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/IM). 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 HIi(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 HIi(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)cof 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.