Cohomological dimension, cofiniteness and Abelian categories of cofinite modules

Let R be a commutative Noetherian ring, I ⊆ J be ideals of R and M be a finitely generated R-module. In this paper it is shown that q ( J , M ) ≤ q ( I , M ) + cd ( J , M / I M ) . Furthermore, it is shown that, for any ideal I of R and any finitely generated R-module M with q ( I , M ) ≤ 1 , the local cohomology modules H I i ( M ) are I-cofinite for all integers i ≥ 0 . As a consequence of this result it is shown that, if q ( I , R ) ≤ 1 , then for any finitely generated R-module M, the local cohomology modules H I i ( M ) are I-cofinite for all integers i ≥ 0 . Finally, it is shown that the category of all I-cofinite R-modules C ( R , I ) c o f is an Abelian subcategory of the category of all R-modules, whenever ( R , m ) is a complete Noetherian local ring and I is an ideal of R with q ( I , R ) ≤ 1 . These assertions answer affirmatively two questions raised by R. Hartshorne in [16] , in the some special cases.