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 are I-cofinite, for all i⩾0, whenever M is I-cofinite and N is finitely generated of dimension d⩽2. Also, we prove that the R-modules are I-cofinite, for all i⩾0, whenever N is finitely generated and M is I-cofinite of dimension d⩽1. This immediately implies that if I has dimension one (i.e., ) then 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 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 are I-weakly cofinite, for all i⩾0, whenever R is local, N is finitely generated and M is I-cofinite of dimension d⩽2.