Continuity of differential operators for nonarchimedean Banach algebras
Feliks Rączka
TL;DR
This work characterizes automatic continuity of Grothendieck differential operators acting between finitely generated Banach modules over a noetherian Banach algebra $A$ on a nonarchimedean field $K$, showing that in characteristic zero the property holds if and only if $[A/\mathfrak{m}:K]<\infty$ for every maximal ideal $\mathfrak{m}\subset A$. The central tool is the nonarchimedean automatic-continuity framework (separating space, continuity ideal, Jewell--Sinclair stabilization) complemented by a detailed analysis of field-extensions and derivations to construct discontinuous operators when the finiteness condition fails. Consequences include automatic continuity for affinoid (Tate) algebras and a precise link between discontinuities and the pair of invariants $\operatorname{Supp}(M)$ and $\operatorname{Ass}_{A}(N)$, with implications for nonarchimedean $\mathcal{D}$-modules. The results clarify the role of residue-field finiteness and associated primes, reveal limitations in positive characteristic and Berkovich settings, and provide a framework for automatic-continuity questions in local nonarchimedean geometry.
Abstract
Given a nonarchimedean field $K$ and a commutative, noetherian, Banach $K$-algebra $A$, we study continuity of $K$-linear differential operators (in the sense of Grothendieck) between finitely generated Banach $A$-modules. When $K$ is of characteristic zero we show that every such operator is continuous if and only if $A/\mathfrak{m}$ is a finite extension of $K$ for every maximal ideal $\mathfrak{m}\subset A$.