Cohomologie syntomique: liens avec les cohomologies étale et rigide

Jean-Yves Etesse (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Syntomic cohomology here defined yields a link between rigid cohomology and etale cohomology, viewing the last one as the fixed points under Frobenius of the former one. Let V be a complete discrete valuation ring, with perfect residue field k = V/m of characteristic p > 0 and fraction field K of characteristic 0. Having defined syntomic cohomology with compact supports of an abelian sheaf G on a k-scheme X, we show that it coincides with etale cohomology with compact supports when G is a lisse sheaf. If moreover the convergent F-isocrystal associated to G comes from an overconvergent isocrystal E, then the rigid cohomology of E expresses as a limit of syntomic cohomologies: then the etale cohomology with compact supports of G is the fixed points of Frobenius acting on the rigid cohomology of E.

Mots clés: cohomologie rigide; cohomologie syntomique; F- isocristaux convergents; F- isocristaux surconvergents; cohomologie cristalline; cohomologie rigide.

Domaines: Géométrie algébrique [math.AG]

Jean-Yves Etesse. Cohomologie syntomique: liens avec les cohomologies étale et rigide. 2009. ⟨hal-00425926⟩