Mathematical Research Letters

Volume 15, Issue 3, May 2008 pp. 563-581.

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

Authors: Shunsuke Takagi (1) and Ryo Takahashi (2)
Author institution: Kyushu University (1) and Shinshu University (2)

Summary: 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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