D-modules on rigid analytic spaces I

Abstract

We introduce a sheaf of infinite order differential operators D 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 Fréchet-Stein algebras, and use this to define co-admissible sheaves of D-modules. We prove analogues of Cartan’s Theorems A and B for co-admissible D-modules.