Algebraic Kasparov K-theory, II

Abstract

A kind of motivic stable homotopy theory of algebras is developed. Explicit fibrant replacements for the $S^1$-spectrum and $\left(S^1,\mathbb{G}\right)$-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 $\left(S^1,\mathbb{G}\right)$-bispectra. Also, explicit fibrant $\left(S^1,\mathbb{G}\right)$-bispectra representing stable algebraic Kasparov $K$-theory and algebraic homotopy $K$-theory are constructed.