Cofiniteness of extension functors of cofinite modules

Let R be a commutative Noetherian ring, I an ideal of R and let M and N be non-zero R-modules. It is shown that the R-modules Ext(R)(I)(N, M) are /-cofinite, for all i >= 0, whenever M is 1-cofinite and N is finitely generated of dimension d <= 2. Also, we prove that the R-modules Ext(I)R(N, M) are I-cofinite, for all i >= 0, whenever N is finitely generated and M is /-cofinite of dimension d <= 1. This immediately implies that if I has dimension one (i.e., dim R/I= 1) then Ext(R)(I)(N, H-I(1)(M)) is I-cofinite for all i >= 0, and all finitely generated R-modules M and N. Also, we prove that if R is local then the R-modules Ext(i)(R)(N, M) are I-weakly cofinite, for all i >= 0, whenever M is I-cofinite and N is finitely generated of dimension d <= 3. Finally, it is shown that the R-modules Ext(R)(i) (N, M) are I-weakly cofinite, for all i >= 0, whenever R is local. N is finitely generated and M is 1-cofinite of dimension d <= 2. Published by Elsevier Inc.