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 nonnegative integer, X be an arbitrary R -module and L be a finitely generated R -module. We characterize when H-a(i) (X and H-a(i) ( L , X) are ( FD<n , 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+2 R -module. As a consequence, we show that Ext(R)(i) ( L, X ) is FD<n for all i and any atorsion 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 Ext(R)(j) ( R / a, X) is FD<1 for all j <= dim X , then Ass(R) ( H-a(i) (L, X )) is finite for all i .