Finiteness of de Rham cohomology in rigid analysis

TL;DR

This paper proves the finite dimensionality of de Rham cohomology for a broad class of smooth dagger spaces, extending finiteness results to non-smooth cases via semi-stable reduction analysis.

Contribution

It establishes finiteness of de Rham cohomology for dagger spaces and extends this to Berthelot's rigid cohomology in non-smooth contexts.

Findings

Finite dimensionality of de Rham cohomology for smooth dagger spaces.

Finiteness of Berthelot's rigid cohomology in non-smooth cases.

Analysis of de Rham cohomology in semi-stable reduction situations.

Abstract

For a big class of smooth dagger spaces --- dagger spaces are 'rigid spaces with overconvergent structure sheaf' --- we prove finite dimensionality of de Rham cohomology. This is enough to obtain finiteness of Berthelot's rigid cohomology also in the non-smooth case. We need a careful study of de Rham cohomology in situations of semi-stable reduction.