# D-modules over rings with finite F-representation type - Mathematics > Commutative Algebra

Abstract: Smith and Van den Bergh introduced the notion of finite F-representation typeas a characteristic $p$ analogue of the notion of finite representation type.In this paper, we prove two finiteness properties of rings with finiteF-representation type. The first property states that if $R=\bigoplus {n \ge0}R n$ 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 withfinite F-representation type, then $H I^nR$ has only finitely many associatedprimes for any ideal $I$ of $R$ and any integer $n$. We also include a resulton the discreteness of F-jumping exponents of ideals of rings with finitegraded F-representation type as an appendix.

Author: Shunsuke Takagi, Ryo Takahashi

Source: https://arxiv.org/