Let R be a commutative Noetherian ring, I an ideal of R, and M an arbitrary R-module. It is shown that the R-module Exti RR/I M is finitely generated, for all i ≥ 0, if and only if the R-module Exti RR/I M is finitely generated, for all 0 ≤ i ≤ araI. As an immediate consequence, we prove that, if R is a Noetherian (resp. R is a Noetherian local) ring of dimension d, then the R-module Exti RR/I M is finitely generated, for all i ≥ 0 if and only if the R-module Exti RR/I M is finitely generated, for all 0 ≤ i ≤ d + 1 (resp. for all 0 ≤ i ≤ d). Also, it is shown that, if I = Rx1 +· · ·+Rxn n ≥ 1 and SuppRM ⊆ VI, then M is I-cofinite if and only if the R-module Exti RR/I M or TorRi R/I M is finitely generated, for all 0 ≤ i ≤ n.
