D-modules over rings with finite F-representation type

Abstract

Smith and Van den Bergh [29] 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≥0 Rn is a Noetherian graded ring with finite (graded) F-representation type, then for every non-zerodivisor x ∈ R, Rx is generated by 1/x as a DR-module. The second one states that if R is a Gorenstein ring with finite F-representation type, then HIn(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. © International Press 2008.