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 $p>0$.

AMS-LaTeX, 8 pages