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A
universal characterization of higher algebraic
$K\mkern-4mu$–theory
Andrew J Blumberg, David Gepner and Gonçalo Tabuada |
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Geometry & Topology 17 (2013) 733–838
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Abstract |
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In this paper we establish a universal characterization of higher algebraic –theory in the setting of small stable –categories. Specifically, we prove that connective algebraic –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 –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 –construction corresponds to the suspension functor and connective and nonconnective algebraic –theory spectra become corepresentable by the noncommutative motive of the sphere spectrum. In particular, the algebraic –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 , we prove moreover that its negative –groups are isomorphic to the negative –groups of the ordinary ring . 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 –theory and the classical definition. As an application of our theory, we obtain a complete classification of the natural transformations from higher algebraic –theory to topological Hochschild homology () and topological cyclic homology (). Notably, we obtain an elegant conceptual description of the cyclotomic trace map. |
Keywords
higher algebraic $K$–theory, homotopy invariance, stable
infinity categories, spectral categories, topological
cyclic homology, cyclotomic trace map
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Mathematical Subject Classification 2010
Primary: 18D20, 19D10, 19D25, 19D55, 55N15
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References |
Publication
Received: 1 January 2011
Revised: 2 October 2012
Accepted: 4 November 2012
Published: 18 April 2013
Proposed: Haynes Miller
Seconded: Ralph Cohen, Paul Goerss
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Authors |
| Andrew J Blumberg | |
| Department of Mathematics University of Texas at Austin 1 University Station C1200 Austin, TX 78712 USA |
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| http://www.math.utexas.edu/users/blumberg/ | |
| David Gepner | |
| Fakultät für Mathematik Universität Regensburg D-93040 Regensburg Germany |
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| http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/gepner/ | |
| Gonçalo Tabuada | |
| Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139 USA and Departamento de Matemática e CMA FCT-UNL Quinta da Torre 2929-516 Caparica Portugal |
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| http://math.mit.edu/~tabuada/ | |
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