# Completed tensor products and a global approach to $p$-adic analytic differential operators

## TL;DR

This paper extends the theory of $p$-adic differential operators by removing smoothness constraints, enabling a more general description of sections and modules using completed tensor products over normed $K$-algebras.

## Contribution

It generalizes the structural results of $p$-adic differential operators to arbitrary affinoid subdomains without requiring smooth Lie lattices.

## Key findings

- Provides exactness criteria for completed tensor products over normed $K$-algebras.
- Generalizes the description of sections of D-cap to all affinoid subdomains.
- Establishes a natural extension of coadmissible D-cap-modules theory.

## Abstract

Ardakov-Wadsley defined the sheaf D-cap of $p$-adic analytic differential operators on a smooth rigid analytic variety $X$ by restricting to the case where $X$ is affinoid and the tangent sheaf admits a smooth Lie lattice. We generalize their results by dropping the assumption of a smooth Lie lattice throughout, which allows us to describe the sections of D-cap for arbitrary affinoid subdomains and not just on a suitable base of the topology. The structural results concerning D-cap and coadmissible D-cap-modules can then be generalized in a natural way. The main ingredient for our proofs is a study of completed tensor products over normed $K$-algebras, for $K$ a discretely valued field of mixed characteristic. Given a normed right module $U$ over a normed $K$-algebra $A$, we provide several exactness criteria for the functor $U\widehat{\otimes}_A-$ applied to complexes of strict morphisms, including a necessary and sufficient condition in the case of short exact sequences.