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 Hⁱ ₀ (X) and Hⁱ ₀ (L, X) are (FD ₍+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 ₍+₃~R-module. As a consequence, we show that ExtⁱR (L, X) is FD ₍+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 (Hⁱ ₀ (L, X) ) is finite for all i.