Opérations sur la K-théorie algébrique et régulateurs via la théorie homotopique des schémas, by Joël Riou

This thesis contributes to the homotopy theory of schemes. In the first part, we carry onward the constructions by Fabien Morel and Vladimir Voevodsky: we define the stable homotopy categories of hanging sites with intervals. This construction is more general than the one of John F. Jardine: this enables us to provide a precise definition of the complex points functors in the homotopy theory of schemes.

In the second part, we prove that in the homotopy category of a regular scheme S, the set of endomorphisms of the infinite Grassmannian (which gives a model for algebraic K-theory by a theorem by Morel and Voevodsky) is naturally isomorphic to the set of natural transformations from the Grothendieck group functor (considered as presheaf of sets on the category of smooth schemes over S) to itself. This enables us to build a special λ-ring structure on higher K-groups and to check that this construction is the same as the ones that were constructed before. Additive operations on algebraic K-theory are studied carefully and stable versions of the theorems are provided either with integer or rational coefficients. The technique also allows us to define Chern classes on higher K-groups with values in motivic cohomology (and other cohomological theories) and to assert the existence of superphantom maps in the homotopy theory of schemes.

Joël Riou <joel.riou@normalesup.org>