Algebraic K-theory of strict ring spectra

John Rognes

Abstract. We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are available at regular primes, but we seek more conceptual answers in terms of localization and descent properties. Calculations for ring spectra related to topological K-theory suggest the existence of a motivic cohomology theory for strictly commutative ring spectra, and we present evidence for arithmetic duality in this theory. To tie motivic cohomology to Galois cohomology we wish to spectrally realize ramified extensions, which is only possible after mild forms of localization. One such mild localization is provided by the theory of logarithmic ring spectra, and we outline recent developments in this area.

Mathematics Subject Classification (2010). Primary 19D10, 55P43; Secondary 19F27, 57R50.

Keywords. Arithmetic duality, automorphisms of manifolds, brave new rings, étale descent, logarithmic ring spectrum, logarithmic topological André–Quillen homology, logarithmic topological Hochschild homology, motivic truncation, replete bar construction, sphere spectrum, tame ramification, topological K-theory.

1. Strict ring spectra

First, let R be an abelian group. Ordinary singular cohomology with coefficients in R is a contravariant homotopy functor that associates to each based space X a graded cohomology group H̃^{∗}(X; R). It is stable, in the sense that there is a natural isomorphism H̃^{∗}(X; R) ≅ H̃^{∗+1}(ΣX; R), and this implies that it extends from the category of based spaces to the category of spectra. The latter is a category of space-like objects, where the suspension is invertible up to homotopy equivalence, and which has all colimits and limits. The extended cohomology functor becomes representable, meaning that there is a spectrum HR, called the Eilenberg– Mac Lane spectrum of R, and a natural isomorphism H̃^{∗}(X; R) ≅ [X, HR]_{−∗}, where [X, HR]_{−n} is the group of homotopy classes of morphisms X → Σ^{n}HR.

Next, let R be a ring. Then the cohomology theory is multiplicative, meaning that there is a bilinear cup product H̃^{∗}(X; R) × H̃^{∗}(X; R) → H̃^{∗}(X; R). This is also representable in the category of spectra, by a morphism μ: HR ∧ HR → HR, where ∧ denotes the smash product of spectra. With the modern models for the category of spectra [26], [40], [48] we may arrange that μ is strictly unital and associative, so that HR is a strict ring spectrum. Equivalent terms are A_{∞} ring spectrum, S-algebra, symmetric ring spectrum and orthogonal ring spectrum.

If R is commutative, then the cup product is graded commutative, which at the representing level means that μτ ≃ μ, where τ: HR ∧ HR → HR ∧ HR denotes the twist isomorphism. In fact, we may arrange that μ is strictly commutative, in the sense that μτ = μ as morphisms of spectra, so that HR is a strictly commutative ring spectrum. Equivalent phrases are E_{∞} ring spectrum, commutative S-algebra, commutative symmetric ring spectrum and commutative orthogonal ring spectrum. This leads to a compatible sequence of Σ_{k}-equivariant morphisms EΣ_{k+} ∧ HR^{∧k} → HR for k ≥ 0. At the represented level these morphisms give rise to power operations in cohomology, including Steenrod’s operations Sq^{i} for R = 𝔽_{2} and β^{ϵ}P^{i} for R = 𝔽_{p}.

One now realizes that the Eilenberg–Mac Lane ring spectra HR exist as special cases within a much wider class of ring spectra. Each spectrum B represents a generalized cohomology theory X ↦ B̃^{∗}(X) = [X, B]_{−∗} and a generalized homology theory X ↦ B̃_{∗}(X) = π_{∗}(B ∧ X). Examples of early interest include the spectrum KU that represents complex topological K-theory, KU^{∗}(X) = [X_{+}, KU]_{−∗}, and the spectrum MU that represents complex bordism, MU_{∗}(X) = π_{∗}(MU∧X_{+}). A fundamental example is given by the sphere spectrum S, which is the image of the based space S^{0} under the stabilization functor from spaces to spectra. It represents stable cohomotopy π_{S}^{∗}(X) = S̃^{∗}(X) and stable homotopy π_{∗}^{S}(X) = S̃_{∗}(X). Each of these three examples, KU, MU and S, is naturally a strictly commutative ring spectrum, representing a multiplicative cohomology theory with power operations, etc. Furthermore, there are interesting multiplicative morphisms connecting these ring spectra to the Eilenberg–Mac Lane ring spectra previously considered, as in the diagram

\xymatrix{ &  & KU \ar[r] & Hℚ \\ S \ar[r] & MU \ar[r] & ku \ar[r] \ar[u] & Hℤ, \ar[u]}

where ku = KU[0, ∞) denotes the connective cover of KU.

By placing the class of traditional rings inside the wider realm of all strict ring spectra, a new world of possibilities opens up. Following Waldhausen [47, p. xiii] we may refer to strict ring spectra as “brave new rings”. If we think in algebrogeometric terms, where commutative rings appear as the rings of functions on pieces of geometric objects, then strictly commutative ring spectra are the functions on affine pieces of brave new geometries, more general than those realized by ordinary schemes.

How vast is this generalization? In the case of connective ring spectra B, i.e., those with π_{i}(B) = 0 for i negative, there is a natural ring spectrum morphism B → Hπ_{0}(B) that induces an isomorphism on π_{0}. This behaves for many purposes like a topologically nilpotent extension, and in geometric terms, B can be viewed as the ring spectrum of functions on an infinitesimal thickening of Spec π_{0}(B).

This infinitesimal thickening can be quite effectively controlled in terms of diagrams of Eilenberg–Mac Lane spectra associated with simplicial rings. The Hurewicz map B ≅ S ∧ B → Hℤ ∧ B is 1-connected, and there is an equivalence Hℤ ∧ B ≃ HR_{•} for some simplicial ring R_{•}. The square

\xymatrix{S ∧ S ∧ B \ar[r] \ar[d] & Hℤ ∧ S ∧ B \ar[d] \\ S ∧ Hℤ ∧ B \ar[r] & Hℤ ∧ Hℤ ∧ B}

induces a 2-connected map from B ≅ S ∧ S ∧ B to the homotopy pullback. More generally, for each n ≥ 1 there is an n-dimensional cubical diagram that induces an n-connected map from the initial vertex B to the homotopy limit of the remainder of the cube, and the terms in that remainder have the form HR_{•} for varying simplicial rings R_{•}. Dundas [24] used a clever strengthening of this statement to prove that relative algebraic K-theory is p-adically equivalent to relative topological cyclic homology for morphisms A → B of connective strict ring spectra, under the assumption that π_{0}(A) → π_{0}(B) is a surjection with nilpotent kernel. He achieved this by reducing to the analogous statement for homomorphisms R_{•} → T_{•} of simplicial rings, which had been established earlier by McCarthy [49]. This confirmed a conjecture of Goodwillie [33], motivated by a similar result for rational K-theory and (negative) cyclic homology [32].

How about the case of non-connective ring spectra? Those that arise as homotopy fixed points B^{hG} = F (EG_{+}, B)^{G} for a group action may be viewed as the functions on an orbit stack for the induced G-action on the geometry associated to B. Those that arise as smashing Bousfield localizations L_{E}B, with respect to a homology theory E_{∗}, may be viewed as open subspaces in a finer topology than the one derived from the Zariski topology on Spec π_{0}(B) [63, §9.3]. In the general case the connection to classical geometry is less clear.

2. Automorphisms of manifolds

Why should we be interested in brave new rings and their ring-theoretic invariants, like algebraic K-theory [55], [26, Ch. VI], other than for the sake of generalization? One good justification comes from the tight connection between the geometric topology of high-dimensional manifolds and the algebraic K-theory of strict ring spectra. This connection is given by the higher simple-homotopy theory initiated by Hatcher [36], which was fully developed by Waldhausen in the context of his algebraic K-theory of spaces [82] and the stable parametrized h-cobordism theorem [83]. On the geometric side, this theory concerns the fundamental problem of finding a parametrized classification of high-dimensional compact manifolds, up to homeomorphism, piecewise-linear homeomorphism or diffeomorphism, as appropriate for the respective geometric category. The set of path components of the resulting moduli space corresponds to the set of isomorphism classes of such manifolds, and each individual path component is a classifying space for the automorphism group Aut(X) of a manifold X in the respective isomorphism class.

This parametrized classification is finer than the one provided by the Browder– Novikov–Sullivan–Wall surgery theory [21], [84], which classifies manifolds up to h-cobordism (or s-cobordism), and whose associated moduli space has path components that classify the block automorphism groups of manifolds, rather than their actual automorphism groups. The difference between these two classifications is controlled by the space H(X) of h-cobordisms (W; X, Y) with a given manifold X at one end. Here W is a compact manifold with ∂W = X ∪ Y, and the inclusions X → W and Y → W are homotopy equivalences. More precisely, there is one h-cobordism space H^{Cat}(X) for each of the three flavors of manifolds mentioned, namely Cat = Top, PL or Diff.

The original h-cobordism theorem enumerates the isomorphism classes of hcobordisms with X at one end, i.e., the set π_{0}H(X) of path components of the h-cobordism space of X, in terms of an algebraic K-group of the integral group ring ℤ[π], where π = π_{1}(X) is the fundamental group of X. One defines the Whitehead group as the quotient Wh_{1}(π) = K_{1}(ℤ[π])/(±π), and associates a Whitehead torsion class τ(W; X, Y) ∈ Wh_{1}(π) to each h-cobordism on X.

Theorem 2.1 (Smale [73], Barden, Mazur, Stallings). Let X be a compact, connected n-manifold with n ≥ 5 and π = π_{1}(X). The Whitehead torsion defines a bijection

π_{0}H(X) ≅ Wh_{1}(π) .

These constructions involve using Morse functions or triangulations to choose a relative CW complex structure on the pair (W, X), or equivalently, a π-equivariant relative CW complex structure on the pair of universal covers (W̃ , X̃), and to study the associated cellular chain complex C_{∗}(W̃ , X̃) of free ℤ[π]-modules. This works fine as long as one is only concerned with a classification of h-cobordisms up to isomorphism, but for the parametrized problem, i.e., the study of the full homotopy type of H(X), the passage from a CW complex structure to the associated cellular chain complex loses too much information. One should remember the actual attaching maps from the boundaries of cells to the preceding skeleta, not just their degrees. In a stable range this amounts to working with maps from spheres to spheres and coefficients in the sphere spectrum S, rather than with degrees and coefficients in the integers ℤ. Likewise, the passage to the π-equivariant universal cover X̃ → X should be replaced to a passage to a G-equivariant principal fibration P → X, where P is contractible and G ≃ ΩX is a topological group that is homotopy equivalent to the loop space of X ≃ BG. To sum up, the parametrized analog of the Whitehead torsion must take values in a Whitehead space that is built from the algebraic K-theory of the spherical group ring S[G] = S ∧ G_{+}, a strict ring spectrum, rather than that of its discrete reduction, the integral group ring π_{0}(S[G]) ≅ ℤ[π].

Waldhausen’s algebraic K-theory of spaces, traditionally denoted A(X), was first introduced without reference to strict ring spectra [80], but can be rewritten as the algebraic K-theory A(X) = K(S[G]) of the strict ring spectrum S[G], cf. [81] and [26, Ch. VI]. This point of view is convenient for the comparison of algebraic K-theory with other ring-theoretic invariants.

In the case of differentiable manifolds, the Whitehead space Wh^{Diff}(X) is defined to sit in a split homotopy fiber sequence of infinite loop spaces

Ω^{∞}(S ∧ X_{+}) ⟶^{ι} K(S[G]) ⟶ Wh^{Diff}(X) .

In the topological case there is a homotopy fiber sequence of infinite loop spaces

Ω^{∞}(K(S) ∧ X_{+}) ⟶^{α} K(S[G]) ⟶ Wh^{Top}(X) ,

where α is known as the assembly map. The piecewise-linear Whitehead space is the same as the topological one. The stable parametrized h-cobordism theorem reads as follows.

Theorem 2.2 (Waldhausen–Jahren–Rognes [83, Thm. 0.1]). Let X be a compact Cat manifold, for Cat = Top, PL or Diff. There is a natural homotopy equivalence

H^{Cat}(X) ≃ Ω Wh^{Cat}(X) ,

where H^{Cat}(X) = colim_{k} H^{Cat}(X × I^{k}) is the stable Cat h-cobordism space of X.

When combined with connectivity results about the dimensional stabilization map H(X) = H^{Cat}(X) → H^{Cat}(X), and here the main result is Igusa’s stability theorem for smooth pseudoisotopies [42], knowledge of K(S) and K(S[G]) gives good general results on the h-cobordism space H(X) and the automorphism group Aut(X) of a high-dimensional manifold X.

Example 2.3. When G is trivial, so that S[G] = S and X is contractible, the π_{0}-isomorphism and rational equivalence S → Hℤ induces a rational equivalence K(S) → K(ℤ). Here π_{∗}K(ℤ) ⊗ ℚ was computed by Borel [20], so

π_{i} Wh^{Diff}(∗) ⊗ ℚ ≅ \begin{cases}ℚ & for i = 4k + 1 ≠ 1, \\ 0 & otherwise. \end{cases}

For X = D^{n}, Farrell–Hsiang [27] used this to show that

π_{i} Diff(D^{n}) ⊗ ℚ ≅ \begin{cases}ℚ & for i = 4k − 1, n odd, \\ 0 & otherwise, \end{cases}

for i up to approximately n/3, where Diff(D^{n}) denotes the group of self-diffeo-morphisms of D^{n} that fix the boundary. For instance, π_{3} Diff(D^{13}) is rationally nontrivial. By contrast, the group Top(D^{n}) of self-homeomorphisms of D^{n} that fix the boundary is contractible. Similar results follow for n-manifolds that are roughly n/3-connected.

The case of spherical space forms, when G is finite with periodic cohomology, has been studied by Hsiang–Jahren [41]. For closed, non-positively curved manifolds X, Farrell–Jones [28] showed that Wh^{Diff}(X) can be assembled from copies of Wh^{Diff}(∗) and Wh^{Diff}(S^{1}), indexed by the points and the closed geodesics in X, respectively. These correspond to the special cases G trivial and G infinite cyclic, respectively, so K(S) and K(S[ℤ]) are of fundamental importance for the parametrized classification of this large class of Riemannian manifolds. In this paper we shall focus on the case of K(S), but see Hesselholt’s paper [37] for the case of K(S[ℤ]), and see Weiss–Williams [86] for a detailed survey about automorphisms of manifolds and algebraic K-theory.

Remark 2.4. More recent papers of Madsen–Weiss [46], Berglund–Madsen [12] and Galatius–Randal-Williams [29] give precise results about automorphism groups of manifolds of a fixed even dimension n = 2d ≠ 4, at the expense of first forming a connected sum with many copies of S^{d} × S^{d}. The latter results are apparently not closely related to the algebraic K-theory of strict ring spectra.

3. Algebraic K-theory of the sphere spectrum

We can strengthen the rational results about A(X) = K(S[G]), Wh^{Diff}(X) and Diff(X) to integral results, or more precisely, to p-adic integral results for each prime p. From here on it will be convenient to think of algebraic K-theory as a spectrum-valued functor, and likewise for the Whitehead theories, so that there are homotopy cofiber sequences of spectra

S ∧ X_{+} ⟶^{ι} K(S[G]) ⟶ Wh^{Diff}(X)
K(S) ∧ X_{+} ⟶^{α} K(S[G]) ⟶ Wh^{Top}(X) ,

and the first one is naturally split.

A key tool for this study is the cyclotomic trace map trc: K(B) → T C(B; p) from algebraic K-theory to the topological cyclic homology of Bökstedt–Hsiang– Madsen [16]. The latter invariant of the strict ring spectrum B can sometimes be calculated by analyzing the S^{1}-equivariant homotopy type of the topological Hochschild homology spectrum T HH(B). Its power is illustrated by the following previously mentioned theorem.

Theorem 3.1 (Dundas [24]). Let B be a connective strict ring spectrum. The square

\xymatrix{K(B) \ar[r] \ar[d]_{trc} & K(π_{0}(B)) \ar[d]^{trc} \\ T C(B; p) \ar[r] & T C(π_{0}(B); p)}

becomes homotopy Cartesian upon p-completion.

In the basic case B = S, when K(S) ≃ S ∨ Wh^{Diff}(∗) determines Diff(D^{n}) for large n, this square takes the form below. Three of the four corners are quite well understood, but for widely different reasons.

\xymatrix{K(S) \ar[r] \ar[d]_{trc} & K(ℤ) \ar[d]^{trc} \\ T C(S; p) \ar[r] & T C(ℤ; p) .}

These reasons were tied together by the author for p = 2 in [61], and for p an odd regular prime in [62], to compute the mod p cohomology

H^{∗}(K(S); 𝔽_{p}) ≅ 𝔽_{p} ⊕ H^{∗}(Wh^{Diff}(∗); 𝔽_{p})

as a module over the Steenrod algebra 𝒜 of stable mod p cohomology operations. This sufficed to determine the E_{2}-term of the Adams spectral sequence

E_{2}^{s,t} = Ext_{𝒜}^{s,t}(H^{∗}(K(S); 𝔽_{p}), 𝔽_{p}) ⟹ π_{t−s}K(S)_{p}

in a large range of degrees, and to determine the homotopy groups π_{i}K(S)_{p} and π_{i} Wh^{Diff}(∗)_{p} in a smaller range of degrees.

The structure of the algebraic K-theory of the integers, K(ℤ), was predicted by the Lichtenbaum–Quillen conjectures [55], which were confirmed for p = 2 by Voevodsky [78] with contributions by Rognes–Weibel [67], and for p odd by Voevodsky [79] with contributions by Rost and Weibel. For p = 2 or p a regular odd prime, this led to a p-complete description of the spectrum K(ℤ) in terms of topological K-theory spectra, which in turn led to an explicit description of the spectrum cohomology H^{∗}(K(ℤ); 𝔽_{p}) as an 𝒜 -module.

The topological cyclic homology of the integers, T C(ℤ; p), was computed for odd primes p by Bökstedt–Madsen [17], [18] and for p = 2 by the author [57], [58], [59], [60], in papers that start with knowledge of the mod p homotopy of the S^{1}-spectrum T HH(ℤ) and inductively determine the mod p homotopy of the C_{p^{n}}-fixed points T HH(ℤ)^{C_{p^{n}}} for n ≥ 1. It is then possible to recognize the pcompleted spectrum level structure by comparisons with known models, using [56] for p odd, and to obtain the 𝒜 -module H^{∗}(T C(ℤ; p); 𝔽_{p}) from this.

The topological cyclic homology of the sphere spectrum, T C(S; p), was determined in the original paper [16]. There is an equivalence of spectra T C(S; p) ≃ S ∨ ΣℂP_{−1}^{∞} after p-completion, where ΣℂP_{−1}^{∞} is the homotopy fiber of the dimensionshifting S^{1}-transfer map t: ΣℂP_{+}^{∞} → S. The mod p cohomology H^{∗}(ΣℂP_{−1}^{∞}; 𝔽_{p}) is well known as an 𝒜 -module. For p = 2 it is cyclic with

H^{∗}(ΣℂP_{−1}^{∞}; 𝔽_{2}) ≅ Σ^{−1}𝒜 /C ,

where the ideal C ⊂ 𝒜 is generated by the admissible Sq^{I} where I = (i_{1}, …, i_{n}) with n ≥ 2 or I = (i) with i odd. The determination of the homotopy groups of T C(S; p) is of comparable difficulty to the computation of the homotopy groups of S, due to our extensive knowledge about the attaching maps in the usual CW spectrum structure on ΣℂP_{−1}^{∞}, cf. Mosher [52].

The linearization map T C(S; p) → T C(ℤ; p) is only partially understood [45], but for p regular the cyclotomic trace map K(ℤ) → T C(ℤ; p) can be controlled by an appeal to global Tate–Poitou duality [76, Thm. 3.1], see [62, Prop. 3.1]. This leads to the following conclusion for p = 2. See [62, Thm. 5.4] for the result at odd regular primes.

Theorem 3.2 ([61, Thm. 4.5]). The mod 2 cohomology of the spectrum Wh^{Diff}(∗) is given by the unique non-trivial extension of 𝒜 -modules

0 → Σ^{−2}C/𝒜 (Sq^{1}, Sq^{3}) ⟶ H^{∗}(Wh^{Diff}(∗); 𝔽_{2}) ⟶ Σ^{3}𝒜 /𝒜 (Sq^{1}, Sq^{2}) → 0 .

Using the Adams spectral sequence and related methods, the author obtained the following explicit calculations. Less complete information, in a larger range of degrees, is provided in the cited references. Previously, Bökstedt–Waldhausen [19, Thm. 1.3] had computed π_{i} Wh^{Diff}(∗) for i ≤ 3.

Theorem 3.3 ([61, Thm. 5.8], [62, Thm. 4.7]). The homotopy groups of Wh^{Diff}(∗) in degrees i ≤ 18 are as follows, modulo p-power torsion for irregular primes p.

\begin{matrix}i & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ π_{i} Wh^{Diff}(∗) & 0 & 0 & 0 & ℤ/2 & 0 & ℤ & 0 & ℤ/2 & 0 & ℤ ⊕ ℤ/2 \end{matrix}

\begin{matrix}i & 10 & 11 & 12 & 13 & 14 \\ π_{i} Wh^{Diff}(∗) & ℤ/8 ⊕ (ℤ/2)^{2} & ℤ/6 & ℤ/4 & ℤ & ℤ/36 ⊕ ℤ/3 \end{matrix}

\begin{matrix}i & 15 & 16 & 17 & 18 \\ π_{i} Wh^{Diff}(∗) & (ℤ/2)^{2} & ℤ/24 ⊕ ℤ/2 & ℤ ⊕ (ℤ/2)^{2} & ℤ/480 ⊕ (ℤ/2)^{3} \end{matrix}

Example 3.4. For X = D^{n} with n sufficiently large, it follows that π_{4p−4} Diff(D^{n}) or π_{4p−4} Diff(D^{n+1}) contains an element of order p, for each regular p ≥ 5, and that π_{9} Diff(D^{n}) or π_{9} Diff(D^{n+1}) contains an element of order 3, see [62, Thm. 6.4]. To get more precise results one needs to investigate the canonical involution on Wh^{Diff}(∗) and apply Weiss–Williams [85, Thm. A].

Remark 3.5. It would be interesting to extend these results to irregular primes. Dwyer–Mitchell [25] described the spectrum K(ℤ)_{p} in terms of the p-primary Iwasawa module of the rationals. It should be possible to turn this into a description of the 𝒜 -module H^{∗}(K(ℤ); 𝔽_{p}). Next one must control the cyclotomic trace map K(ℤ) → T C(ℤ; p), or the closely related completion map K(ℤ) → K(ℤ_{p}), whose behavior is governed by special values of p-adic L-functions, cf. Soulé [74, Thm. 3].

4. Algebraic K-theory of topological K-theory

The calculations reviewed in the previous section extracted detailed information about π_{∗}K(S) ≅ π_{∗}(S) ⊕ π_{∗} Wh^{Diff}(∗) from our knowledge of π_{∗}(ℂP_{−1}^{∞}). However, this understanding was not presented to us in as conceptual a way as the understanding we have of K(ℤ), say in terms of Quillen’s localization sequence

K(𝔽_{p}) ⟶ K(ℤ) ⟶ K(ℤ[1/p])

and the étale descent property

π_{i}K(ℤ[1/p])_{p} ⟶^{≅} K_{i}^{ét}(ℤ[1/p]; ℤ_{p})

for i > 0, cf. [54, §5] and [55, §9]. It would be desirable to have a similarly conceptual understanding of K(S)_{p} in terms of a comparison with K(B)_{p} for suitably local strict ring spectra B, a descent property describing K(B)_{p} as a homotopy limit of K(C)_{p} for appropriate extensions B → C, and a simple description of K(Ω)_{p} for a sufficiently large such extension B → Ω.

To explore this problem, we first simplify the number theory involved by working with the p-adic integers ℤ_{p} in place of the rational integers ℤ, and then seek a conceptual understanding of K(B)_{p} for some of the strictly commutative ring spectra B that are closest to Hℤ_{p}, namely the p-complete connective complex K-theory spectrum ku_{p} and its Adams summand ℓ_{p}. Here π_{∗}(ku_{p}) = ℤ_{p}[u] and π_{∗}(ℓ_{p}) = ℤ_{p}[v_{1}], with |u| = 2 and |v_{1}| = 2p − 2. Let KU_{p} and L_{p} denote the associated periodic spectra, with π_{∗}(KU_{p}) = ℤ_{p}[u^{±1}] and π_{∗}(L_{p}) = ℤ_{p}[v_{1}^{±1}]. There are multiplicative morphisms

\xymatrix{ & L_{p} \ar[r]^{ϕ} & KU_{p} \\ S_{p} \ar[r] & ℓ_{p} \ar[r]^{ϕ} \ar[u] & ku_{p} \ar[r] \ar[u] & Hℤ_{p}}  (1)

of strictly commutative ring spectra, where ϕ_{∗}(v_{1}) = u^{p−1}. The group Δ ≅ 𝔽_{p}^{×} of p-adic roots of unity acts by Adams operations on KU_{p}, and ϕ: L_{p} → KU_{p} is a Δ-Galois extension in the sense of [63, p. 3].

Definition 4.1. Let V (1) = S ∪_{p} e^{1} ∪_{α_{1}} e^{2p−1} ∪_{p} e^{2p} be the type 2 Smith–Toda complex, defined as the mapping cone of the Adams self-map v_{1} : Σ^{2p−2}S/p → S/p of the mod p Moore spectrum S/p = S ∪_{p} e^{1}. It is a ring spectrum up to homotopy for p ≥ 5, which we now assume. We write V (1)_{∗}B = π_{∗}(V (1) ∧ B) for the “mod p and v_{1} homotopy” of any spectrum B. It is naturally a module over the polynomial ring 𝔽_{p}[v_{2}], where v_{2} ∈ π_{2p^{2}−2}V (1).

The mod p and v_{1} homotopy of the topological cyclic homology of the connective Adams summand ℓ was computed by Ausoni and the author, by starting with knowledge of V (1)_{∗}T HH(ℓ) from [51] and inductively determining the mod p and v_{1} homotopy of the fixed points T HH(ℓ)^{C_{p^{n}}} for n ≥ 1. The calculations were later extended to the full connective complex K-theory spectrum ku by Ausoni. To avoid introducing too much notation, we only describe the most striking features of the answers, referring to the original papers for more precise statements.

Theorem 4.2 (Ausoni–Rognes [5, Thm. 0.3, Thm. 0.4]). V (1)_{∗}T C(ℓ; p) is a finitely generated free 𝔽_{p}[v_{2}]-module on 4(p + 1) generators, which are located in degrees −1 ≤ ∗ ≤ 2p^{2} + 2p − 2. There is an exact sequence of 𝔽_{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 .

Theorem 4.3 (Ausoni [4, Thm. 7.9, Thm. 8.1]). V (1)_{∗}T C(ku; p) is a finitely generated free 𝔽_{p}[v_{2}]-module on 4(p − 1)(p + 1) generators, which are located in degrees −1 ≤ ∗ ≤ 2p^{2} + 2p − 2. There is an exact sequence of 𝔽_{p}[v_{2}]-modules

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

and the natural map K(ℓ_{p}) → K(ku_{p})^{hΔ} is a p-adic equivalence.

Blumberg–Mandell [14] constructed homotopy cofiber sequences

K(ℤ_{p}) ⟶ K(ℓ_{p}) ⟶ K(L_{p})  (2)

and K(ℤ_{p}) → K(ku_{p}) → K(KU_{p}), which lead to calculations of V (1)_{∗}K(L_{p}) and V (1)_{∗}K(KU_{p}), cf. [4, Thm. 8.3]. The natural map K(L_{p}) → K(KU_{p})^{hΔ} is also a p-adic equivalence, which confirms the étale descent property for algebraic K-theory in this particular case. Remark 4.4. The examples discussed above are the case n = 1 of a series of approximations to S associated with the Lubin–Tate spectra E_{n}, with coefficient rings π_{∗}E_{n} = 𝕎𝔽_{p^{n}}[[u_{1}, …, u_{n−1}]][u^{±1}], which are known to be strictly commutative ring spectra by the Goerss–Hopkins–Miller obstruction theory [31]. There are multiplicative morphisms

\xymatrix{L̂_{n}S \ar[r] & L̂_{n}E(n) \ar[r]^{ϕ} & E_{n} \\ L_{n}S \ar[u] \ar[r] & BP ⟨n⟩ \ar[u] \ar[r] & e_{n} \ar[u] \ar[r] & Hπ_{0}(e_{n}) ,}

where L_{n} and L̂_{n} denote Bousfield localization with respect to the Johnson–Wilson spectrum E(n) and the Morava K-theory spectrum K(n), respectively, and BP ⟨n⟩ is the truncated Brown–Peterson spectrum. The n-th extended Morava stabilizer group 𝔾_{n} acts on E_{n}, and L̂_{n}E(n) → E_{n} is an H-Galois extension for H ≅ 𝔽_{p^{n}}^{×} ⋊ ℤ/n. We write e_{n} for the connective cover of E_{n}.

The algebraic K-theory computations above provide evidence for the chromatic redshift conjecture, see [5, p. 7] and [6], predicting that the algebraic K-theory K(B) of a purely v_{n}-periodic strictly commutative ring spectrum B, such as E_{n}, is purely v_{n+1}-periodic in sufficiently high degrees.

5. Motivic truncation and arithmetic duality

The proven Lichtenbaum–Quillen conjectures subsume a spectral sequence

E_{s,t}^{2} = H_{ét}^{−s}(R; ℤ_{p}(t/2)) ⟹ π_{s+t}K(R)_{p} ,

which converges for reasonable R and s+t sufficiently large. Here H_{ét}^{∗} denotes étale cohomology, R is a commutative ℤ[1/p]-algebra, and ℤ_{p}(t/2) = π_{t}(KU_{p}) is ℤ_{p}(m) when t = 2m is even, and 0 otherwise. For instance, we may take R = 𝒪_{F} [1/p] to be the ring of p-integers in a number field F, or R may be a p-adic field, i.e., a finite extension of ℚ_{p}.

The proven Beilinson–Lichtenbaum conjectures, cf. [75] and [30], provide a more precise convergence statement. For each field F containing 1/p there is a spectral sequence

E_{s,t}^{2} = H_{mot}^{−s}(F; ℤ(t/2)) ⟹ π_{s+t}K(F) ,

converging in all degrees, and similarly with mod p coefficients. Here H_{mot}^{∗} denotes motivic cohomology, which satisfies

H_{mot}^{r}(F; ℤ/p(m)) ≅ \begin{cases}H_{ét}^{r}(F; ℤ/p(m)) & for 0 ≤ r ≤ m, \\ 0 & otherwise. \end{cases}  (3)

In terms of Bloch’s higher Chow groups [13], the vanishing of these groups for r > m expresses the fact that there are no codimension r subvarieties of affine m-space over Spec F. Conversely,

H_{ét}^{r}(F; ℤ/p(∗)) ≅ v_{1}^{−1}H_{mot}^{r}(F; ℤ/p(∗))  (4)

with v_{1} ∈ H_{mot}^{0}(F; ℤ/p(p−1)). We refer to the aspects (3) and (4) of the Beilinson– Lichtenbaum conjectures as the motivic truncation property for the field F.

The following prediction expresses a similar conceptual description of K(B) for some strictly commutative ring spectra, and should in particular apply for B = ℓ_{p}, L_{p}, ku_{p} and KU_{p}.

Conjecture 5.1. For purely v_{1}-periodic strictly commutative ring spectra B there is a spectral sequence

E_{s,t}^{2} = H_{mot}^{−s}(B; 𝔽_{p^{2}}(t/2)) ⟹ V (1)_{s+t}K(B) ,

converging for s + t sufficiently large.

Here H_{mot}^{∗} denotes a currently undefined form of motivic cohomology for strictly commutative ring spectra. The coefficient 𝔽_{p^{2}}(t/2) may be interpreted as V (1)_{t}E_{2}, where E_{2} is the Lubin–Tate ring spectrum [31] with π_{∗}E_{2} = 𝕎𝔽_{p^{2}}[[u_{1}]][u^{±1}].

More generally one might consider purely v_{n}-periodic ring spectra B, replace V (1) by any type n + 1 finite spectrum F (n + 1), see [39], and replace V (1)_{t}E_{2} and V (1)_{s+t}K(B) by F (n + 1)_{t}E_{n+1} and F (n + 1)_{s+t}K(B), respectively. Example 5.2. Based on the detailed calculations behind Theorem 4.2, it is fairly evident that the E^{2}-term for the spectral sequence conjectured to converge to V (1)_{∗}K(ℓ_{p}) will be concentrated in the four columns −3 ≤ s ≤ 0, and that the free 𝔽_{p}[v_{2}]-module generators are located in the groups H_{mot}^{r}(ℓ_{p}; 𝔽_{p^{2}}(m)) where 0 ≤ r ≤ 3 and r ≤ m < r + p^{2} + p − 1. This presumes that the spectral sequence collapses at the E^{2}-term, for p ≥ 5. In addition, there is a sporadic copy of 𝔽_{p} in V (1)_{2p−3}K(ℓ_{p}).

••  2p^{2} • •  •  − •  •  − •  •  − •  • ••  ••  2p •  • •  • •  • ••  •  0 −3  −2  −1  0  s\t

Figure 1. 𝔽_{p}[v_{2}]-generators of E_{s,t}^{2} = H_{mot}^{−s}(L_{p}; 𝔽_{p^{2}}(t/2)) ⟹ V (1)_{s+t}K(L_{p})

The class v_{2} ∈ V (1)_{2p^{2}−2}K(ℓ_{p}) is represented in bidegree (s, t) = (0, 2p^{2} − 2), corresponding to (r, m) = (0, p^{2} − 1). The presence of 𝔽_{p}[v_{2}]-module generators in the range r + p^{2} − 1 ≤ m shows that H_{mot}^{r}(ℓ_{p}; 𝔽_{p^{2}}(∗)) is not isomorphic to v_{2}^{−1}H_{mot}^{r}(ℓ_{p}; 𝔽_{p^{2}}(∗)) in several bigradings (r, m) with r ≤ m < r + p. In other words, the motivic truncation property fails for ℓ_{p}. However, this is to be expected, since ℓ_{p} has the residue ring spectrum Hℤ_{p} and should not behave as a field.

Example 5.3. Turning instead to V (1)_{∗}K(L_{p}), as determined from V (1)_{∗}K(ℤ_{p}) and V (1)_{∗}K(ℓ_{p}) by the homotopy cofiber sequence (2), free 𝔽_{p}[v_{2}]-module generators for the E^{2}-term in Conjecture 5.1 would be concentrated in the groups H_{mot}^{r}(L_{p}; 𝔽_{p^{2}}(m)) with 0 ≤ r ≤ 3 and r ≤ m < r + p^{2} − 1. The (s, t)-bidegrees of these 4p + 4 generators is displayed for p = 5 in Figure 1, lying in a fundamental domain in the shape of a parallelogram, of width 3 and height 2p^{2} −2. In addition, there are sporadic copies of 𝔽_{p} in V (1)_{2p−3}K(L_{p}) and V (1)_{2p−2}K(L_{p}).

In this case, the motivic truncation property for L_{p} is perfectly satisfied, in the sense that

H_{mot}^{r}(L_{p}; 𝔽_{p^{2}}(m)) ≅ \begin{cases}H_{ét}^{r}(L_{p}; 𝔽_{p^{2}}(m)) & for 0 ≤ r ≤ m, \\ 0 & otherwise, \end{cases}

where, by definition,

H_{ét}^{r}(L_{p}; 𝔽_{p^{2}}(∗)) = v_{2}^{−1}H_{mot}^{r}(L_{p}; 𝔽_{p^{2}}(∗))

is free over the Laurent polynomial ring 𝔽_{p}[v_{2}^{±1}] on the same generators as in Figure 1. The (additive) Euler characteristic

χ(L_{p}; 𝔽_{p^{2}}(m)) = ∑_{r=0}^{3}(−1)^{r} dim_{𝔽_{p}} H_{mot}^{r}(L_{p}; 𝔽_{p^{2}}(m))

is zero for each m, cf. [76, Thm. 2.2]. To the eyes of algebraic K-theory and the hypothetical motivic cohomology, the strictly commutative ring spectrum L_{p} behaves much like a brave new field. We discuss the role of its (non-commutative) residue ring L/p in the next section.

The étale cohomology of a p-adic field F satisfies local Tate–Poitou duality [76, Thm. 2.1]. In the case of mod p coefficients, this is a perfect pairing

H_{ét}^{r}(F; ℤ/p(m)) ⊗ H_{ét}^{2−r}(F; ℤ/p(1 − m)) ⟶^{∪} H_{ét}^{2}(F; ℤ/p(1)) ≅ ℤ/p

for each r and m. For general p-power torsion coefficients there is a perfect pairing taking values in the larger group H_{ét}^{2}(F; ℤ/p^{∞}(1)) ≅ ℤ/p^{∞}, cf. [72, p. 130]. The multiplicative structure on V (1)_{∗}K(L_{p}) is compatible with an algebra structure on H_{mot}^{∗}(L_{p}; 𝔽_{p^{2}}(∗)) such that the resulting multiplicative structure on H_{ét}^{∗}(L_{p}; 𝔽_{p^{2}}(∗)) also satisfies arithmetic duality. This can be seen as a rotational symmetry about (s, t) = (−3/2, p + 1) in the variant of Figure 1 where v_{2} has been inverted.

Conjecture 5.4. For finite extensions B of L_{p} there is a perfect pairing

H_{ét}^{r}(B; 𝔽_{p^{2}}(m)) ⊗ H_{ét}^{3−r}(B; 𝔽_{p^{2}}(p + 1 − m)) ⟶^{∪} H_{ét}^{3}(B; 𝔽_{p^{2}}(p + 1)) ≅ ℤ/p

for each r and m.

Remark 5.5. The dependence of the twist in 𝔽_{p^{2}}(p + 1) on the prime p may be an artifact of the passage to mod p and v_{1} coefficients. Let E_{2}/(p^{∞}, u_{1}^{∞}) be the E_{2}-module spectrum defined by the homotopy cofiber sequences E_{2} → p^{−1}E_{2} → E_{2}/p^{∞} and E_{2}/p^{∞} → u_{1}^{−1}E_{2}/p^{∞} → E_{2}/(p^{∞}, u_{1}^{∞}). Its homotopy groups π_{∗}E_{2}/(p^{∞}, u_{1}^{∞}) = 𝕎𝔽_{p^{2}}[[u_{1}]]/(p^{∞}, u_{1}^{∞})[u^{±1}] are Pontryagin dual to those of E_{2}. Then

𝔽_{p^{2}}(p + 1) = V (1)_{2p+2}E_{2} ≅ V (1)_{2p+3}(E_{2}/p^{∞}) ≅ V (1)_{2p+4}E_{2}/(p^{∞}, u_{1}^{∞})

and

V (1)_{2p+4}E_{2}/(p^{∞}, u_{1}^{∞}) ⊂ (S/p)_{5}E_{2}/(p^{∞}, u_{1}^{∞})
⊂ π_{4}E_{2}/(p^{∞}, u_{1}^{∞}) = 𝕎𝔽_{p^{2}}[[u_{1}]]/(p^{∞}, u_{1}^{∞})(2) .

The conjectured arithmetic duality for mod p and v_{1} coefficients may be a special case of a duality for p- and u_{1}-power torsion coefficients, taking values in

H_{ét}^{3}(B; 𝕎𝔽_{p^{2}}[[u_{1}]]/(p^{∞}, u_{1}^{∞})(2)) .

It would be desirable to find a canonical identification of this group, like the Hasse invariant in the classical case of p-adic fields and Kato’s work [43] on the Galois cohomology of higher-dimensional local fields.

6. Fraction fields and ramified extensions

The étale cohomology of a field is, by construction, the same as its Galois cohomology, i.e., the continuous group cohomology of its absolute Galois group. There is no such direct description of H_{ét}^{r}(L_{p}; 𝔽_{p^{2}}(m)), since according to Baker–Richter [8] the maximal connected pro-Galois extension of L_{p} is the composite

L_{p} ⟶^{ϕ} KU_{p} ⟶ KU_{p}^{nr} ,

where π_{∗}(KU_{p}^{nr}) = 𝕎𝔽̄_{p}[u^{±1}]. The unramified extensions of π_{0}(KU_{p}) = ℤ_{p} are spectrally realized, using the methods of Schwänzl–Vogt–Waldhausen [71, Thm. 3] or Goerss–Hopkins–Miller [31], but the associated Galois group only has p-cohomo-logical dimension 1, whereas L_{p} would have p-cohomological dimension 3. Likewise, the maximal connected pro-Galois extension of E_{n} is E_{n}^{nr}, with π_{∗}(E_{n}^{nr}) = 𝕎𝔽̄[[u_{1}, …, u_{n−1}]][u^{±1}], of p-cohomological dimension 1 over E_{n} and L̂_{n}E(n).

To allow for ramification at p, one might simply invert that prime. However, the resulting strictly commutative ring spectrum p^{−1}L_{p}, with π_{∗}(p^{−1}L_{p}) = ℚ_{p}[v_{1}^{±1}], is an algebra over p^{−1}S_{p} = Hℚ_{p}, so V (1)_{∗}K(p^{−1}L_{p}) is an algebra over V (1)_{∗}K(ℚ_{p}), where v_{2} acts trivially. Hence H_{ét}^{r}(p^{−1}L_{p}; 𝔽_{p^{2}}(∗)) would be zero.

A milder form of localization may be appropriate. By Waldhausen’s localization theorem [82], the homotopy fiber of K(L_{p}) → K(p^{−1}L_{p}) is given by the algebraic K-theory of the category with cofibrations of finite cell L_{p}-modules with p-power torsion homotopy, equipped with the usual weak equivalences. We might instead step back to a category with cofibrations of coherent L/p^{ν}-modules (i.e., having degreewise finite homotopy groups, see Barwick–Lawson [9]), for some natural number ν, and suppose that these have the same algebraic K-theory as the category with cofibrations of finite cell L/p-modules. Here L/p = K(1) is the first Morava K-theory, which by Angeltveit [2] is a strict ring spectrum, but not strictly commutative. By Davis–Lawson [22, Cor. 6.4] the tower {L/p^{ν}}_{ν} is as commutative as possible in the category of pro-spectra:

L/p ⟵ {L/p^{ν}}_{ν} ⟵ L_{p} ⟶ p^{−1}L_{p} .

Definition 6.1. Let K(ff L_{p}) be defined by the homotopy cofiber sequence

K(L/p) ⟶^{i_{∗}} K(L_{p}) ⟶ K(ff L_{p}) ,

where i_{∗} is the transfer map associated to i: L_{p} → L/p.

We think of K(ff L_{p}) as the algebraic K-theory of a hypothetical fraction field of L_{p}, intermediate between L_{p} and p^{−1}L_{p}, and similar to the 2-dimensional local field ℚ_{p}((u)). Its mod p and v_{1} homotopy groups can be calculated using the following result, in combination with the homotopy cofiber sequence K(𝔽_{p}) → K(ℓ/p) → K(L/p).

Theorem 6.2 (Ausoni–Rognes [7, Thm. 7.6, Thm. 7.7]). V (1)_{∗}T C(ℓ/p; p) is a finitely generated free 𝔽_{p}[v_{2}]-module on 2p^{2} − 2p + 8 generators, which are located in degrees −1 ≤ ∗ ≤ 2p^{2} + 2p − 2. There is an exact sequence of 𝔽_{p}[v_{2}]-modules

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

Example 6.3. The expected E^{2}-term for a spectral sequence

E_{s,t}^{2} = H_{mot}^{−s}(ff L_{p}; 𝔽_{p^{2}}(t/2)) ⟹ V (1)_{s+t}K(ff L_{p})

is displayed for p = 5 in Figure 2. In addition there are four sporadic copies of 𝔽_{p}, in degrees 2p − 3, 2p − 2, 2p − 2 and 2p − 1. The motivic truncation properties for ff L_{p}, analogous to (3) and (4), are clearly visible, and conjecturally there is now a perfect arithmetic duality pairing

H_{ét}^{r}(ff L_{p}; 𝔽_{p^{2}}(m)) ⊗ H_{ét}^{3−r}(ff L_{p}; 𝔽_{p^{2}}(2 − m)) ⟶^{∪} H_{ét}^{3}(ff L_{p}; 𝔽_{p^{2}}(2)) ≅ ℤ/p .

After such localization of L_{p} away from L/p, it may be possible to construct enough Galois extensions of ff L_{p} to realize its v_{2}-localized motivic cohomology as continuous group cohomology

H_{ét}^{r}(ff L_{p}; 𝔽_{p^{2}}(m)) ≅ H_{gp}^{r}(G_{ff L_{p}}; 𝔽_{p^{2}}(m)) ,

for an absolute Galois group G_{ff L_{p}} of p-cohomological dimension 3, corresponding to some maximal extension ff L_{p} → Ω_{1}. If each Galois extension of ℚ_{p} can be lifted to an extension of ff L_{p}, we get a short exact sequence

1 → I_{v_{1}} ⟶ G_{ff L_{p}} ⟶ G_{ℚ_{p}} → 1 ,

with I_{v_{1}} the inertia group over (v_{1}). Here G_{ℚ_{p}} has p-cohomological dimension 2, and I_{v_{1}} will have p-cohomological dimension 1.

In the less structured setting of ring spectra up to homotopy it is possible to construct totally ramified extensions of KU_{p}, complementary to the unramified extension KU_{p}^{nr}, without inverting p. Torii [77, Thm. 2.5] shows that for each r ≥ 1 the homotopy cofiber

F (Bℤ/p_{+}^{r−1}, KU_{p}) ⟶^{τ_{r}^{#}} F (Bℤ/p_{+}^{r}, KU_{p}) ⟶ KU_{p}[ζ_{p^{r}}] ,

of the map of function spectra induced by the stable transfer map τ_{r}: Bℤ/p_{+}^{r} → Bℤ/p_{+}^{r−1}, is a ring spectrum up to homotopy with π_{∗}KU_{p}[ζ_{p^{r}}] ≅ ℤ_{p}[ζ_{p^{r}}][u^{±1}], where ζ_{p^{r}} denotes a primitive p^{r}-th root of unity. (He has similar realization

•  •••  2p^{2} •  •• •  • •  • •  • •  •  − •  • •  • •  • •  • •  •  − •  • •  • •  • •  • •  •  − •  • •  • •  • •  • •  •  2p •  • •  • ••  • •••  •  0 −3  −2  −1  0  s\t

Figure 2. 𝔽_{p}[v_{2}]-generators of E_{s,t}^{2} = H_{mot}^{−s}(ff L_{p}; 𝔽_{p^{2}}(t/2)) ⟹ V (1)_{s+t}K(ff L_{p})

results in the K(n)-local category.) However, it does not make sense to talk about the algebraic K-theory of a ring spectrum up to homotopy, so these constructions are only helpful if they can be made strict.

It is not possible to realize KU_{p}[ζ_{p^{r}}] as a strictly commutative ring spectrum. Angeltveit [1, Rem. 5.18] uses the identity ψ^{p}(x) = x^{p} + pθ(x) among power operations in π_{0} of a K(1)-local strictly commutative ring spectrum to show that if −p admits a k-th root in such a ring, with k ≥ 2, then p is invertible in that ring. For r = 1 we have ℤ_{p}[ζ_{p}] = ℤ_{p}[ξ] where ξ^{p−1} = −p, so for p odd this proves that adjoining ζ_{p} to KU_{p} in a strictly commutative context will also invert p.

It is, however, possible to adjoin ζ_{p} to π_{0} of the connective cover ku_{p}, in the category of strictly commutative ring spectra, without fully inverting p. Instead, one must make some positive power of the Bott element u ∈ π_{2}(ku_{p}) singly divisible by p. If one thereafter inverts u, it follows that p has also become invertible. To achieve this, we modify Torii’s construction for r = 1 by replacing the transfer map with a norm map. This leads to the G-Tate construction

B^{tG} = t_{G}(B)^{G} = [ẼG ∧ F (EG_{+}, i_{∗}B)]^{G}

for a spectrum B with G-action, cf. Greenlees–May [35, p. 3]. This construction preserves strictly commutative ring structures, see McClure [50, Thm. 1].

Example 6.4. Let

KU′_{p}[ξ] = (ku_{p})^{tℤ/p}

denote the ℤ/p-Tate construction on the spectrum ku_{p} with trivial ℤ/p-action, and let ku′_{p}[ξ] = KU′_{p}[ξ][0, ∞) be its connective cover. Additively, these are generalized Eilenberg–Mac Lane spectra, cf. Davis–Mahowald [23] and [35, Thm. 13.5]. Multiplicatively, π_{∗}(KU′_{p}[ξ]) ≅ ℤ_{p}[ξ][v^{±1}] where p + ξ^{p−1} = 0 and |v| = 2. Furthermore,

π_{∗}(ku′_{p}[ξ]) ≅ ℤ_{p}[ξ][v] ,

and a morphism ku_{p} → ku′_{p}[ξ] of strictly commutative ring spectra induces the ring homomorphism ℤ_{p}[u] → ℤ_{p}[ξ][v] that maps u to ξ · v. There is no multiplicative morphism KU_{p} → KU′_{p}[ξ], but

KU_{p} ∧_{ku_{p}} ku′_{p}[ξ] = u^{−1}ku′_{p}[ξ] ≃ KUℚ_{p}(ξ) = KUℚ_{p}(ζ_{p})

is a totally ramified extension of KUℚ_{p} = p^{−1}KU_{p}. We get a diagram of strictly commutative ring spectra

\xymatrix{(Hℤ_{p})^{tℤ/p} & KU′_{p}[ξ] \ar[l] \ar[r] & KUℚ_{p}(ξ) \\ (Hℤ_{p})^{tℤ/p}[0, ∞) \ar[d] \ar[u] & ku′_{p}[ξ] \ar[l] \ar[r] \ar[d] \ar[u] & kuℚ_{p}(ξ) \ar[d] \ar[u] \\ Hℤ/p & Hℤ_{p}[ξ] \ar[l] \ar[r] & Hℚ_{p}(ξ)}

with horizontal maps reducing modulo or inverting ξ, and vertical maps reducing modulo or inverting v. Here π_{∗}((Hℤ_{p})^{tℤ/p}) ≅ Ĥ^{−∗}(ℤ/p; ℤ_{p}) = ℤ/p[v^{±1}]. We view ku′_{p}[ξ] as an integral model for a 2-dimensional local field close to KUℚ_{p}(ξ), but note that ku′_{p}[ξ] is not finite as a ku_{p}-module.

Example 6.5. Let (ℤ/p)^{×} act on the group ℤ/p by multiplication, hence also on the ℤ/p-Tate construction KU′_{p}[ξ] = (ku_{p})^{tℤ/p}. Let

KU′_{p} = (KU′_{p}[ξ])^{h(ℤ/p)^{×}}

be the homotopy fixed points, and let ku′_{p} = KU′_{p}[0, ∞) be its connective cover. These are strictly commutative ring spectra, with π_{∗}(KU′_{p}) ≅ ℤ_{p}[u, w^{±1}]/(pw + u^{p−1}) where |w| = 2p − 2, and π_{∗}(ku′_{p}) ≅ ℤ_{p}[u, w]/(pw + u^{p−1}). Multiplicative morphisms ku_{p} → ku′_{p} → ku′_{p}[ξ] induce the inclusions ℤ_{p}[u] → ℤ_{p}[u, w]/(pw + u^{p−1}) → ℤ_{p}[ξ][v] where w maps to v^{p−1}.

The morphism KU′_{p} → KU′_{p}[ξ] is a (ℤ/p)^{×}-Galois extension in the sense of [63]. The morphism ku′_{p} → ku′_{p}[ξ] becomes (ℤ/p)^{×}-Galois after inverting p or w. It remains to be determined whether V (1)_{∗}K(ku′_{p}) remains purely v_{2}-periodic, i.e., whether the multiplicative approximation ku_{p} → ku′_{p} counters the additive splitting of ku′_{p} as a sum of suspended Eilenberg–Mac Lane spectra.

Example 6.6. More generally, for r ≥ 1 let G = ℤ/p^{r}, let 𝒫 be the family of proper subgroups of G, and let KU_{G}[0, ∞) be the “brutal” truncation of G-equivariant periodic K-theory, cf. [34, p. 129]. Define

KU′_{p}[ζ_{p^{r}}] = (KU_{G}[0, ∞))^{t𝒫} = [Ẽ𝒫 ∧ F (E𝒫_{+}, KU_{G}[0, ∞))]^{G}

to be the 𝒫-Tate construction, as in Greenlees–May [35, §17], and let ku′_{p}[ζ_{p^{r}}] be its connective cover. Then π_{∗}(KU′_{p}[ζ_{p^{r}}]) ≅ ℤ_{p}[ζ_{p^{r}}][v^{±1}] and

π_{∗}(ku′_{p}[ζ_{p^{r}}]) ≅ ℤ_{p}[ζ_{p^{r}}][v] .

The map (KU_{G}[0, ∞))^{t𝒫} → (KU_{G})^{t𝒫} induces the inclusion ℤ_{p}[ζ_{p^{r}}] ⊂ ℚ_{p}[ζ_{p^{r}}] in each even degree. To prove this, one can compute Amitsur–Dress homology for the family 𝒫, use the generalized Tate spectral sequence from [35, §22], and then compare with the calculation in [35, §19] of the periodic case. The cyclotomic extension ℤ_{p}[ζ_{p^{r}}] arises as (R(G)/J′𝒫)_{J𝒫}^{∧} , where R(G) is the representation ring, J𝒫 is the kernel of the restriction map R(G) → R(H), and J′𝒫 is the image of the induction map R(H) → R(G), where H is the index p subgroup in G.

7. Logarithmic ring spectra

The heuristics from the last two sections suggest that we should attempt to construct ramified finite extensions B → C of strictly commutative ring spectra B like ℓ_{p}, ku_{p} and e_{n}. The Goerss–Hopkins–Miller obstruction theory [31] for strictly commutative B-algebra structures on such spectra C has vanishing obstruction groups in the case of unramified extensions, but appears to be less useful in the case of ramification over (p), due to the presence of nontrivial (topological) André– Quillen cohomology groups [11].

The same heuristics also suggest that we should approach the extension problem by passing to mildly local versions of B, intermediate between B and p^{−1}B. In arithmetic algebraic geometry, one such intermediary is provided by logarithmic geometry, cf. Kato [44]. An affine pre-log scheme (Spec R, M) is a scheme Spec R, a commutative monoid M, and a homomorphism α: M → (R, ·) to the underlying multiplicative monoid of R. More precisely, M and α live étale locally on Spec R. In this wider context, there is a factorization

Spec R[M^{−1}] ⟶ (Spec R, M) ⟶ Spec R

of the natural inclusion, and the right-hand map is often a well-behaved proper replacement for the composite open immersion. Logarithmic structures on valuation rings in p-adic fields were successfully used by Hesselholt–Madsen [38] to analyze the topological cyclic homology and algebraic K-theory of these classical rings.

A theory of logarithmic structures on strictly commutative ring spectra was started by the author in [64], and developed further in joint work with Sagave and Schlichtkrull. To present it, we take the category CSp^{Σ} of commutative symmetric ring spectra [40], with the positive stable model structure [48, §14], as our model for strictly commutative ring spectra.

By the graded underlying space of a symmetric spectrum A we mean a diagram

Ω^{J} (A): (n_{1}, n_{2}) ⟼ Ω^{n_{2}}A_{n_{1}}

of spaces, where (n_{1}, n_{2}) ranges over the objects in a category J. We call such a diagram a J -space. Following Sagave, the natural category J to consider turns out to be isomorphic to Quillen’s construction Σ^{−1}Σ, where Σ is the permutative groupoid of finite sets and bijections. Its nerve BJ ≅ B(Σ^{−1}Σ) is homotopy equivalent to QS^{0} = Ω^{∞}S. For any J -space X, the homotopy colimit X_{hJ} = hocolim_{J} X is augmented over BJ, so we say that X is QS^{0}-graded. A map X → Y of J -spaces is called a J -equivalence if the induced map X_{hJ} → Y_{hJ} is a weak equivalence.

If A is a commutative symmetric ring spectrum, then Ω^{J} (A) is a commutative monoid with respect to a convolution product in the category of J -spaces. The category CS^{J} of commutative J -space monoids has a positive projective model structure [70, §4], with the J -equivalences as the weak equivalences, and is Quillen equivalent to a category of E_{∞} spaces over BJ. The functor Ω^{J} : CSp^{Σ} → CS^{J} admits a left adjoint M ↦ S^{J} [M], and (S^{J} [−], Ω^{J}) is a Quillen adjunction.

There is a commutative submonoid of graded homotopy units ι: GL_{1}^{J} (A) ⊂ Ω^{J} (A). A pre-log ring spectrum (A, M, α) is a commutative symmetric ring spectrum A with a pre-log structure (M, α), i.e., a commutative J -space monoid M and a map α: M → Ω^{J} (A) in CS^{J}. If the pullback α^{−1}(GL_{1}^{J} (A)) → GL_{1}^{J} (A) is a J -equivalence we call (M, α) a log structure and (A, M, α) a log ring spectrum. We often omit α from the notation.

In order to classify extensions (A, M) → (B, N) of pre-log ring spectra, one is led to study infinitesimal deformations and derivations in this category. Derivations are corepresented by a logarithmic version TAQ(A, M) of topological André– Quillen homology, defined by a pushout

\xymatrix{A ∧_{S^{J} [M]} TAQ(S^{J} [M]) \ar[r]^{ψ} \ar[d]_{α} & A ∧ γ(M) \ar[d] \\ TAQ(A) \ar[r] & TAQ(A, M)}

of A-module spectra, cf. [64, Def. 11.19] and [69, Def. 5.20]. Here TAQ(A) is the ordinary topological André–Quillen homology, as defined by Basterra [10], and γ(M) is the connective spectrum associated to the E_{∞} space M_{hJ}. A morphism (A, M) → (B, N) is formally log étale if B ∧_{A} TAQ(A, M) → TAQ(B, N) is an equivalence.

Let j: e → E be a fibration of commutative symmetric ring spectra. The direct image of the trivial log structure on E is the log structure (j_{∗}GL_{1}^{J} (E), α) on e given by the pullback

\xymatrix{j_{∗}GL_{1}^{J} (E) \ar[r] \ar[d]_{α} & GL_{1}^{J} (E) \ar[d]^{ι} \\ Ω^{J} (e) \ar[r]^{Ω^{J} (j)} & Ω^{J} (E)}

in CS^{J}. Applying this natural construction to the vertical maps in (1), we get the following example. Note that ℓ_{p} → ku_{p} is not étale, while L_{p} → KU_{p} is Δ-Galois, hence étale.

Theorem 7.1 (Sagave [69, Thm. 6.1]). The morphism

ϕ: (ℓ_{p}, j_{∗}GL_{1}^{J} (L_{p})) ⟶ (ku_{p}, j_{∗}GL_{1}^{J} (KU_{p}))

is log étale.

In order to approximate algebraic K-theory, one is likewise led to study logarithmic topological Hochschild homology and logarithmic topological cyclic homology. The former is defined by a pushout

\xymatrix{S^{J} [B^{cy}(M)] \ar[r]^{ρ} \ar[d]_{α} & S^{J} [B^{rep}(M)] \ar[d] \\ T HH(A) \ar[r] & T HH(A, M)}

in CSp^{Σ}, cf. [65, §4]. Here T HH(A) is the ordinary topological Hochschild homology of A, given by the cyclic bar construction of A in CSp^{Σ}. The cyclic bar construction B^{cy}(M) is formed in CS^{J}, and is naturally augmented over M. The replete bar construction B^{rep}(M) can be viewed as a fibrant replacement of B^{cy}(M) over M in a group completion model structure on CS^{J}, cf. [68, Thm. 1.6], but also has a more direct description as the (homotopy) pullback in the right hand square below:

\xymatrix{B^{cy}(M) \ar[r]^{ρ} \ar[d]_{ϵ} & B^{rep}(M) \ar[r] \ar[d] & B^{cy}(M^{gp}) \ar[d]^{ϵ} \\ M \ar[r]^{=} & M \ar[r]^{η} & M^{gp} .}

Here η: M → M^{gp} is a group completion in CS^{J}, which means that (M^{gp})_{hJ} is a group completion of the E_{∞} space M_{hJ}. The role of repletion in homotopy theory is similar to that of working within the subcategory of fine and saturated logarithmic structures in the discrete setting [44, §2].

A morphism (A, M) → (B, N) is formally log thh-étale if B ∧_{A} T HH(A, M) → T HH(B, N) is an equivalence. The following theorem strengthens the previous result.

Theorem 7.2 (Rognes–Sagave–Schlichtkrull [66, Thm. 1.5]). The morphism

ϕ: (ℓ_{p}, j_{∗}GL_{1}^{J} (L_{p})) ⟶ (ku_{p}, j_{∗}GL_{1}^{J} (KU_{p}))

is log thh-étale.

Remark 7.3. These results harmonize with the classical correspondence between tamely ramified extensions and log étale extensions. By Noether’s theorem [53], tame ramification corresponds locally to the existence of a normal basis. This conforms with the observation that ku_{p} is a retract of a finite cell ℓ_{p}[Δ]-module, so that ℓ_{p} → ku_{p} is tamely ramified. By contrast, ku_{2} is not a retract of a finite cell ko_{2}[C_{2}]-module, e.g. because (ku_{2})^{tC_{2}} is nontrivial, so ko_{2} → ku_{2} is wildly ramified.

We say that a commutative symmetric ring spectrum E is d-periodic if d is the minimal positive integer such that π_{∗}(E) contains a unit in degree d.

Theorem 7.4 (Rognes–Sagave–Schlichtkrull [65, Thm. 1.5]). Let E in CSp^{Σ} be d-periodic, with connective cover j: e → E. There is a natural homotopy cofiber sequence

T HH(e) ⟶^{ρ} T HH(e, j_{∗}GL_{1}^{J} (E)) ⟶^{∂} ΣT HH(e[0, d)) ,

where e[0, d) is the (d − 1)-th Postnikov section of e.

These results allow us to realize the strategy outlined in [3, §10] to compute the V (1)-homotopy of T HH(ku_{p}) by way of T HH(ℓ_{p}), T HH(ℓ_{p}, j_{∗}GL_{1}^{J} (L_{p})) and T HH(ku_{p}, j_{∗}GL_{1}^{J} (KU_{p})). The details are given in [66, §7, §8].

When e[0, d) = Hπ_{0}(e) with π_{0}(e) regular, Blumberg–Mandell [15, Thm. 4.2.1] have constructed a map of horizontal homotopy cofiber sequences

\xymatrix{K(π_{0}(e)) \ar[r]^{i_{∗}} \ar[d] & K(e) \ar[r]^{j^{∗}} \ar[d] & K(E) \ar[d] \\ T HH(π_{0}(e)) \ar[r]^{i_{∗}} & T HH(e) \ar[r]^{j^{∗}} & W T HH^{Γ}(e|E)}

where the vertical arrows are trace maps.

Conjecture 7.5. There is an equivalence of cyclotomic spectra

T HH(e, j_{∗}GL_{1}^{J} (E)) ≃ W T HH^{Γ}(e|E) ,

compatible with the maps from T HH(e) and to ΣT HH(π_{0}(e)).

The author hopes that a logarithmic analog of the Goerss–Hopkins–Miller obstruction theory [31] can be developed to classify log extensions of log ring spectra,

22 John Rognes

and that in the case of log étale extensions the obstruction groups will vanish in such a way as to enable the construction of interesting examples. The underlying strictly commutative ring spectra should then provide novel examples of tamely ramified extensions, and realize a larger part of motivic cohomology as a case of Galois cohomology.

References

[1] Vigleik Angeltveit, Topological Hochschild homology and cohomology of A_{∞} ring spectra, Geom. Topol. 12 (2008), no. 2, 987–1032.

[2] , Uniqueness of Morava K-theory, Compos. Math. 147 (2011), no. 2, 633–648.

[3] Christian Ausoni, Topological Hochschild homology of connective complex K-theory, Amer. J. Math. 127 (2005), no. 6, 1261–1313.

[4] , On the algebraic K-theory of the complex K-theory spectrum, Invent. Math. 180 (2010), no. 3, 611–668.

[5] Christian Ausoni and John Rognes, Algebraic K-theory of topological K-theory, Acta Math. 188 (2002), no. 1, 1–39.

[6] , The chromatic red-shift in algebraic K-theory, Monographie de L’Enseignement Mathématique 40 (2008), 13–15.

[7] , Algebraic K-theory of the first Morava K-theory, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 4, 1041–1079.

[8] Andrew Baker and Birgit Richter, Galois extensions of Lubin–Tate spectra, Homology, Homotopy Appl. 10 (2008), no. 3, 27–43.

[9] Clark Barwick and Tyler Lawson, Regularity of structured ring spectra and localization in K-theory (2014), available at arXiv:1402.6038v2[math.KT].

[10] M. Basterra, André-Quillen cohomology of commutative S-algebras, J. Pure Appl. Algebra 144 (1999), no. 2, 111–143.

[11] Maria Basterra and Birgit Richter, (Co-)homology theories for commutative (S-)algebras, Structured ring spectra, London Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 115–131.

[12] A. Berglund and I. Madsen, Rational homotopy theory of automorphisms of highly connected manifolds (2014), available at arXiv:1401.4096[math.AT].

[13] Spencer Bloch, Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), no. 3, 267– 304.

[14] Andrew J. Blumberg and Michael A. Mandell, The localization sequence for the algebraic K-theory of topological K-theory, Acta Math. 200 (2008), no. 2, 155–179.

[15] , Localization for T HH(ku) and the topological Hochschild and cyclic homology of Waldhausen categories (2014), available at arXiv:1111.4003v3[math.KT].

[16] M. Bökstedt, W. C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), no. 3, 465–539.

[17] M. Bökstedt and I. Madsen, Topological cyclic homology of the integers, Astérisque 226 (1994), 7–8, 57–143. K-theory (Strasbourg, 1992).

[18] , 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.

[19] Marcel Bökstedt and Friedhelm Waldhausen, The map BSG → A(∗) → QS^{0}, Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 418–431.

[20] Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272 (1975).

[21] William Browder, Surgery on simply-connected manifolds, Springer-Verlag, New York, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65.

[22] Daniel G. Davis and Tyler Lawson, Commutative ring objects in pro-categories and generalized Moore spectra (2013), available at arXiv:1208.4519v3[math.AT].

[23] Donald M. Davis and Mark Mahowald, The spectrum (P ∧ bo)_{−∞}, Math. Proc. Cambridge Philos. Soc. 96 (1984), no. 1, 85–93.

[24] Bjørn Ian Dundas, Relative K-theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 223–242.

[25] W. G. Dwyer and S. A. Mitchell, On the K-theory spectrum of a ring of algebraic integers, K-Theory 14 (1998), no. 3, 201–263.

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

[27] F. T. Farrell and W. C. Hsiang, On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 325–337.

[28] F. T. Farrell and L. E. Jones, Stable pseudoisotopy spaces of compact non-positively curved manifolds, J. Differential Geom. 34 (1991), no. 3, 769–834.

[29] S. Galatius and O. Randal-Williams, Stable moduli spaces of high dimensional manifolds

(2012), available at arXiv:1201.3527v2[math.AT].

[30] Thomas Geisser and Marc Levine, The Bloch–Kato conjecture and a theorem of Suslin– Voevodsky, J. Reine Angew. Math. 530 (2001), 55–103.

[31] 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.

[32] Thomas G. Goodwillie, Relative algebraic K-theory and cyclic homology, Ann. of Math. (2) 124 (1986), no. 2, 347–402.

[33] , The differential calculus of homotopy functors, Proceedings of the International Congress of Mathematicians, Vol. II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, pp. 621– 630.

[34] J. P. C. Greenlees, Equivariant connective K-theory for compact Lie groups, J. Pure Appl. Algebra 187 (2004), no. 1-3, 129–152.

[35] J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178.

[36] A. E. Hatcher, Higher simple homotopy theory, Ann. of Math. (2) 102 (1975), no. 1, 101–137.

[37] Lars Hesselholt, On the Whitehead spectrum of the circle, Algebraic topology, Abel Symp., vol. 4, Springer, Berlin, 2009, pp. 131–184.

[38] Lars Hesselholt and Ib Madsen, On the K-theory of local fields, Ann. of Math. (2) 158 (2003), no. 1, 1–113.

[39] Michael J. Hopkins and Jeffrey H. Smith, Nilpotence and stable homotopy theory. II, Ann. of Math. (2) 148 (1998), no. 1, 1–49.

[40] Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208.

[41] W. C. Hsiang and B. Jahren, A note on the homotopy groups of the diffeomorphism groups of spherical space forms, Algebraic K-theory, Part II (Oberwolfach, 1980), Lecture Notes in Math., vol. 967, Springer, Berlin, 1982, pp. 132–145.

24 John Rognes

[42] Kiyoshi Igusa, The stability theorem for smooth pseudoisotopies, K-Theory 2 (1988), no. 1-2, vi+355.

[43] Kazuya Kato, Galois cohomology of complete discrete valuation fields, Algebraic K-theory, Part II (Oberwolfach, 1980), Lecture Notes in Math., vol. 967, Springer, Berlin, 1982, pp. 215–238.

[44] , Logarithmic structures of Fontaine–Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224.

[45] John R. Klein and John Rognes, The fiber of the linearization map A(∗) → K(Z), Topology 36 (1997), no. 4, 829–848.

[46] Ib Madsen and Michael Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture, Ann. of Math. (2) 165 (2007), no. 3, 843–941.

[47] Mark Mahowald and Stewart Priddy (eds.), Algebraic topology, Contemporary Mathematics, vol. 96, American Mathematical Society, Providence, RI, 1989.

[48] M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441–512.

[49] Randy McCarthy, Relative algebraic K-theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 197–222.

[50] J. E. McClure, E_{∞}-ring structures for Tate spectra, Proc. Amer. Math. Soc. 124 (1996), no. 6, 1917–1922.

[51] J. E. McClure and R. E. Staffeldt, On the topological Hochschild homology of bu. I, Amer. J. Math. 115 (1993), no. 1, 1–45.

[52] Robert E. Mosher, Some stable homotopy of complex projective space, Topology 7 (1968), 179–193.

[53] E. Noether, Normalbasis bei Körpern ohne höhere Verzweigung, J. Reine Angew. Math. 167 (1932), 147–152.

[54] Daniel Quillen, Higher algebraic K-theory. I, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973.

[55] , Higher algebraic K-theory, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., 1975, pp. 171–176.

[56] John Rognes, Characterizing connected K-theory by homotopy groups, Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 1, 99–102.

[57] , Trace maps from the algebraic K-theory of the integers (after Marcel Bökstedt), J. Pure Appl. Algebra 125 (1998), no. 1-3, 277–286.

[58] , The product on topological Hochschild homology of the integers with mod 4 coefficients, J. Pure Appl. Algebra 134 (1999), no. 3, 211–218.

[59] , Topological cyclic homology of the integers at two, J. Pure Appl. Algebra 134 (1999), no. 3, 219–286.

[60] , Algebraic K-theory of the two-adic integers, J. Pure Appl. Algebra 134 (1999), no. 3, 287–326.

[61] , Two-primary algebraic K-theory of pointed spaces, Topology 41 (2002), no. 5, 873– 926.

[62] , The smooth Whitehead spectrum of a point at odd regular primes, Geom. Topol. 7 (2003), 155–184.

[63] , Galois extensions of structured ring spectra. Stably dualizable groups, Mem. Amer. Math. Soc. 192 (2008), no. 898, viii+137.

[64] , Topological logarithmic structures, New topological contexts for Galois theory and algebraic geometry (BIRS 2008), Geom. Topol. Monogr., vol. 16, Geom. Topol. Publ., Coventry, 2009, pp. 401–544.

[65] John Rognes, Steffen Sagave, and Christian Schlichtkrull, Localization sequences for logarithmic topological Hochschild homology (2014), available at arXiv:1402.1317[math.AT].

[66] , Logarithmic topological Hochschild homology of topological K-theory spectra (2014).

[67] J. Rognes and C. Weibel, Two-primary algebraic K-theory of rings of integers in number fields, J. Amer. Math. Soc. 13 (2000), no. 1, 1–54. Appendix A by Manfred Kolster.

[68] Steffen Sagave, Spectra of units for periodic ring spectra and group completion of graded E_{∞} spaces (2013), available at arXiv:1111.6731v2[math.AT].

[69] , Logarithmic structures on topological K-theory spectra, Geom. Topol. 18 (2014), no. 1, 447–490.

[70] Steffen Sagave and Christian Schlichtkrull, Diagram spaces and symmetric spectra, Adv. Math. 231 (2012), no. 3-4, 2116–2193.

[71] R. Schwänzl, R. M. Vogt, and F. Waldhausen, Adjoining roots of unity to E_{∞} ring spectra in good cases—a remark, Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 245–249.

[72] J.-P. Serre, Local class field theory, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 128–161.

[73] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399.

[74] Christophe Soulé, On higher p-adic regulators, Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math., vol. 854, Springer, Berlin, 1981, pp. 372–401.

[75] Andrei Suslin and Vladimir Voevodsky, Bloch–Kato conjecture and motivic cohomology with finite coefficients, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 117–189.

[76] John Tate, Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 288–295.

[77] Takeshi Torii, Topological realization of the integer ring of local field, J. Math. Kyoto Univ. 38 (1998), no. 4, 781–788.

[78] Vladimir Voevodsky, Motivic cohomology with ℤ/2-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104.

[79] , On motivic cohomology with ℤ/l-coefficients, Ann. of Math. (2) 174 (2011), no. 1, 401–438.

[80] Friedhelm Waldhausen, Algebraic K-theory of topological spaces. I, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 35–60.

[81] , Algebraic K-theory of spaces, localization, and the chromatic filtration of stable homotopy, Algebraic topology, Aarhus 1982 (Aarhus, 1982), Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 173–195.

[82] , 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.

[83] Friedhelm Waldhausen, Bjørn Jahren, and John Rognes, Spaces of PL manifolds and categories of simple maps, Annals of Mathematics Studies, vol. 186, Princeton University Press, Princeton, NJ, 2013.

[84] C. T. C. Wall, Surgery on compact manifolds, Academic Press, London, 1970. London Mathematical Society Monographs, No. 1.

[85] Michael Weiss and Bruce Williams, Automorphisms of manifolds and algebraic K-theory. I, K-Theory 1 (1988), no. 6, 575–626.

[86] , Automorphisms of manifolds, Surveys on surgery theory, Vol. 2, Ann. of Math. Stud., vol. 149, Princeton Univ. Press, Princeton, NJ, 2001, pp. 165–220.

Department of Mathematics, University of Oslo, Norway E-mail: rognes@math.uio.no