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We construct a version of Beilinson’s regulator as a map of sheaves of commutative ring spectra and use it to define a multiplicative variant of differential algebraic -theory. We use this theory to give an interpretation of Bloch’s construction of -classes and the relation with dilogarithms. Furthermore, we provide a relation to Arakelov theory via the arithmetic degree of metrized line bundles, and we give a proof of the formality of the algebraic -theory of number rings.
Let be a quasicompact algebraic stack with quasifinite and separated diagonal. We classify the thick -ideals of . If is tame, then we also compute the Balmer spectrum of the -triangulated category of perfect complexes on . In addition, if admits a coarse space , then we prove that the Balmer spectra of and are naturally isomorphic.
A kind of motivic stable homotopy theory of algebras is developed. Explicit fibrant replacements for the -spectrum and -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 -motives. It is established that the triangulated category of Cortiñas and Thom (J. Reine Angew. Math. 610 (2007), 71–123) can be identified with the -motives of algebras. It is proved that the triangulated category of -motives is a localisation of the triangulated category of -bispectra. Also, explicit fibrant -bispectra representing stable algebraic Kasparov -theory and algebraic homotopy -theory are constructed.
For a reciprocity functor we consider the local symbol complex
where is a smooth complete curve over an algebraically closed field with generic point and is the product of Mackey functors. We prove that if satisfies certain assumptions, then the homology of this complex is isomorphic to the -group of reciprocity functors .