D-modules over rings with finite F-representation type

TL;DR

This paper explores finiteness properties of rings with finite F-representation type, demonstrating their implications for D-module generation and local cohomology, and discusses the discreteness of F-jumping exponents.

Contribution

It proves new finiteness properties for rings with finite F-representation type, including D-module generation and finiteness of associated primes in local cohomology.

Findings

For graded rings, localizations are generated by 1/x as D-modules.

Gorenstein rings with finite F-representation type have finitely many associated primes in local cohomology.

F-jumping exponents are discrete in rings with finite F-representation type.

Abstract

Smith and Van den Bergh introduced the notion of finite F-representation type as a characteristic $p$ analogue of the notion of finite representation type. In this paper, we prove two finiteness properties of rings with finite F-representation type. The first property states that if $R=\bigoplus_{n \ge 0}R_n$ is a Noetherian graded ring with finite (graded) F-representation type, then for every non-zerodivisor $x \in R$, $R_x$ is generated by $1/x$ as a $D_{R}$-module. The second one states that if $R$ is a Gorenstein ring with finite F-representation type, then $H_I^n(R)$ has only finitely many associated primes for any ideal $I$ of $R$ and any integer $n$. We also include a result on the discreteness of F-jumping exponents of ideals of rings with finite (graded) F-representation type as an appendix.