Auslander regularity of completed rings of $p$-adic differential operators

TL;DR

This paper demonstrates that the completed rings of differential operators on smooth rigid analytic varieties have Auslander regularity, enabling new projection and adjunction formulas for coadmissible modules.

Contribution

It establishes Auslander regularity for the Banach algebras in the Fréchet–Stein presentation of these rings, a novel result in p-adic differential operator theory.

Findings

Banach algebras are Auslander regular

Projection formulae for coadmissible D-modules proven

Adjunction results for coadmissible D-modules established

Abstract

We prove that any smooth rigid analytic variety $X$ admits an affinoid covering $\{U_i\}$ such that the Banach algebras involved in the Fr\'echet--Stein presentation of the completed ring of differential operators D-cap$(U_i)$ are Auslander regular for each $i$. We use this result to prove projection formulae and adjunction results for coadmissible D-cap-modules.