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 Supp R (M) ⊆ V(a), then M is a-weakly cofinite if (and only if) Hom R (R/a, M), Ext 1 R (R/a, M) and Ext 2 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 ExtiR (R/a, M) is weakly Laskerian for all i, then Hi a (M) is a-weakly cofinite for all i if (and only if) Hom R (R/a, Hi a (M)) is weakly Laskerian for all i.
