The de Rham comparison theorem for Deligne-Mumford stacks

TL;DR

This paper extends the de Rham comparison theorem from varieties to proper, smooth Deligne-Mumford stacks, providing two approaches—one incomplete but insightful, and the other complete—using Weil cohomologies and simplicial methods respectively.

Contribution

It introduces the first complete proof of the de Rham comparison theorem for Deligne-Mumford stacks, expanding the scope of p-adic Hodge theory.

Findings

Complete proof using simplicial methods based on Kisin and Tsuji's work

Partial approach employing Weil cohomologies highlighting compatibility issues

Extension of the de Rham comparison theorem to 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.