Beilinson Motives and Algebraic K-Theory

Abstract

Section 12 is a recollection on the basic results of stable homotopy theory of schemes, after Morel and Voevodsky. In particular, we recall the theory of orientations in a motivic cohomology theory. Section 13 is a recollection of the fundamental results on algebraic K-theory which we translate into results within stable homotopy theory of schemes. In particular, Quillen’s localization theorem is seen as an absolute purity theory for the K-theory spectrum. In Section 14, we introduce the fibred category of Beilinson motives as an appropriate Verdier quotient of the motivic stable homotopy category. Using the Adams filtration on K-theory, we prove that Beilinson motives have the properties of h-descent and of absolute purity. In Section 15, we prove that the six operations preserve constructibility in the context of Beilinson motives, following a strategy of Gabber. Grothendieck-Verdier duality is also established for constructible Beilinson motives. In Section 16, we compare Beilinson motives with all the other candidates for a good theory of mixed motives with integral coefficients. Finally, in Section 17, we explain that a large class of examples of motivic categories can be described as the compact objects in the fibred category of modules over a ring object within Beilinson motives. We then explain how the main examples of mixed Weil cohomologies fit into this picture, and how this recovers many classical comparison results (such as Grothendieck’s comparison of algebraic vs analytic de Rham cohomology, or the Bloch-Ogus isomorphism) with a new motivic proof.