Abstract
In this work, I extend the theory of motives, as developed by Voevodsky and Morel-Voevodsky, to the context of rigid analytic geometry over a complete non archimedean field. The first chapter deals with the homotopical approach of Morel and Voevodsky. One finds there the construction of the motivic stable homotopy category of rigid analytic varieties and a complete description of this category in terms of algebraic motives when the base field has equal characteristic zero and its valuation is discrete. The second chapter deals with Voevodsky's approach based on transfers. One finds there the construction of the triangulated category of rigid analytic motives, and an extension to rigid analytic geometry of a large number of Voevodsky's fundamental results such as his theory of homotopy invariants presheaves with transfers. This is said, the present work is a lot more than just a mere copy of the classical theory and the reader will find a lot of results that are new and specific to rigid analytic geometry.
Ayoub, Joseph