Coherent rings of differential operators

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 $p > 0$.