On the Vanishing Properties of Local Cohomology Modules Defined by a Pair of Ideals

Abstract

As a generalization of the ordinary local cohomology modules, recently some authors introduced the local cohomology modules with respect to a pair of ideals. In this paper, we get some results on Artinianness, vanishing, finiteness and other properties of these modules. Let

 

R be a commutative Noetherian ring, I , J two ideals of R and M a finitely generated R-module such that dimR M = n. We prove that Hn I ,J (M)/JHn I ,J (M) is I -cofinite Artinian and Hn I ,J (M)/IHn I ,J (M) has finite length. Also we show that, if R is local with dimR/I + J = 0 and dimR M/JM = d > 0, then Hd I ,J (M) is not finitely generated.