Abelian Category of Weakly Cofinite Modules and Local Cohomology

Let R be a commutative Noetherian ring, a an ideal of R, and M an R-module. We prove that the category of a-weakly cofinite modules is a Melkersson subcategory of R-modules whenever dim R <= 1 and is an Abelian subcategory whenever dim R <= 2. We also prove that if (R, m) is a local ring with dim R/a <= 2 and SuppR(M) subset of V(a), then M is a-weakly cofinite if (and only if) Hom(R)(R/a, M), Ext(R)(1)(R/a, M) and Ext(R)(2) (R/a, M) are weakly Laskerian. In addition, we prove that if (R, m) is a local ring with dim R/a <= 2 and n is an element of N-0, such that Ext(R)(i) (R/a, M) is weakly Laskerian for all i, then H-a(i) (M) is a-weakly cofinite for all i if (and only if) Hom(R)(R/a, H-a(i) (M)) is weakly Laskerian for all i