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) ⊆ V(a), then M is a-weakly cofinite if (and only if) HomR(R/a, M), Ext1 R(R/a, M) and Ext2 R(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 ∈ N0, such that Exti R(R/a, M) is weakly Laskerian for all i , then Hi a(M) is a-weakly cofinite for all i if (and only if) HomR(R/a,Hi a(M)) is weakly Laskerian for all i