Rigid analytic spaces with overconvergent structure sheaf

TL;DR

This paper introduces dagger spaces, a new category of rigid analytic spaces with overconvergent structure sheaves, providing a suitable framework for de Rham cohomology and establishing duality theorems and relations to rigid cohomology.

Contribution

It defines dagger spaces as a new category for rigid analysis, enabling better study of de Rham cohomology and connecting to Berthelot's rigid cohomology.

Findings

Established Serre and Poincaré duality theorems for dagger spaces

Compared dagger spaces with usual rigid spaces

Clarified the relation to Berthelot's rigid cohomology

Abstract

We introduce a category of 'rigid spaces with overconvergent structure sheaf' which we call dagger spaces --- this is the correct category in which de Rham cohomology in rigid analysis should be studied. We compare it with the (usual) category of rigid spaces, give Serre and Poincar\'e duality theorems and explain the relation with Berthelot's rigid cohomology.