K-theory Spectrum Research Articles

Global algebraic K‐theory

We introduce a global equivariant refinement of algebraic K-theory; here `global equivariant' refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global $\Omega$-spectrum that keeps track of genuine $G$-equivariant infinite loop spaces, for all finite groups $G$. The resulting global algebraic K-theory spectrum is a rigid way of packaging the representation K-theory, or `Swan K-theory' into one highly structured object.

Naive-commutative ring structure on rational equivariant K-theory for abelian groups

In this paper, we calculate the image of the connective and periodic rational equivariant complex K-theory spectrum in the algebraic model for naive-commutative ring G-spectra given by Barnes, Greenlees and Kędziorek for finite abelian G. Our calculations show that these spectra are unique as naive-commutative ring spectra in the sense that they are determined up to weak equivalence by their homotopy groups. We further deduce a structure theorem for module spectra over rational equivariant complex K-theory.

Suspension spectra of matrix algebras, the rank filtration, and rational noncommutative CW-spectra

In a companion paper (Arone et al. in Noncommutative CW-spectra as enriched presheaves on matrix algebras, arXiv:2101.09775, 2021) we introduced the stable infty -category of noncommutative CW-spectra, which we denoted mathtt {NSp}. Let {mathcal {M}} denote the full spectrally enriched subcategory of mathtt {NSp} whose objects are the non-commutative suspension spectra of matrix algebras. In Arone et al. (2021) we proved that mathtt {NSp} is equivalent to the infty -category of spectral presheaves on {mathcal {M}}. In this paper we investigate the structure of {mathcal {M}}, and derive some consequences regarding the structure of mathtt {NSp}. To begin with, we introduce a rank filtration of {mathcal {M}}. We show that the mapping spectra of {mathcal {M}} map naturally to the connective K-theory spectrum ku, and that the rank filtration of {mathcal {M}} is a lift of the classical rank filtration of ku. We describe the subquotients of the rank filtration in terms of spaces of direct-sum decompositions which also arose in the study of K-theory and of Weiss’s orthogonal calculus. We prove that the rank filtration stabilizes rationally after the first stage. Using this we give an explicit model of the rationalization of mathtt {NSp} as presheaves of rational spectra on the category of finite-dimensional Hilbert spaces and unitary transformations up to scaling. Our results also have consequences for the p-localization and the chromatic localization of {mathcal {M}}.

Brauer groups and Galois cohomology of commutative ring spectra

In this paper we develop methods for classifying Baker, Richter, and Szymik's Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing and classifying these algebras and their automorphisms with Goerss–Hopkins obstruction theory, and give descent-theoretic tools, applying Lurie's work on$\infty$-categories to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra$E$, we find that the ‘algebraic’ Azumaya algebras whose coefficient ring is projective are governed by the Brauer–Wall group of$\pi _0(E)$, recovering a result of Baker, Richter, and Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin–Tate spectra have either four or two Morita equivalence classes, depending on whether the prime is odd or even, that all algebraic Azumaya algebras over the complex K-theory spectrum$KU$are Morita trivial, and that the group of the Morita classes of algebraic Azumaya algebras over the localization$KU[1/2]$is$\mathbb {Z}/8\times \mathbb {Z}/2$. Using our descent results and an obstruction theory spectral sequence, we also study Azumaya algebras over the real K-theory spectrum$KO$which become Morita-trivial$KU$-algebras. We show that there exist exactly two Morita equivalence classes of these. The nontrivial Morita equivalence class is realized by an ‘exotic’$KO$-algebra with the same coefficient ring as$\mathrm {End}_{KO}(KU)$. This requires a careful analysis of what happens in the homotopy fixed-point spectral sequence for the Picard space of$KU$, previously studied by Mathew and Stojanoska.

Higher topological Hochschild homology of periodic complex K-theory

We describe the topological Hochschild homology of the periodic complex K-theory spectrum, THH(KU), as a commutative KU-algebra: it is equivalent to KU[K(Z,3)] and to F(ΣKUQ), where F is the free commutative KU-algebra functor on a KU-module. Moreover, F(ΣKUQ)≃KU∨ΣKUQ, a square-zero extension. In order to prove these results, we first establish that topological Hochschild homology commutes, as an algebra, with localization at an element.Then, we prove that THHn(KU), the n-fold iteration of THH(KU), i.e. Tn⊗KU, is equivalent to KU[G] where G is a certain product of integral Eilenberg-Mac Lane spaces, and to a free commutative KU-algebra on a rational KU-module. We prove that Sn⊗KU is equivalent to KU[K(Z,n+2)] and to F(ΣnKUQ). We describe the topological André-Quillen homology of KU as KUQ.

Twisted iterated algebraic K-theory and topological T-duality for sphere bundles

We introduce a periodic form of the iterated algebraic K-theory of ku, the (connective) complex K-theory spectrum, as well as a natural twisting of this cohomology theory by higher gerbes. Furthermore, we prove a form of topological T-duality for sphere bundles oriented with respect to this theory.

Derived ℓ-adic zeta functions

Let K0(Vk) be the Grothendieck group of k-varieties. Campbell and Zakharevich have constructed a higher algebraic K-theory spectrum K(Vk) such that π0K(Vk)=K0(Vk). In this paper we construct non-trivial classes in the higher homotopy groups of K(Vk) when k is finite or a subfield of C. To do this we give a recipe for lifting motivic measures K0(Vk)→K0(E) to maps of spectra K(Vk)→K(E). We consider two special cases: the classical local zeta function, thought of as a homomorphism K0(VFq)→K0(End(Qℓ)), and the compactly-supported Euler characteristic, thought of as a homomorphism K0(VC)→K0(Q). We use lifts of these motivic measures to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map S→K(Vk) is nontrivial in higher dimensions when k is finite or a subfield of C, and, moreover, that when k is finite this map is not surjective on higher homotopy groups.

On very effective hermitian K-theory

We argue that the very effective cover of hermitian K-theory in the sense of motivic homotopy theory is a convenient algebro-geometric generalization of the connective real topological K-theory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over the motivic Steenrod algebra, and that its motivic Adams and slice spectral sequences are amenable to calculations.

A multiplicative K-theoretic model of Voevodsky’s motivic K-theory spectrum

Voevodsky defined a motivic spectrum representing algebraic K-theory, and Panin, Pimenov, and Rondigs described its ring structure up to homotopy. We construct a motivic symmetric spectrum with a strict ring structure. Then we show that these spectra are stably equivalent and that their ring structures are compatible up to homotopy.

Waldhausen Additivity: classical and quasicategorical

We use a simplicial product version of Quillen’s Theorem A to prove classical Waldhausen Additivity of \(wS_\bullet \), which says that the “subobject” and “quotient” functors of cofiber sequences induce a weak equivalence \(wS_\bullet {\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}) \rightarrow wS_\bullet {\mathcal {A}}\times wS_\bullet {\mathcal {B}}\). A consequence is Additivity for the Waldhausen K-theory spectrum of the associated split exact sequence, namely a stable equivalence of spectra \({\mathbf {K}}({\mathcal {A}}) \vee {\mathbf {K}}({\mathcal {B}}) \rightarrow {\mathbf {K}}({\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}))\). This paper is dedicated to transferring these proofs to the quasicategorical setting and developing Waldhausen quasicategories and their sequences. We also give sufficient conditions for a split exact sequence to be equivalent to a standard one. These conditions are always satisfied by stable quasicategories, so Waldhausen K-theory sends any split exact sequence of pointed stable quasicategories to a split cofiber sequence. Presentability is not needed. In an effort to make the article self-contained, we recall all the necessary results from the theory of quasicategories, and prove a few quasicategorical results that are not in the literature.

The Beilinson regulator is a map of ring spectra

We prove that the Beilinson regulator, which is a map from K-theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of E∞-ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over R. The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual K-theory spectrum is the source. To prove this result we compute the space of maps from the motivic K-theory spectrum to the motivic spectrum that represents absolute Hodge cohomology using the motivic Snaith theorem. We identify those maps which admit an E∞-refinement and prove a uniqueness result for these refinements.

A higher chromatic analogue of the image of J

We prove a higher chromatic analogue of Snaith's theorem which identifies the K-theory spectrum as the localisation of the suspension spectrum of CP^\infty away from the Bott class; in this result, higher Eilenberg-MacLane spaces play the role of CP^\infty = K(Z,2). Using this, we obtain a partial computation of the part of the Picard-graded homotopy of the K(n)-local sphere indexed by powers of a spectrum which for large primes is a shift of the Gross-Hopkins dual of the sphere. Our main technical tool is a K(n)-local notion generalising complex orientation to higher Eilenberg-MacLane spaces. As for complex-oriented theories, such an orientation produces a one-dimensional formal group law as an invariant of the cohomology theory. As an application, we prove a theorem that gives evidence for the chromatic redshift conjecture.

The derived category of complex periodic K-theory localized at an odd prime

We prove that for an odd prime p, the derived category D(KU(p)) of the p-local complex periodic K-theory spectrum KU(p) is triangulated equivalent to the derived category of its homotopy ring π⁎KU(p). This implies that if p is an odd prime, the triangulated category D(KU(p)) is algebraic.

An explicit KO‐degree map and applications

The goal of this note is to study the analog in unstable A 1 -homotopy theory of the unit map from the motivic sphere spectrum to the Hermitian K-theory spectrum, that is, the degree map in Hermitian K-theory. We show that ‘Suslin matrices’, which are explicit maps from odd-dimensional split smooth affine quadrics to geometric models of the spaces appearing in Bott periodicity in Hermitian K-theory, stabilize in a suitable sense to the unit map. As applications, we deduce that K i M W ( F ) = G W i i ( F ) for i ⩽ 3 , which can be thought of as an extension of Matsumoto's celebrated theorem describing K 2 of a field. These results provide the first step in a program aimed at computing the sheaf π n A 1 ( A n ∖ 0 ) for n ⩾ 4 .

Equivariant algebraic K-theory of G-rings

A group action on the input ring or category induces an action on the algebraic K-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic K-theory is, for example, that the map of spectra with G-action induced by a G-map of G-rings is not equivariant. We define a version of equivariant algebraic K-theory which encodes a group action on the input in a functorial way to produce a genuine algebraic K-theory G-spectrum for a finite group G. The main technical work lies in studying coherent actions on the input category. A payoff of our approach is that it builds a unifying framework for equivariant topological K-theory, Atiyah’s Real K-theory, and existing statements about algebraic K-theory spectra with G-action. We recover the map from the Quillen–Lichtenbaum conjecture and the representational assembly map studied by Carlsson and interpret them from the perspective of equivariant stable homotopy theory.

The K-theory of assemblers

In this paper we introduce the notion of an assembler, which formally encodes “cutting and pasting” data. An assembler has an associated K-theory spectrum, in which π0 is the free abelian group of objects of the assembler modulo the cutting and pasting relations, and in which the higher homotopy groups encode further geometric invariants. The goal of this paper is to prove structural theorems about this K-theory spectrum, including analogs of Quillen's localization and dévissage theorems. We demonstrate the uses of these theorems by analyzing the assembler associated to the Grothendieck ring of varieties and the assembler associated to scissors congruence groups of polytopes.

The mod 2 Hopf ring for connective Morava K-theory

This paper examines the mod 2 homology of the spaces in the Omega-spectrum for connective Morava K-theory, i.e., the mod 2 Hopf ring for connective Morava K-theory. A natural set of generators for this Hopf ring arising from the homology and homotopy of the connective Morava K-theory spectrum is calculated and the non-trivial circle product relations among the generators arising from homology and homotopy are determined. This Hopf ring calculation is accomplished using Dieudonne ring theory and Adams spectral sequences for the connective Morava K-theory of Brown–Gitler spectra.

Algebraic K-Theory of ∞-Operads

AbstractThe theory of dendroidal sets has been developed to serve as a combinatorial model for homotopy coherent operads, see [MW07, CM13b]. An ∞-operad is a dendroidal setDsatisfying certain lifting conditions.In this paper we give a definition of K-groupsKn(D) for a dendroidal setD. These groups generalize the K-theory of symmetric monoidal (resp. permutative) categories and algebraic K-theory of rings. We establish some useful properties like invariance under the appropriate equivalences and long exact sequences which allow us to compute these groups in some examples. Using results from [Heu11b] and [BN12] we show that theK-theory groups ofDcan be realized as homotopy groups of a K-theory spectrum.

K-theory, reality, and duality

AbstractWe present a new proof of Anderson's result that the realK-theory spectrum is Anderson self-dual up to a fourfold suspension shift; more strongly, we show that the Anderson dual of the complexK-theory spectrumKUisC2-equivariantly equivalent to Σ4KU, whereC2acts by complex conjugation. We give an algebro-geometric interpretation of this result in spectrally derived algebraic geometry and apply the result to calculate 2-primary Gross-Hopkins duality at height 1. From the latter we obtain a new computation of the group of exotic elements of theK(1)-local Picard group.

A twisted Bass–Heller–Swan decomposition for the algebraic K-theory of additive categories

Abstract We prove a twisted Bass–Heller–Swan decomposition for both the connective and the non-connective K-theory spectrum of additive categories.