Mathematical Research Letters

Volume 15 (2008)

Number 3

$D$-modules over rings with finite F-representation type

Pages: 563 – 581

DOI: http://dx.doi.org/10.4310/MRL.2008.v15.n3.a15

Authors

Shunsuke Takagi (Kyushu University)

Ryo Takahashi (Shinshu University)

Abstract

Smith and Van den Bergh \cite{SvdB} introduced the notion of finite F-representa- tion 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.

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