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

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.