Six operations for D-cap-modules on rigid analytic spaces

TL;DR

This paper develops a comprehensive six-functor formalism for D-cap-modules on smooth rigid analytic spaces, extending classical algebraic results to the non-Archimedean analytic setting.

Contribution

It introduces the six operations for D-cap-modules on rigid analytic spaces and establishes key analogues of classical theorems in this new context.

Findings

Established Kashiwara's equivalence for D-cap-modules.

Proved stability of coadmissibility under inverse and direct images.

Computed relative de Rham cohomology in the rigid analytic setting.

Abstract

We introduce all six operations for D-cap-modules on smooth rigid analytic spaces by considering the derived category of complete bornological D-cap-modules. We then focus on a full subcategory which should be thought of as consisting of complexes with coadmissible cohomology, and establish analogues of some classical results: Kashiwara's equivalence, stability of coadmissibility under extraordinary inverse image for smooth morphisms and direct image for projective morphisms, as well as the computation of relative de Rham cohomology.