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

Andrew J Blumberg, David Gepner and Gonçalo Tabuada

Geometry & Topology 17 (2013) 733–838

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.

higher algebraic $K$–theory, homotopy invariance, stable infinity categories, spectral categories, topological cyclic homology, cyclotomic trace map
Mathematical Subject Classification 2010
Primary: 18D20, 19D10, 19D25, 19D55, 55N15
Received: 1 January 2011
Revised: 2 October 2012
Accepted: 4 November 2012
Published: 18 April 2013
Proposed: Haynes Miller
Seconded: Ralph Cohen, Paul Goerss
Andrew J Blumberg
Department of Mathematics
University of Texas at Austin
1 University Station C1200
Austin, TX 78712
David Gepner
Fakultät für Mathematik
Universität Regensburg
D-93040 Regensburg
Gonçalo Tabuada
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
Departamento de Matemática e CMA
Quinta da Torre
2929-516 Caparica