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Theses

modules homotopiques avec transferts et motifs génériques

Abstract : In this thesis, the (arithmetic) theory of cycle modules of M.Rost is related to the (more geometrical) theory of homotopy invariant sheaves with transfers of V. Voevodsky: the latter category is a localization of the category of cycle modules. Moreover, inspired by the construction of spectra in algebraic topology, homotopy module with transfers is introduced; these are defined as certain graded modules in the category of homotopy invariant sheaves with transfers. The category formed by these modules is equivalent to the category of cycle modules, thus extending the relation concerning homotopy invariant sheaves with transfers. This allows another proof to be given, using results of M. Rost on Chow groups with coefficients in cycle modules, that homotopy invariant sheaves with transfers have cohomology groups which are homotopy invariant, a result first obtained by V. Voevodsky. In addition, we deduce from the equivalence of categories above that the category of cycle modules is a Grothendieck abelian category equipped with a monoïdal structure for which Milnor K-theory is the unit. In addition, we show how the techniques used can be generalized to the derived category of motives defined by V. Voevodsky, thus obtaining formulae which bring into play the Gysin distinguished triangles. For example, we give a way to interpret ramification in the case of discrete valuation rings of equal characteristic by using deformation to the normal cone. The work concludes with the definition of certain pro-motives, which we baptize generic motives. These are pro-objects of the derived category of motives, which are associated to finite-type extensions of the base field (which is assumed to be perfect). The 'twists' of these motives by the motive Z(1)[1], or one of its powers by any relative integer, are also considered. In a surprising way, each part of the data of a cycle pre-module has an analogue which is a morphism of generic motives. Moreover, the structural relations on the data of a cycle pre-module hold in the category of generic motives, thus giving a geometrical incarnation of the rather arithmetic axioms of cycle pre-modules.
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Contributor : Frédéric Déglise <>
Submitted on : Wednesday, March 19, 2003 - 9:15:04 AM
Last modification on : Tuesday, December 8, 2020 - 3:35:53 AM
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  • HAL Id : tel-00002562, version 1

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Frédéric Déglise. modules homotopiques avec transferts et motifs génériques. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2002. Français. ⟨tel-00002562⟩

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