Cofiniteness and associated primes of local cohomology modules

Abstract

Let R be a d-dimensional regular local ring, I an ideal of R, and M a finitely generated R-module of dimension n. We prove that the set of associated primes of ExtRi(R/I,HIj(M)) is finite for all i and j in the following cases: (a) dimM⩽3; (b) dimR⩽4; (c) dimM/IM⩽2 and M satisfies Serre's condition Sn−3; (d) dimM/IM⩽3, annRM=0, R is unramified, and M satisfies Sn−3. In these cases we also prove that HIi(M)p is Ip-cofinite for all but finitely many primes p of R. Additionally, we show that if dimR/I⩾2 and SpecR/I−{m/I} is disconnected then HId−1(R) is not I-cofinite, generalizing a result due to Huneke and Koh.