On some Rings of differentiable type

TL;DR

This paper constructs a field within regular affine domains over characteristic zero fields, ensuring the differential operator ring is Noetherian with finite global dimension and analyzing its Bernstein class and localizations.

Contribution

It introduces a method to embed a field into regular affine domains so that their differential operator rings have desirable algebraic properties, including Noetherianity and finite global dimension.

Findings

The ring of differential operators is Noetherian with finite global dimension.

The Bernstein class is stable under localization.

The differential operators modulo a prime ideal are isomorphic to the injective hull of the residue field.

Abstract

Let $K$ be a field of characteristic 0 and $S=K[x_1,\ldots,x_m]/I$ be an affine domain. Consider $R=S_P$ where $P\in Spec(S)$ such that $R$ is regular. In this paper we construct a field $F$ which is contained in $R$ such that (1) The residue field of $R$ is a finite extension of $F$. (2) $D_F(R)$, the ring of $F$-linear differential operators on $R$ is left and right Noetherian with finite global dimension. (3) The Bernstein class of $D_F(R)$ is closed under localization at one element of $R$. We also prove a similar result for $R^h$, the Henselization of $R$. As an application we prove that $\frac{D_F(R)}{D_F(R)P}\cong E(\kappa(P))$ where $E(\kappa(P))$ is the injective hull of the residue field of $R$.