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2016 Algebraic Kasparov K-theory, II
Grigory Garkusha
Ann. K-Theory 1(3): 275-316 (2016). DOI: 10.2140/akt.2016.1.275


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.


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Grigory Garkusha. "Algebraic Kasparov K-theory, II." Ann. K-Theory 1 (3) 275 - 316, 2016.


Received: 12 January 2015; Revised: 26 August 2015; Accepted: 26 August 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1375.19013
MathSciNet: MR3529093
Digital Object Identifier: 10.2140/akt.2016.1.275

Primary: 19D25 , 19D50 , 19K35
Secondary: 55P99

Keywords: bivariant algebraic $K$-theory , homotopy theory of algebras , triangulated categories

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.1 • No. 3 • 2016
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