Coherent rings of differential operators

TL;DR

This paper investigates the coherence of rings of differential operators, proving coherence in positive characteristic for smooth algebras and conjecturing it in characteristic zero for certain singular cases, with implications for $D$-modules.

Contribution

It proves that rings of differential operators are coherent over smooth, connected algebras in characteristic p > 0 and conjectures this extends to characteristic zero for specific singular cases.

Findings

Ring $D$ is coherent for smooth, connected algebras in characteristic p > 0.

Ring $D$ is not necessarily Noetherian or finitely generated in positive characteristic.

Conjecture that $D$ is coherent for the cubic cone in characteristic zero.

Abstract

We consider the following question: When are rings of differential operators coherent? If $A$ is a finitely generated smooth domain over a field $k$ of characteristic $0$, then the ring $D$ of differential operators on $A$ is a Noetherian ring and a finitely generated $k$-algebra. However, when $k$ has characteristic $p > 0$ or when $A$ is singular, this is no longer true. In fact, Bernstein, Gelfand and Gelfand showed that for the cubic cone $A = k[x,y,z]/(x^3 + y^3 + z^3)$, the ring $D$ is neither Noetherian nor finitely generated if $k$ has characteristic $0$, and the same is true for the polynomial ring $A = k[x_1, \dots, x_n]$ if $k$ has characteristic $p > 0$. In this paper, we prove that the ring $D$ of differential operators on a finitely generated, smooth and connected algebra $A$ over a field $k$ of characteristic $p > 0$ is coherent, and conjecture that same holds for the cubic cone in characteristic $0$. We argue that the question of coherence is the more fundamental one, and use some interesting results of Bavula to study holonomic $D$-modules on $A = k[x_1, \dots, x_n]$ in characteristic $p > 0$.