D-modules over 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 $R={⨁}_{n\ge 0}{R}_n$ 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.

19 pages; v.2: minor changes, to appear in Math. Res. Lett

References in corpus (1)

• F-thresholds and Bernstein-Sato polynomials