2}] are nonzero infinite cycles in the target Adams E_{2}-term. They have Adams filtration one, hence cannot be boundaries. Thus they are permanent cycles, and the classes i_{1}i_{0}(α_{1}) and v_{2} have nonzero images under the composite V (1)_{∗} → V (1)_{∗}(T^{0}) → V (1)_{∗}(T^{0}/T^{2}). Thus they are also detected in V (1)_{∗}(T^{0}), in filtration s ≥ −2. For bidegree reasons the only possibility is that i_{1}i_{0}(α_{1}) is detected in the V (1)-homotopy spectral sequence E^{∞}(S^{1}) as tλ_{1}, and v_{2} is detected as tμ.

Next consider the filtration subquotient T^{0}/T^{p+1} = F (S_{+}^{2p+1}, T HH(ℓ))^{S^{1}}. The unit map V (1) → V (1) ∧ (T^{0}/T^{p+1}) again induces a map of Adams spectral sequences, taking the permanent cycle [ξ̄_{1}^{p}] in the source Adams spectral sequence to an infinite cycle with the same name. Again this is not a boundary in the cobar complex for the A_{∗}-comodule H_{∗}(T^{0}/T^{p+1}; 𝔽_{p}), because of the differential d^{2p}(ξ̄_{1}^{p}) = t^{p}λ_{2} that is present in the (p + 1) column spectral sequence converging to H_{∗}(T^{0}/T^{p+1}; 𝔽_{p}).

Thus [ξ̄_{1}^{p}] is a nonzero infinite cycle in the target Adams E_{2}-term, of Adams filtration one. Hence the class i_{1}(β′_{1}) has a nonzero image under the composite V (1)_{∗} → V (1)_{∗}(T^{0}) → V (1)_{∗}(T^{0}/T^{p+1}). Thus it is also detected in V (1)_{∗}(T^{0}) in filtration s ≥ −2p. Again for bidegree reasons the only possibility is that i_{1}(β′_{1}) is detected in the V (1)-homotopy spectral sequence E^{∞}(S^{1}) as t^{p}λ_{2}. □

5. The homotopy limit property

5.1. Homotopy fixed point and Tate spectral sequences. For closed subgroups G ⊆ S^{1} we will consider the (second quadrant) G homotopy fixed point spectral sequence

E_{s,t}^{2}(G) = H^{−s}(G, V (1)_{t}T HH(ℓ))
⟹ V (1)_{s+t}T HH(ℓ)^{hG} .

We also consider the (upper half plane) G Tate spectral sequence

Ê_{s,t}^{2}(G) = Ĥ^{−s}(G, V (1)_{t}T HH(ℓ))
⟹ V (1)_{s+t}ℍ̂(G, T HH(ℓ)) .

When G = S^{1} we have

E_{∗∗}^{2}(S^{1}) = E(λ_{1}, λ_{2}) ⊗ P (t, μ)

since H^{∗}(S^{1}; 𝔽_{p}) = P (t), and

Ê_{∗∗}^{2}(S^{1}) = E(λ_{1}, λ_{2}) ⊗ P (t, t^{−1}, μ)

since Ĥ^{∗}(S^{1}; 𝔽_{p}) = P (t, t^{−1}). When G = C_{p^{n}} we have

E_{∗∗}^{2}(C_{p^{n}}) = E(u_{n}, λ_{1}, λ_{2}) ⊗ P (t, μ)

since H^{∗}(C_{p^{n}}; 𝔽_{p}) = E(u_{n}) ⊗ P (t), while

Ê_{∗∗}^{2}(C_{p^{n}}) = E(u_{n}, λ_{1}, λ_{2}) ⊗ P (t, t^{−1}, μ)

since Ĥ^{∗}(C_{p^{n}}; 𝔽_{p}) = E(u_{n}) ⊗ P (t, t^{−1}).

The homotopy restriction map R^{h} induces a map of spectral sequences

E^{∗}(R^{h}): E^{∗}(G) → Ê^{∗}(G) ,

which on E^{2}-terms identifies E^{2}(G) with the restriction of Ê^{2}(G) to the second quadrant.

The Frobenius and Verschiebung maps F and V are compatible under Γ_{n} and Γ_{n−1} with homotopy Frobenius and Verschiebung maps F^{h} and V^{h} that induce maps of homotopy fixed point spectral sequences

E^{∗}(F^{h}): E^{∗}(C_{p^{n}}) → E^{∗}(C_{p^{n−1}})

and

E^{∗}(V^{h}): E^{∗}(C_{p^{n−1}}) → E^{∗}(C_{p^{n}}) .

Here E^{2}(F^{h}) maps the even columns isomorphically and the odd columns trivially. On the other hand, E^{2}(V^{h}) maps the odd columns isomorphically and the even columns trivially. This pattern persists to higher E^{r}-terms, until a differential of odd length appears in either spectral sequence. Thus the spectral sequences E^{∗}(C_{p^{n}}) and E^{∗}(C_{p^{n−1}}) are abstractly isomorphic up to and including the E^{r}- term, where r is the length of the first odd differential in either spectral sequence. The same remarks apply for the Tate spectral sequences.

5.2. Input for Tsalidis’ theorem.

Definition 5.3. A map A_{∗} → B_{∗} of graded groups is k-coconnected if it is an isomorphism in all dimensions > k and injective in dimension k.

Theorem 5.4. The canonical map

Γ̂_{1} : T HH(ℓ) → ℍ̂(C_{p}, T HH(ℓ))

induces a (2p − 2)-coconnected map on V (1)-homotopy.

Proof. Consider diagram (3.4) in the case n = 1. The classes i_{1}i_{0}(α_{1}), i_{1}(β′_{1}) and v_{2} in V (1)_{∗} map through V (1)_{∗}K(ℓ_{p}) and Γ_{1} ∘ tr_{1} to classes in V (1)_{∗}T HH(ℓ)^{hC_{p}} that are detected by tλ_{1}, t^{p}λ_{2} and tμ in E^{∞}(C_{p}), respectively. Continuing by R^{h} to V (1)_{∗}ℍ̂(C_{p}, T HH(ℓ)) these classes factor through V (1)_{∗}T HH(ℓ), where they pass through zero groups. Hence the images of tλ_{1}, t^{p}λ_{2} and tμ in Ê^{∞}(C_{p}) must be zero, i.e., these infinite cycles in Ê^{2}(C_{p}) are boundaries. For dimension reasons the only possibilities are

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 .

The classes i_{1}i_{0}(λ_{1}^{K}) and i_{1}(λ_{2}^{K}) in V (1)_{∗}K(ℓ_{p}) map by Γ_{1} ∘ tr_{1} to classes in V (1)_{∗}T HH(ℓ)^{hC_{p}} that have Frobenius images λ_{1} and λ_{2} in V (1)_{∗}T HH(ℓ), and hence survive as permanent cycles in E_{0,∗}^{∞}(C_{p}). Thus their images λ_{1} and λ_{2} in Ê^{∗}(C_{p}) are infinite cycles.

Hence the various E^{r}-terms of the C_{p} Tate spectral sequence are:

Ê^{2}(C_{p}) = E(u_{1}, λ_{1}, λ_{2}) ⊗ P (t, t^{−1}, tμ)
Ê^{2p+1}(C_{p}) = E(u_{1}, λ_{1}, λ_{2}) ⊗ P (t^{p}, t^{−p}, tμ)
Ê^{2p^{2}+1}(C_{p}) = E(u_{1}, λ_{1}, λ_{2}) ⊗ P (t^{p^{2}}, t^{−p^{2}}, tμ)
Ê^{2p^{2}+2}(C_{p}) = E(λ_{1}, λ_{2}) ⊗ P (t^{p^{2}}, t^{−p^{2}}) .

For bidegree reasons there are no further differentials, so Ê^{2p^{2}+2}(C_{p}) = Ê^{∞}(C_{p}) and the classes λ_{1}, λ_{2} and t^{±p^{2}} are permanent cycles.

The map Γ̂_{1} : T HH(ℓ) → ℍ̂(C_{p}, T HH(ℓ)) induces on V (1)-homotopy the homomorphism

E(λ_{1}, λ_{2}) ⊗ P (μ) ⟶ E(λ_{1}, λ_{2}) ⊗ P (t^{p^{2}}, t^{−p^{2}})

that maps λ_{1} ↦ λ_{1}, λ_{2} ↦ λ_{2} and μ ↦ t^{−p^{2}}. For the classes i_{1}i_{0}(λ_{1}^{K}) and i_{1}(λ_{2}^{K}) in V (1)_{∗}K(ℓ_{p}) map by tr to λ_{1} and λ_{2} in V (1)_{∗}T HH(ℓ), and by R^{h} ∘ Γ_{1} ∘ tr_{1} to the classes in V (1)_{∗}ℍ̂(C_{p}, T HH(ℓ)) represented by λ_{1} and λ_{2}. The class μ in V (1)_{∗}T HH(ℓ) must have nonzero image in V (1)_{∗}ℍ̂(C_{p}, T HH(ℓ)), since its pth v_{1}-Bockstein β_{1,p}(μ) = λ_{1} has nonzero image there. Thus μ maps to the class represented by t^{−p^{2}}, up to a unit multiple which we ignore. So V (1)_{∗}Γ̂_{1} is an isomorphism in dimensions greater than |λ_{1}λ_{2}t^{p^{2}}| = 2p − 2, and is injective in dimension 2p − 2. □

5.5. The homotopy limit property.

Theorem 5.6. The canonical maps

Γ_{n}: T HH(ℓ)^{C_{p^{n}}} → T HH(ℓ)^{hC_{p^{n}}}
Γ̂_{n}: T HH(ℓ)^{C_{p^{n−1}}} → ℍ̂(C_{p^{n}}, T HH(ℓ))

and

Γ: T F (ℓ; p) → T HH(ℓ)^{hS^{1}}
Γ̂: T F (ℓ; p) → ℍ̂(S^{1}, T HH(ℓ))

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

Proof. The claims for Γ_{n} and Γ̂_{n} follows from 5.4 and a theorem of Tsalidis [Ts]. The claims for Γ and Γ̂ follow by passage to homotopy limits, using the p-adic homotopy equivalence T HH(ℓ)^{hS^{1}} ≃ holim_{n,F} T HH(ℓ)^{hC_{p^{n}}} and its analogue for the Tate constructions. □

6. Higher fixed points

Let [k] = 1 when k is odd, and [k] = 2 when k is even. Let λ′_{[k]} = λ_{[k+1]}, so that {λ_{[k]}, λ′_{[k]}} = {λ_{1}, λ_{2}} for all k. We write v_{p}(k) for the p-valuation of k, i.e., the exponent of the greatest power of p that divides k. By convention, v_{p}(0) = +∞. Recall the integers r(n) from 2.5.

Theorem 6.1. In the C_{p^{n}} Tate spectral sequence Ê^{∗}(C_{p^{n}}) there are differentials

d^{2r(k)}(t^{p^{k−1}−p^{k}}) = λ_{[k]}t^{p^{k−1}}(tμ)^{r(k−2)}

for all 1 ≤ k ≤ 2n, and

d^{2r(2n)+1}(u_{n}t^{−p^{2n}}) = (tμ)^{r(2n−2)+1} .

The classes λ_{1}, λ_{2} and tμ are infinite cycles.

We shall prove this by induction on n, the case n = 1 being settled in the previous section. Hence we assume the theorem holds for one n ≥ 1 and we will establish its assertions for n + 1.

The terms of the Tate spectral sequence are

Ê^{2r(m)+1}(C_{p^{n}}) = E(u_{n}, λ_{1}, λ_{2}) ⊗ P (t^{p^{m}}, t^{−p^{m}}, tμ)

⊕ ⨁_{k=3}^{m} E(u_{n}, λ′_{[k]}) ⊗ P_{r(k−2)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}t^{i} | v_{p}(i) = k − 1}

for 1 ≤ m ≤ 2n. Next

Ê^{2r(2n)+2}(C_{p^{n}}) = E(λ_{1}, λ_{2}) ⊗ P_{r(2n−2)+1}(tμ) ⊗ P (t^{p^{2n}}, t^{−p^{2n}})

⊕ ⨁_{k=3}^{2n} E(u_{n}, λ′_{[k]}) ⊗ P_{r(k−2)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}t^{i} | v_{p}(i) = k − 1} .

For bidegree reasons the remaining differentials are zero, so Ê^{2r(2n)+2}(C_{p^{n}}) = Ê^{∞}(C_{p^{n}}), and the classes t^{±p^{2n}} are permanent cycles.

Proposition 6.2. The associated graded of V (1)_{∗}ℍ̂(C_{p^{n}}, T HH(ℓ)) is

Ê^{∞}(C_{p^{n}}) = E(λ_{1}, λ_{2}) ⊗ P_{r(2n−2)+1}(tμ) ⊗ P (t^{p^{2n}}, t^{−p^{2n}})

⊕ ⨁_{k=3}^{2n} E(u_{n}, λ′_{[k]}) ⊗ P_{r(k−2)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}t^{i} | v_{p}(i) = k − 1} .

Comparing E^{∗}(C_{p^{n}}) with Ê^{∗}(C_{p^{n}}) via the homotopy restriction map R^{h}, we obtain the following:

Proposition 6.3. In the C_{p^{n}} homotopy fixed point spectral sequence E^{∗}(C_{p^{n}}) there are differentials

d^{2r(k)}(t^{p^{k−1}}) = λ_{[k]}t^{p^{k}+p^{k−1}}(tμ)^{r(k−2)}

for all 1 ≤ k ≤ 2n, and

d^{2r(2n)+1}(u_{n}) = t^{p^{2n}}(tμ)^{r(2n−2)+1} .

The classes λ_{1}, λ_{2} and tμ are infinite cycles.

We also consider the μ-inverted spectral sequences μ^{−1}E^{∗}(G) for G closed in S^{1}, obtained by inverting tμ in Ê^{∗}(G) and restricting to the left half plane. The E^{2}- term μ^{−1}E^{2}(G) is obtained from E^{2}(G) by inverting μ. At each term, the natural map E^{∗}(G) → μ^{−1}E^{∗}(G) is an isomorphism in total degrees greater than 2p − 2, and an injection in total degree 2p − 2.

Proposition 6.4. In the μ-inverted spectral sequence μ^{−1}E^{∗}(C_{p^{n}}) there are differentials

d^{2r(k)}(μ^{p^{k}−p^{k−1}}) = λ_{[k]}(tμ)^{r(k)}μ^{−p^{k−1}}

for all 1 ≤ k ≤ 2n, and

d^{2r(2n)+1}(u_{n}μ^{p^{2n}}) = (tμ)^{r(2n)+1} .

The classes λ_{1}, λ_{2} and tμ are infinite cycles.

The terms of the μ-inverted spectral sequence are

μ^{−1}E^{2r(m)+1}(C_{p^{n}}) = E(u_{n}, λ_{1}, λ_{2}) ⊗ P (μ^{p^{m}}, μ^{−p^{m}}, tμ)

⊕ ⨁_{k=1}^{m} E(u_{n}, λ′_{[k]}) ⊗ P_{r(k)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}μ^{j} | v_{p}(j) = k − 1}

for 1 ≤ m ≤ 2n. Next

μ^{−1}E^{2r(2n)+2}(C_{p^{n}}) = E(λ_{1}, λ_{2}) ⊗ P_{r(2n)+1}(tμ) ⊗ P (μ^{p^{2n}}, μ^{−p^{2n}})

⊕ ⨁_{k=1}^{2n} E(u_{n}, λ′_{[k]}) ⊗ P_{r(k)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}μ^{j} | v_{p}(j) = k − 1} .

Again μ^{−1}E^{2r(2n)+2}(C_{p^{n}}) = μ^{−1}E^{∞}(C_{p^{n}}) for bidegree reasons, and the classes μ^{±p^{2n}} are permanent cycles.

Proposition 6.5. The associated graded E^{∞}(C_{p^{n}}) of V (1)_{∗}T HH(ℓ)^{hC_{p^{n}}} maps by a (2p − 2)-coconnected map to

μ^{−1}E^{∞}(C_{p^{n}}) = E(λ_{1}, λ_{2}) ⊗ P_{r(2n)+1}(tμ) ⊗ P (μ^{p^{2n}}, μ^{−p^{2n}})

⊕ ⨁_{k=1}^{2n} E(u_{n}, λ′_{[k]}) ⊗ P_{r(k)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}μ^{j} | v_{p}(j) = k − 1} .

Proof of Theorem 6.1. By our inductive hypothesis, the abutment μ^{−1}E^{∞}(C_{p^{n}}) contains summands

P_{r(2n−1)}(tμ){λ_{1}μ^{p^{2n−2}}}, P_{r(2n)}(tμ){λ_{2}μ^{p^{2n−1}}} and P_{r(2n)+1}(tμ){μ^{p^{2n}}}

representing elements in V (1)_{∗}T HH(ℓ)^{C_{p^{n}}}. By inspection there are no permanent cycles in the same total degree and of lower s-filtration in μ^{−1}E^{∞}(C_{p^{n}}) than (tμ)^{r(2n−1)} ·λ_{1}μ^{p^{2n−2}}, (tμ)^{r(2n)} ·λ_{2}μ^{p^{2n−1}} and (tμ)^{r(2n)+1} ·μ^{p^{2n}}, respectively. So the three homotopy classes represented by λ_{1}μ^{p^{2n−2}}, λ_{2}μ^{p^{2n−1}} and μ^{p^{2n}} are v_{2}-torsion classes of orders precisely r(2n − 1), r(2n) and r(2n) + 1, respectively.

Consider the commutative diagram

\xymatrix{T HH(ℓ)^{hC_{p^{n}}} \ar[d]^{F^{n}} & T HH(ℓ)^{C_{p^{n}}} \ar[d]^{F^{n}} \ar[l]_{Γ_{n}} \ar[r]^{Γ̂_{n+1}} & ℍ̂(C_{p^{n+1}}, T HH(ℓ)) \ar[d]^{F^{n}} \\ T HH(ℓ) & T HH(ℓ) \ar[l]_{Γ_{0}}^{=} \ar[r]^{Γ̂_{1}} & ℍ̂(C_{p}, T HH(ℓ)) .}

Here F^{n} is the n-fold Frobenius map forgetting C_{p^{n}}-invariance. The right hand diagram commutes because Γ̂_{n+1} is constructed as the C_{p^{n}}-invariant part of an S^{1}-equivariant model for Γ̂_{1}.

The above three homotopy classes in V (1)_{∗}T HH(ℓ)^{C_{p^{n}}} map by the middle F^{n} to homotopy classes in V (1)_{∗}T HH(ℓ) with the same names, and by Γ̂_{1} to homotopy classes in V (1)_{∗}ℍ̂(C_{p}, T HH(ℓ)) represented by λ_{1}t^{−p^{2n}}, λ_{2}t^{−p^{2n+1}} and t^{−p^{2n+2}} in Ê^{∞}(C_{p}), respectively. Hence they map by Γ̂_{n+1} to permanent cycles in Ê^{∗}(C_{p^{n+1}}) with these images under the right hand F^{n}.

By comparison over homotopy Frobenius and Verschiebung maps, there are isomorphisms Ê^{r}(C_{p^{n}}) ≅ Ê^{r}(C_{p^{n+1}}) for all r ≤ 2r(2n) + 1, taking u_{n} to u_{n+1}. This determines the d^{r}-differentials and E^{r}-terms of Ê^{∗}(C_{p^{n+1}}) up to and including the E^{r}-term with r = 2r(2n) + 1:

Ê^{2r(2n)+1}(C_{p^{n+1}}) = E(u_{n+1}, λ_{1}, λ_{2}) ⊗ P (t^{p^{2n}}, t^{−p^{2n}}, tμ)

⊕ ⨁_{k=3}^{2n} E(u_{n+1}, λ′_{[k]}) ⊗ P_{r(k−2)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}t^{i} | v_{p}(i) = k − 1} .

By inspection there are no permanent cycles in the same total degree and of higher s-filtration in Ê^{∗}(C_{p^{n+1}}) than λ_{1}t^{−p^{2n}}, λ_{2}t^{−p^{2n+1}} and t^{−p^{2n+2}}, respectively. So also the equivalence Γ̂_{n+1}Γ_{n}^{−1} takes the homotopy classes represented by λ_{1}μ^{p^{2n−2}}, λ_{2}μ^{p^{2n−1}} and μ^{p^{2n}} to homotopy classes represented by λ_{1}t^{−p^{2n}}, λ_{2}t^{−p^{2n+1}} and t^{−p^{2n+2}}, respectively.

Since Γ̂_{n+1}Γ_{n}^{−1} induces an isomorphism on V (1)-homotopy in dimensions > (2p− 2), it preserves the v_{2}-torsion order of these classes. Thus the infinite cycles

(tμ)^{r(2n−1)} · λ_{1}t^{−p^{2n}}, (tμ)^{r(2n)} · λ_{2}t^{−p^{2n+1}} and (tμ)^{r(2n)+1} · t^{−p^{2n+2}}

are all boundaries in Ê^{∗}(C_{p^{n+1}}). These are all tμ-periodic classes in Ê^{r}(C_{p^{n+1}}) for r = 2r(2n) + 1, hence cannot be hit by differentials on the tμ-torsion classes in this E^{r}-term.

This leaves the tμ-periodic part E(u_{n+1}, λ_{1}, λ_{2}) ⊗ P (t^{p^{2n}}, t^{−p^{2n}}, tμ), where all the generators above the horizontal axis are infinite cycles. Hence the differentials hitting (tμ)^{r(2n−1)} · λ_{1}t^{−p^{2n}}, (tμ)^{r(2n)} · λ_{2}t^{−p^{2n+1}} and (tμ)^{r(2n)+1} · t^{−p^{2n+2}} must originate on the horizontal axis, and the only possibilities are

d^{2r(2n+1)}(t^{−p^{2n}−p^{2n+1}}) = (tμ)^{r(2n−1)} · λ_{1}t^{−p^{2n}}
d^{2r(2n+2)}(t^{−p^{2n+1}−p^{2n+2}}) = (tμ)^{r(2n)} · λ_{2}t^{−p^{2n+1}}
d^{2r(2n+2)+1}(u_{n+1}t^{−2p^{2n+2}}) = (tμ)^{r(2n)+1} · t^{−p^{2n+2}} .

The algebra structure on Ê^{∗}(C_{p^{n+1}}) allows us to rewrite these differentials as the remaining differentials asserted by case n + 1 of 6.1. □

Passing to the limit over the Frobenius maps, we obtain:

Theorem 6.6. The associated graded of V (1)_{∗}ℍ̂(S^{1}, T HH(ℓ)) is

Ê^{∞}(S^{1}) = E(λ_{1}, λ_{2}) ⊗ P (tμ)

⊕ ⨁_{k≥3} E(λ′_{[k]}) ⊗ P_{r(k−2)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}t^{i} | v_{p}(i) = k − 1} .

Theorem 6.7. The associated graded E^{∞}(S^{1}) of V (1)_{∗}T HH(ℓ)^{hS^{1}} maps by a (2p − 2)-coconnected map to

μ^{−1}E^{∞}(S^{1}) = E(λ_{1}, λ_{2}) ⊗ P (tμ)

⊕ ⨁_{k≥1} E(λ′_{[k]}) ⊗ P_{r(k)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}μ^{j} | v_{p}(j) = k − 1} .

Each of these E^{∞} terms compute V (1)_{∗}T F (ℓ; p) in dimensions > (2p − 2), by way of the (2p − 2)-coconnected maps Γ and Γ̂, respectively.

7. The restriction map

We now evaluate the homomorphism

R_{∗}: V (1)_{∗}T F (ℓ; p) → V (1)_{∗}T F (ℓ; p)

induced on V (1)-homotopy by the restriction map R, in dimensions > (2p − 2). The source and target are both identified with V (1)_{∗}T HH(ℓ)^{hS^{1}} via Γ_{∗}. Then R_{∗} is identified with the composite homomorphism (ΓΓ̂^{−1})_{∗} ∘ R_{∗}^{h}, where R^{h} is the homotopy restriction map. The latter induces a map of spectral sequences

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

where the E^{∞} terms are given in 6.6 and 6.7.

Proposition 7.1. In dimensions > (2p − 2) the homomorphism E^{∞}(R^{h}) maps: (a) E(λ_{1}, λ_{2})⊗P (tμ) in E^{∞}(S^{1}) isomorphically to E(λ_{1}, λ_{2})⊗P (tμ) in Ê^{∞}(S^{1}).

(b) E(λ′_{[k]})⊗P_{r(k)}(tμ)⊗𝔽_{p}{λ_{[k]}μ^{−ep^{k−1}}} in E^{∞}(S^{1}) onto E(λ′_{[k]})⊗P_{r(k−2)}(tμ)⊗

𝔽_{p}{λ_{[k]}t^{ep^{k−1}}} in Ê^{∞}(S^{1}), for k ≥ 3 and 0 < e < p.

(c) the remaining terms in E^{∞}(S^{1}) to zero.

Proof. Case (a) is clear. For (b) and (c) note that E^{∞}(R^{h}) maps the term

E(λ′_{[k]}) ⊗ P_{r(k)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}μ^{−ep^{k−1}}}

in E^{∞}(S^{1}) to the term

E(λ′_{[k]}) ⊗ P_{r(k−2)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}t^{ep^{k−1}}}

in Ê^{∞}(S^{1}). Here e is prime to p. For e > p the source and target are in negative dimensions, while for e < 0 the source and target are concentrated in disjoint dimensions. The cases 0 < e < p remain, when the map is a surjection since r(k) − ep^{k−1} > r(k − 2). □

This identifies the image of R_{∗}^{h}, by the following lemma from [BM1, §2].

Lemma 7.2. The representatives in E^{∞}(S^{1}) of the kernel of R_{∗}^{h} equal the kernel of E^{∞}(R^{h}). Hence the image of R_{∗}^{h} is isomorphic to the image of E^{∞}(R^{h}).

The composite equivalence ΓΓ̂^{−1} does not induce a map of spectral sequences. Nonetheless it induces an isomorphism of E(λ_{1}, λ_{2}) ⊗ P (v_{2})-modules on V (1)- homotopy in dimensions > (2p − 2). Here v_{2} acts by multiplication in V (1)_{∗}, while multiplications by λ_{1} and λ_{2} are realized by the images of λ_{1}^{K} and λ_{2}^{K}, since both Γ and Γ̂ are ring spectrum maps.

Proposition 7.3. In dimensions > (2p − 2) the composite equivalence ΓΓ̂^{−1} induces an isomorphism

V (1)_{∗}ℍ̂(S^{1}, T HH(ℓ)) ≅ V (1)_{∗}T HH(ℓ)^{hS^{1}}

of P (v_{2})-modules, taking all classes represented by λ_{1}^{ϵ_{1}}λ_{2}^{ϵ_{2}}(tμ)^{m}t^{i} in Ê^{∞}(S^{1}) to classes represented by λ_{1}^{ϵ_{1}}λ_{2}^{ϵ_{2}}(tμ)^{m}μ^{j} in E^{∞}(S^{1}) with i+p^{2}j = 0. Here 0 ≤ ϵ_{1}, ϵ_{2} ≤ 1 and m ≥ 0.

Proof. Let T̂ = ℍ̂(C_{p}, T HH(ℓ)). The S^{1}-equivariant map Γ̂_{1} : T HH(ℓ) → T̂ induces a map of S^{1} homotopy fixed points, which realizes the localization homomorphism E^{∗}(S^{1}) → μ^{−1}E^{∗}(S^{1}) on the level of spectral sequences. It follows that there is a homotopy equivalence V (1)∧ℍ̂(S^{1}, T HH(ℓ)) ≃ V (1)∧T̂^{hS^{1}} which agrees with ΓΓ̂^{−1} on (2p − 2)-connected covers.

The v_{2}-indivisible elements in V (1)_{∗}ℍ̂(S^{1}, T HH(ℓ)) are represented in Ê^{∞}(S^{1}) by the classes

Ê^{∞}(S^{1})/(tμ) = E(λ_{1}, λ_{2})

⊕ ⨁_{k≥3} E(λ′_{[k]}) ⊗ 𝔽_{p}{λ_{[k]}t^{i} | v_{p}(i) = k − 1} .

The v_{2}-indivisible elements in V (1)_{∗}T̂^{hS^{1}} are represented in μ^{−1}E^{∞}(S^{1}) by the classes

μ^{−1}E^{∞}(S^{1})/(tμ) = E(λ_{1}, λ_{2})

⊕ ⨁_{k≥1} E(λ′_{[k]}) ⊗ 𝔽_{p}{λ_{[k]}μ^{j} | v_{p}(j) = k − 1} .

The asserted homotopy equivalence induces an isomorphism between these two terms, which by a dimension count must be given by

λ_{1}^{ϵ_{1}}λ_{2}^{ϵ_{2}}t^{i} ⟼ λ_{1}^{ϵ_{1}}λ_{2}^{ϵ_{2}}μ^{j}

with i + p^{2}j = 0. Hence the same formulas hold modulo multiples of v_{2} on V (1)- homotopy. Taking the P (v_{2})-module structure into account, the corresponding formulas including factors (tμ)^{m} also hold, and express the isomorphism

V (1)_{∗}ℍ̂(S^{1}, T HH(ℓ)) ≅ V (1)_{∗}T̂^{hS^{1}}

which agrees with ΓΓ̂^{−1} in dimensions > (2p − 2). □

Definition 7.4. Let A = E(λ_{1}, λ_{2}) ⊗ P (tμ),

B_{k} = E(λ′_{[k]}) ⊗ P_{r(k)}(tμ) ⊗ 𝔽_{p}{λ_{[k]}μ^{−ep^{k−1}} | 0 < e < p}

and B = ⨁_{k≥1} B_{k}. Let C be the span of the remaining monomial terms in μ^{−1}E^{∞}(S^{1}). Then E^{∞}(S^{1}) = A ⊕ B ⊕ C in dimensions > (2p − 2).

Theorem 7.5. In dimensions > (2p−2) there are subgroups Ã = E(λ_{1}, λ_{2})⊗P (v_{2}), B̃_{k} and C̃ of V (1)_{∗}T F (ℓ; p) represented by A, B_{k} and C in E^{∞}(S^{1}), respectively, such that

(a) R_{∗} is the identity on Ã.

(b) R_{∗} maps B̃_{k+2} onto B̃_{k} for all k ≥ 1.

(c) R_{∗} is zero on B̃_{1}, B̃_{2} and C̃. In these dimensions V (1)_{∗}T F (ℓ; p) = Ã ⊕ B̃ ⊕ C̃, with B̃ = ∏_{k≥1} B̃_{k}.

Proof. At the level of E^{∞}(S^{1}), the composite map ΓΓ̂^{−1} ∘ E^{∞}(R^{h}) is the identity on A, maps B_{k+2} onto B_{k} for all k ≥ 1 and is zero on B_{1}, B_{2} and C, by 7.1 and 7.3. The task is to find lifts of these groups to V (1)_{∗}T F (ℓ; p) such that R_{∗} has similar properties.

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

By 7.1 we have C ⊂ ker E^{∞}(R^{h}). Thus by 7.2 there is a subgroup C̃ in ker(R_{∗}) ≅ ker(R_{∗}^{h}) represented by C. Then R_{∗} is zero on C̃.

Note that im(R_{∗}) and ker(R_{∗}) span V (1)_{∗}T F (ℓ; p). For by 7.1 the representatives of im(R_{∗}) span A ⊕ B, and the representatives of the subgroup C̃ in ker(R_{∗}) span C. Thus the classes in im(R_{∗}) and ker(R_{∗}) have representatives spanning E^{∞}(S^{1}), and therefore span all of V (1)_{∗}T F (ℓ; p). Hence the image of R_{∗} on V (1)_{∗}T F (ℓ; p) equals the image of its restriction to im(R_{∗}).

Consider the subgroup

B_{k}^{0} = B_{k} ∩ ker E^{∞}(R^{h})

= E(λ′_{[k]}) ⊗ ⨁_{0<e<p} P_{r(k−2)+ep^{k−1}}^{r(k)−1}(tμ) ⊗ 𝔽_{p}{λ_{[k]}μ^{−ep^{k−1}}} .

This can be lifted to im(R_{∗}) by 7.1, and to ker(R_{∗}) by 7.2. By an argument using the strong convergence of the spectral sequence E^{∗}(S^{1}) (exercise !), it can be simultaneously lifted to a subgroup of im(R_{∗})∩ker(R_{∗}). Let B̃_{k}^{0} ⊂ im(R_{∗})∩ker(R_{∗}) be such a lift.

Inductively for n ≥ 1 let B_{k}^{n} ⊂ B_{k+2n} ⊂ E^{∞}(S^{1}) be the subgroup generated by the monomials mapped by E^{∞}(R^{h}) and ΓΓ̂^{−1} to the monomials generating B_{k}^{n−1}. Then B_{k} is the span of all B_{k−2n}^{n} for n ≥ 0.

Suppose inductively that we have chosen a lift B̃_{k}^{n} ⊂ im(R_{∗}) of B_{k}^{n} which maps by R_{∗} to B̃_{k}^{n−1} for n ≥ 1 and to zero for n = 0. Then choose classes in im(R_{∗}) mapping by R_{∗} to generators of B̃_{k}^{n} and let B̃_{k}^{n+1} be the subgroup they generate. Then B̃_{k}^{n+1} is a lift of B_{k}^{n+1} by 7.1 and 7.3.

Let B̃_{k} ⊂ V (1)_{∗}T F (ℓ; p) be the span of all B̃_{k−2n}^{n} for n ≥ 0. Then B̃_{k} is represented by all of B_{k}, R_{∗} maps B̃_{k+2} onto B̃_{k} for k ≥ 1, and B̃_{1} and B̃_{2} lie in ker(R_{∗}). □

8. Topological cyclic homology

Recall from 3.1 the long exact sequence

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

Proposition 8.2. In dimensions > (2p − 2) there are isomorphisms

ker(R_{∗} − 1) ≅ E(λ_{1}, λ_{2}) ⊗ P (v_{2})
⊕ E(λ_{2}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{1}t^{e} | 0 < e < p}
⊕ E(λ_{1}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{2}t^{ep} | 0 < e < p}

and

cok(R_{∗} − 1) ≅ E(λ_{1}, λ_{2}) ⊗ P (v_{2}) .

Proof. By 7.5 the homomorphism R_{∗} − 1 is zero on Ã = E(λ_{1}, λ_{2}) ⊗ P (v_{2}) and an isomorphism on C̃. The remainder of V (1)_{∗}T F (ℓ; p) decomposes as

B̃ = ∏_{k odd} B̃_{k} ⊕ ∏_{k even} B̃_{k}

and R_{∗} takes B̃_{k+2} to B̃_{k} for k ≥ 1, forming two sequential limit systems. Hence there is an exact sequence

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

and a corresponding one for k even. The right derived limit vanishes since each B̃_{k} has finite type. Hence it remains to prove that in dimensions > (2p − 2),

lim_{k odd} B̃_{k} ≅ E(λ_{2}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{1}t^{e} | 0 < e < p}

and

lim_{k even} B̃_{k} ≅ E(λ_{1}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{2}t^{ep} | 0 < e < p} .

Each B̃_{k} ≅ B_{k} is a sum of (2p − 2) finite cyclic P (v_{2})-modules. The restriction homomorphisms R_{∗} respect this sum decomposition, and map each cyclic module surjectively onto the next. Hence their limit is a sum of (2p − 2) cyclic modules, and it remains to check that these are infinite cyclic, i.e., not bounded above.

For k odd the ‘top’ class λ_{1}λ_{2}(tμ)^{r(k)−1}μ^{−ep^{k−1}} in B_{k} is in dimension 2p^{k+1}(p−e). For k even the corresponding class in B_{k} is in dimension 2p^{k+1}(p − e) + 2p − 2p^{2}. In both cases the dimension grows to +∞ for 0 < e < p as k grows.

For k odd each infinite cyclic P (v_{2})-module contains a class in non-negative degree with nonzero image in B̃_{1} ≅ B_{1}, namely the classes λ_{1}t^{e} and λ_{1}λ_{2}t^{e} for 0 < e < p. Hence we take these as generators for lim_{k odd} B̃_{k}. Likewise there are generators in non-negative degrees for lim_{k even} B̃_{k} with nonzero image in B̃_{2} ≅ B_{2}, namely the classes λ_{2}t^{ep} and λ_{1}λ_{2}t^{ep} for 0 < e < p. □

Let e ∈ π_{2p−1}T C(ℤ; p) be the image of e^{K} ∈ K_{2p−1}(ℤ_{p}), and let ∂ ∈ π_{−1}T C(ℤ; p) be the image of 1 ∈ π_{0}T F (ℤ; p) under ∂: Σ^{−1}T F (ℤ; p) → T C(ℤ; p). We recall from [BM1], [BM2] the calculation of the mod p homotopy of T C(ℤ; p).

Theorem 8.3 (Bökstedt–Madsen).

V (0)_{∗}T C(ℤ; p) ≅ E(e, ∂) ⊗ P (v_{1}) ⊕ P (v_{1}) ⊗ 𝔽_{p}{et^{i} | 0 < i < p} .

Hence

V (1)_{∗}T C(ℤ; p) ≅ E(e, ∂) ⊕ 𝔽_{p}{et^{i} | 0 < i < p} .

The (2p − 2)-connected map ℓ_{p} → Hℤ_{p} induces a (2p − 1)-connected map K(ℓ_{p}) → K(ℤ_{p}), and thus a (2p − 1)-connected map T C(ℓ; p) → T C(ℤ; p) after p-adic completion, by [Du]. This brings us to our main theorem.

Theorem 8.4. There is an isomorphism of E(λ_{1}, λ_{2}) ⊗ P (v_{2})-modules

V (1)_{∗}T C(ℓ; p) ≅ E(λ_{1}, λ_{2}, ∂) ⊗ P (v_{2})
⊕ E(λ_{2}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{1}t^{e} | 0 < e < p}
⊕ E(λ_{1}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{2}t^{ep} | 0 < e < p}

with |λ_{1}| = 2p − 1, |λ_{2}| = 2p^{2} − 1, |v_{2}| = 2p^{2} − 2, |∂| = −1 and |t| = −2.

Proof. This follows in dimensions > (2p − 2) from 8.2 and the exact sequence (8.1). It follows in dimensions ≤ (2p − 2) from 8.3 and the (2p − 1)-connected map V (1)_{∗}T C(ℓ; p) → V (1)_{∗}T C(ℤ; p). It remains to check that the module structures are compatible for multiplications crossing dimension (2p − 2).

The classes E(λ_{1})⊕𝔽_{p}{λ_{1}t^{e} | 0 < e < p} in V (1)_{∗}T C(ℓ; p) map to E(e)⊕𝔽_{p}{et^{i} | 0 < i < p} in V (1)_{∗}T C(ℤ; p), and map by Γ ∘ π to classes with the same names in the S^{1} homotopy fixed point spectral sequence for T HH(ℤ). By naturality, the given classes in V (1)_{∗}T C(ℓ; p) map by Γ ∘ π to classes with the same names in E^{∞}(S^{1}). Here these classes generate free E(λ_{2}) ⊗ P (tμ)-modules. For degree reasons multiplication by λ_{1} is zero on each λ_{1}t^{e}. Hence the E(λ_{1}, λ_{2}) ⊗ P (v_{2})- module structure on the given classes is as claimed.

Finally the class ∂ in V (1)_{−1}T C(ℓ; p) is the image under the connecting homomorphism ∂ of the class 1 in V (1)_{∗}T F (ℓ; p), which generates the free E(λ_{1}, λ_{2}) ⊗ P (v_{2})-module cok(R_{∗} − 1) of 8.2. Hence also the module structure on ∂ and λ_{1}∂ is as claimed. □

A very important feature of this calculational result is that V (1)_{∗}T C(ℓ; p) is a finitely generated free P (v_{2})-module. Thus T C(ℓ; p) is a fp-spectrum of fptype 2 in the sense of [MR]. Notice that V (1)_{∗}T F (ℓ; p) is not a free P (v_{2})-module. On the other hand we have the following calculation for the companion functor T R(ℓ; p) = holim_{n,R} T HH(ℓ)^{C_{p^{n}}}, showing that V (1)_{∗}T R(ℓ; p) is a free but not finitely generated P (v_{2})-module.

Theorem 8.5. There is an isomorphism of E(λ_{1}, λ_{2}) ⊗ P (v_{2})-modules

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

⊕ ⨁_{n≥1} E(u, λ_{2}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{1}t^{e} | 0 < e < p}

⊕ ⨁_{n≥1} E(u, λ_{1}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{2}t^{ep} | 0 < e < p} .

The nth summand classes u^{δ}λ_{1}t^{e} and u^{δ}λ_{2}t^{ep} for 0 ≤ δ ≤ 1 and 0 < e < p are detected in V (1)_{∗}T HH(ℓ)^{C_{p^{n}}} by the classes representing u_{n}^{δ}λ_{1}t^{e} and u_{n}^{δ}λ_{2}t^{ep} in E^{∞}(C_{p^{n}}), respectively.

We omit the proof.

9. Algebraic K-theory

We are now in a position to describe the V (1)-homotopy of the algebraic Ktheory spectrum of the p-completed Adams summand of connective topological K-theory, i.e., V (1)_{∗}K(ℓ_{p}). We use the cyclotomic trace map to largely identify it with the corresponding topological cyclic homology. Hence we will identify the algebraic K-theory classes λ_{1}^{K} and λ_{2}^{K} with their cyclotomic trace images λ_{1} and λ_{2}, in this section.

Theorem 9.1. There is an exact sequence of E(λ_{1}, λ_{2}) ⊗ P (v_{2})-modules

0 → Σ^{2p−3}𝔽_{p} ⟶ V (1)_{∗}K(ℓ_{p}) ⟶^{trc} V (1)_{∗}T C(ℓ; p) → Σ^{−1}𝔽_{p} → 0

taking the degree 2p − 3 generator in Σ^{2p−3}H𝔽_{p} to a class a ∈ V (1)_{2p−3}K(ℓ_{p}), and taking the class ∂ in V (1)_{−1}T C(ℓ; p) to the degree −1 generator in Σ^{−1}H𝔽_{p}. Hence

V (1)_{∗}K(ℓ_{p}) ≅ E(λ_{1}, λ_{2}) ⊗ P (v_{2})
⊕ P (v_{2}) ⊗ 𝔽_{p}{∂λ_{1}, ∂v_{2}, ∂λ_{2}, ∂λ_{1}λ_{2}}
⊕ E(λ_{2}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{1}t^{e} | 0 < e < p}
⊕ E(λ_{1}) ⊗ P (v_{2}) ⊗ 𝔽_{p}{λ_{2}t^{pe} | 0 < e < p}
⊕ 𝔽_{p}{a} .

Proof. By [HM1] the map ℓ_{p} → Hℤ_{p} induces a map of horizontal cofiber sequences of p-complete spectra:

[figure: K(ℓ_{p})_{p}; trc; T C(ℓ; p); Σ^{−1}Hℤ_{p}; K(ℤ_{p})_{p}; trc; T C(ℤ; p); Σ^{−1}Hℤ_{p} .]

Here V (1)_{∗}Σ^{−1}Hℤ_{p} is 𝔽_{p} in degrees −1 and 2p − 2, and 0 otherwise. Clearly ∂ in V (1)_{∗}T C(ℓ; p) maps to the generator in degree −1, since K(ℓ_{p})_{p} is a connective spectrum. The connecting map in V (1)-homotopy for the lower cofiber sequence takes the generator in degree (2p−2) to the nonzero class i_{1}(∂v_{1}) in V (1)_{2p−3}K(ℤ_{p}). By naturality it factors through V (1)_{2p−3}K(ℓ_{p}), where we let a be its image. □

Hence also K(ℓ_{p})_{p} is an fp-spectrum of fp-type 2. By [MR, 3.2] its mod p spectrum cohomology is finitely presented as a module over the Steenrod algebra, hence is induced up from a finite module over a finite subalgebra of the Steenrod algebra. In particular, K(ℓ_{p}) is closely related to elliptic cohomology.

We conclude with some comments on the v_{1}-Bockstein spectral sequence leading from the V (1)-homotopy of K(ℓ_{p}) to its V (0)-homotopy, i.e., its mod p homotopy. For any X, classes in the image of i_{1} : V (0)_{∗}X → V (1)_{∗}X are called mod p classes, while classes in the image of i_{1}i_{0} : π_{∗}X_{p} → V (1)_{∗}X are called integral classes.

Lemma 9.2. The classes 1, ∂λ_{1}, λ_{1} and λ_{1}t^{e} for 0 < e < p are integral classes both in V (1)_{∗}K(ℓ_{p}) and V (1)_{∗}T C(ℓ; p). Also ∂ is integral in V (1)_{∗}T C(ℓ; p), while a is integral in V (1)_{∗}K(ℓ_{p}).

The classes ∂λ_{2}, λ_{2}, ∂λ_{1}λ_{2}, λ_{1}λ_{2}, λ_{1}λ_{2}t^{e}, λ_{2}t^{ep} and λ_{1}λ_{2}t^{ep} for 0 < e < p are mod p classes in both V (1)_{∗}K(ℓ_{p}) and V (1)_{∗}T C(ℓ; p).

We are not excluding the possibility that some of the mod p classes are actually integral classes.

Proof. Each v_{1}-Bockstein β_{1,r} lands in a trivial group when applied to the classes ∂, 1, a and λ_{1}t^{e} for 0 < e < p in V (1)_{∗}K(ℓ_{p}) or V (1)_{∗}T C(ℓ; p). Hence these are at least mod p classes.

Since 1 maps to an element of infinite order in π_{0}T C(ℤ; p) ≅ ℤ_{p} and the other classes sit in odd degrees, all mod p^{r} Bocksteins on these classes are zero. Hence they are integral classes. The class λ_{1} is integral by construction, hence so is the product ∂λ_{1}.

The mod p homotopy operation (P^{p−e})^{∗} takes λ_{1}t^{e} in integral homotopy to λ_{2}t^{ep} in mod p homotopy, for 0 < e < p. Hence these are all mod p classes, as is λ_{2} by construction. The remaining classes listed are then products of established integral and mod p classes, and are therefore mod p classes. □

The classes listed in this lemma generate V (1)_{∗}K(ℓ_{p}) and V (1)_{∗}T C(ℓ; p) as P (v_{2})-modules. But v_{2} itself is not a mod p class.

Lemma 9.3. Let x be a mod p (or integral) class of V (1)_{∗}K(ℓ_{p}) or V (1)_{∗}T C(ℓ; p) and let t ≥ 0. Then

β_{1,1}(v_{2}^{t} · x) = tv_{2}^{t−1}i_{1}(β′_{1}) · x .

In particular i_{1}(β′_{1}) · 1 = t^{p}λ_{2} and i_{1}(β′_{1}) · λ_{1} = t^{p}λ_{1}λ_{2}.

We expect that i_{1}(β′_{1}) · t^{p^{2}−p}λ_{2} = ∂λ_{2} and i_{1}(β′_{1}) · t^{p^{2}−p}λ_{1}λ_{2} = ∂λ_{1}λ_{2}, by symmetry considerations.

Proof. The v_{1}-Bockstein β_{1,1} = i_{1}j_{1} acts as a derivation by [Ok]. By definition j_{1}(v_{2}) = β′_{1} = [h_{11}], which is detected as t^{p}λ_{2} by 4.8. Clearly j_{1}(x) = 0 for mod p classes x. □

In V (1)_{∗} the powers v_{2}^{t} support nonzero differentials β_{1,1}(v_{2}^{t}) = tv_{2}^{t−1}i_{1}(β′_{1}) for p ∤ t. The first nonzero differential on v_{2}^{p} is β_{1,p}:

Lemma 9.4. β_{1,p}(v_{2}^{p}) = [h_{12}] ≠ 0 in V (1)_{∗}.

We refer to [Ra2, §4.4] for background for the following calculation.

Proof. In the BP -based Adams–Novikov spectral sequence for V (0) the relation j_{1}(v_{2}^{p}) = v_{1}^{p−1}β′_{p/p} holds, where β′_{p/p} is the class represented by h_{12} + v_{1}^{p^{2}−p}h_{11} in degree 1 of the cobar complex. Its integral image β_{p/p} = j_{0}(β′_{p/p}) is represented by b_{11}, and supports the Toda differential d_{2p−1}(β_{p/p}) = α_{1}β_{1}^{p}. This differential lifts to d_{2p−1}(β′_{p/p}) = v_{1}β_{1}^{p} in the Adams–Novikov spectral sequence for V (0). Consider the image of β′_{p/p} under i_{1} in the Adams–Novikov spectral sequence for V (1), which is represented by h_{12} in the cobar complex. Then d_{2p−1}(i_{1}(β′_{p/p})) = i_{1}(v_{1}β_{1}^{p}) = 0. By sparseness and the vanishing line there are no further differentials, and i_{1}(β′_{p/p}) = [h_{12}] represents a nonzero element of V (1)_{∗}. Hence β_{1,p}(v_{2}^{p}) = [h_{12}], as claimed. □

Remark 9.5. We would like to obtain the mod p homotopy groups V (0)_{∗}T C(ℓ; p) by means of the v_{1}-Bockstein spectral sequence. This requires, first, that we compute the product with β_{1,1}(v_{2}) = i_{1}(β′_{1}) in the remaining V (1)-homotopy groups. Next we must identify the image of β_{1,p}(v_{2}^{p}) = [h_{12}] in V (1)_{∗}T C(ℓ; p). Imaginably there is a homology differential in (4.3) on [ξ̄^{p^{2}}] hitting (tμ)^{p}λ_{1}t^{p^{2}}, if (P^{p^{2}})^{∗}(λ_{2}) = (tμ)^{p}λ_{1}. Then most likely [h_{12}] is detected by (tμ)^{p}λ_{1}t^{p^{2}} in V (1)_{∗}T F (ℓ; p), which is identified under R with the class (tμ)^{p}λ_{1}μ^{−1} generating V (1)_{∗}T C(ℓ; p) in this degree. The general picture appears to be complicated.

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Department of Mathematics, ETH Zürich, Switzerland

   E-mail address: ausoni@math.ethz.ch

Department of Mathematics, University of Oslo, Norway

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