The de Rham comparison theorem for Deligne-Mumford stacks

Abstract

The de Rham comparison theorem for varieties, first proved by Faltings, gives the de Rham cohomology of a variety in terms of its p-adic etale cohomology. We extend this theorem to proper, smooth Deligne-Mumford stacks. Two approaches are given, which both in the end reduce the problem to the already known comparison theorem for varieties. The first approach employs the formalism of Weil cohomologies. Unfortunately, this does not result in a complete proof of the comparison theorem, as the author was unable to prove the required compatibility between intersections and cup products. Nevertheless, the author thought the results that are obtained and the method that is suggested interesting enough to include them. The second approach uses simplicial methods and is based on versions of the comparison theorem by Kisin and Tsuji. The latter approach does result in a complete proof of the extended comparison theorem.