The Overconvergent Site II. Cohomology

TL;DR

This paper establishes that rigid cohomology can be computed via an overconvergent site, providing a functorial approach that unifies crystalline and Monsky-Washnitzer cohomology.

Contribution

It proves the equivalence of cohomology computed on the overconvergent site with existing theories, enhancing functoriality and unification of cohomological methods.

Findings

Rigid cohomology is computed via the overconvergent site.

Cohomology on the overconvergent site matches existing theories.

Enhances functoriality in rigid cohomology.

Abstract

We prove that rigid cohomology can be computed as the cohomology of a site analogous to the crystalline site. Berthelot designed rigid cohomology as a common generalization of crystalline and Monsky-Washnitzer cohomology. Unfortunately, unlike the former, the functoriality of the theory is not built-in. We defined somewhere else the "overconvergent site" which is functorially attached to an algebraic variety and proved that the category of modules of finite presentation on this ringed site is equivalent to the category of over- convergent isocrystals on the variety. We show here that their cohomology also coincides.