D-modules on rigid analytic spaces I

TL;DR

This paper introduces a sheaf of infinite order differential operators on smooth rigid analytic spaces, establishing foundational properties and analogues of classical theorems for these modules.

Contribution

It defines a new sheaf of differential operators D-cap as a rigid analytic quantisation and develops the theory of co-admissible D-cap-modules.

Findings

Sections over affinoid varieties are Fréchet-Stein algebras

Established analogues of Cartan's Theorems A and B for co-admissible modules

Provided a framework for D-cap-modules on rigid analytic spaces

Abstract

We introduce a sheaf of infinite order differential operators D-cap on smooth rigid analytic spaces that is a rigid analytic quantisation of the cotangent bundle. We show that the sections of this sheaf over sufficiently small affinoid varieties are Fr\'echet-Stein algebras, and use this to define co-admissible sheaves of D-cap-modules. We prove analogues of Cartan's Theorems A and B for co-admissible D-cap-modules.