Vol. 1, No. 3, 2016

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Algebraic Kasparov K-theory, II

Grigory Garkusha

Vol. 1 (2016), No. 3, 275–316
Abstract

A kind of motivic stable homotopy theory of algebras is developed. Explicit fibrant replacements for the S1-spectrum and (S1, G)-bispectrum of an algebra are constructed. As an application, unstable, Morita stable and stable universal bivariant theories are recovered. These are shown to be embedded by means of contravariant equivalences as full triangulated subcategories of compact generators of some compactly generated triangulated categories. Another application is the introduction and study of the symmetric monoidal compactly generated triangulated category of K-motives. It is established that the triangulated category kk of Cortiñas and Thom (J. Reine Angew. Math. 610 (2007), 71–123) can be identified with the K-motives of algebras. It is proved that the triangulated category of K-motives is a localisation of the triangulated category of (S1, G)-bispectra. Also, explicit fibrant (S1, G)-bispectra representing stable algebraic Kasparov K-theory and algebraic homotopy K-theory are constructed.

Keywords
bivariant algebraic $K$-theory, homotopy theory of algebras, triangulated categories
Mathematical Subject Classification 2010
Primary: 19D25, 19D50, 19K35
Secondary: 55P99
Milestones
Received: 12 January 2015
Revised: 26 August 2015
Accepted: 26 August 2015
Published: 18 July 2016
Authors
Grigory Garkusha
Department of Mathematics
Swansea University
Singleton Park
Swansea, SA2 8PP
United Kingdom