Short course "Motivic Cohomology". Lecture 3: Applications and perspectives

Lecture 3/3.

We introduce the construction by Morel and Voevodsky of A1-homotopy theory and briefly describe how Voevodsky’s triangulated category of motives fits into this picture via the motivic Eilenberg-MacLane spectrum. We discuss two applications of A1-homotopy theory to motivic cohomology: the slice spectral sequence and the solution of the Bloch-Kato conjectures.

The second part of this lecture is devoted to extensions of the theory to more general base-schemes. This includes the Déglise-Cisinski category of Beilinson motives, and it’s use by Spitzweck in constructing a motivic cohomology spectrum over an arbitrary base. We conclude with a description of Hoyois’ construction of this motivic cohomology spectrum, which relies on the theory of framed correspondences.