Geometry & Topology

A universal characterization of higher algebraic $K\mkern-4mu$–theory

Andrew J Blumberg, David Gepner, and Gonçalo Tabuada

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Abstract

In this paper we establish a universal characterization of higher algebraic K–theory in the setting of small stable –categories. Specifically, we prove that connective algebraic K–theory is the universal additive invariant, ie the universal functor with values in spectra which inverts Morita equivalences, preserves filtered colimits and satisfies Waldhausen’s additivity theorem. Similarly, we prove that nonconnective algebraic K–theory is the universal localizing invariant, ie the universal functor that moreover satisfies the Thomason–Trobaugh–Neeman Localization Theorem.

To prove these results, we construct and study two stable –categories of “noncommutative motives”; one associated to additivity and another to localization. In these stable –categories, Waldhausen’s S–construction corresponds to the suspension functor and connective and nonconnective algebraic K–theory spectra become corepresentable by the noncommutative motive of the sphere spectrum. In particular, the algebraic K–theory of every scheme, stack and ring spectrum can be recovered from these categories of noncommutative motives. In the case of a connective ring spectrum R, we prove moreover that its negative K–groups are isomorphic to the negative K–groups of the ordinary ring π0R.

In order to work with these categories of noncommutative motives, we establish comparison theorems between the category of spectral categories localized at the Morita equivalences and the category of small idempotent-complete stable –categories. We also explain in detail the comparison between the –categorical version of Waldhausen K–theory and the classical definition.

As an application of our theory, we obtain a complete classification of the natural transformations from higher algebraic K–theory to topological Hochschild homology (THH) and topological cyclic homology (TC). Notably, we obtain an elegant conceptual description of the cyclotomic trace map.

Article information

Source
Geom. Topol., Volume 17, Number 2 (2013), 733-838.

Dates
Received: 1 January 2011
Revised: 2 October 2012
Accepted: 4 November 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732567

Digital Object Identifier
doi:10.2140/gt.2013.17.733

Mathematical Reviews number (MathSciNet)
MR3070515

Zentralblatt MATH identifier
1267.19001

Subjects
Primary: 18D20: Enriched categories (over closed or monoidal categories) 19D10: Algebraic $K$-theory of spaces 19D25: Karoubi-Villamayor-Gersten $K$-theory 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60] 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX}

Keywords
higher algebraic $K$–theory homotopy invariance stable infinity categories spectral categories topological cyclic homology cyclotomic trace map

Citation

Blumberg, Andrew J; Gepner, David; Tabuada, Gonçalo. A universal characterization of higher algebraic $K\mkern-4mu$–theory. Geom. Topol. 17 (2013), no. 2, 733--838. doi:10.2140/gt.2013.17.733. https://projecteuclid.org/euclid.gt/1513732567


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