A characterization of cofinite local cohomology modules in a certain Serre class

Let R be a commutative Noetherian ring, a be an ideal of R, n be a non-negative integer, X be an arbitrary R-module and L be a finitely generated R-module. We characterize when Hⁱₐ(X) and Hⁱₐ(L,X) are (FD≤ n+1, a)-cofinite for all i, whenever one of the following statements holds: (a) ara( a)≤ 1, (b) dim R/ a ≤ n+1, (c) dim R≤ n+2 or (d) X is an FD≤ n+3~R-module. As a consequence, we show that ExtⁱR(L,X) is FD≤ n+1 for all i and any a-torsion finitely generated R-module L with dim L≤ n+1 if one of the above statements holds. Moreover, we obtain that, if R is a semi-local ring with dim R/ a≤ 2 and X is an R-module such that ExtʲR(R/ a, X) is FD≤ 2 for all j≤ dim X, then AssR(Hⁱₐ(L, X)) is finite for all i.