Homotopy Modules with Transfers and Generic Motives (in french)
Thesis from the University of Paris VII, December 2002
supervised by Fabien Morel
In this thesis, a construction by Markus Rost is connected to a construction by Vladimir Voevodsky, which forms the basis of the derived category of mixed motives. More specifically, the primarily arithmetic theory of Rost's cycle modules is linked to the more geometric theory of Voevodsky's homotopy invariant sheaves with transfers. We demonstrate precisely that the latter category is a localization of the category of cycle modules.
Furthermore, inspired by the construction of spectra in algebraic topology, we introduce the notion of homotopy modules with transfers based on homotopy invariant sheaves with transfers. The category formed by these modules is equivalent to the category of cycle modules, thus extending the statement regarding homotopy sheaves.
This allows us to reprove, using Rost's results, that homotopy invariant sheaves with transfers possess homotopy invariant cohomology, a result already obtained by Voevodsky. Additionally, we deduce that the category of cycle modules is a Grothendieck abelian category, equipped with a monoidal structure such that Milnor's K-theory is the neutral element.
Moreover, we show how the techniques employed can be extended to the category of motives, thereby obtaining formulas involving Gysin triangles. We also provide a lemma that offers a geometric interpretation of ramification in the sense of discrete valuation rings of equal characteristics.
The work concludes with the definition of certain pro-motives that we have named generic motives. These are pro-objects in the derived category of mixed motives, associated with finitely generated extensions of the base field (assumed to be perfect). It is also considered that these motives can be "twisted" by the motive \(\mathbb Z(1)[1]\) or one of its powers by any relative integer n. Surprisingly, each of the data from the pre-cycle modules has its analogue as a morphism of generic motives. Furthermore, the structural relations on the data from the pre-cycle modules hold true in the category of generic motives, thus realizing the geometric embodiment of the rather arithmetic axioms of the pre-cycle modules.