Cohomologie syntomique: liens avec les cohomologies \'etale et rigide

TL;DR

This paper establishes a deep connection between syntomic, etale, and rigid cohomologies, showing how syntomic cohomology bridges rigid and etale theories through Frobenius fixed points, especially for sheaves arising from overconvergent isocrystals.

Contribution

It defines syntomic cohomology with compact supports and proves its equivalence to etale cohomology for lisse sheaves, linking rigid cohomology to syntomic cohomology via Frobenius fixed points.

Findings

Syntomic cohomology coincides with etale cohomology for lisse sheaves.

Rigid cohomology of overconvergent isocrystals is expressed as a limit of syntomic cohomologies.

Etale cohomology with compact supports is identified as Frobenius fixed points in rigid cohomology.

Abstract

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.