Abelian Category of Weakly Cofinite Modules and Local Cohomology

Abstract

Let R be a commutative Noetherian ring, $${\mathfrak {a}}$$ an ideal of R, and M an R-module. We prove that the category of $${\mathfrak {a}}$$ -weakly cofinite modules is a Melkersson subcategory of R-modules whenever $$\dim R\le 1$$ and is an Abelian subcategory whenever $$\dim R\le 2$$ . We also prove that if $$(R,{\mathfrak {m}})$$ is a local ring with $$\dim R/{\mathfrak {a}}\le 2$$ and $${{\,\mathrm{Supp}\,}}_R(M)\subseteq {{\,\mathrm{V}\,}}({\mathfrak {a}})$$ , then M is $${\mathfrak {a}}$$ -weakly cofinite if (and only if) $${{\,\mathrm{Hom}\,}}_R(R/{\mathfrak {a}}, M)$$ , $${{\,\mathrm{Ext}\,}}_{R}^{1}(R/{\mathfrak {a}}, M)$$ and $${{\,\mathrm{Ext}\,}}_{R}^{2}(R/{\mathfrak {a}}, M)$$ are weakly Laskerian. In addition, we prove that if $$(R,{\mathfrak {m}})$$ is a local ring with $$\dim R/{\mathfrak {a}}\le 2$$ and $$n\in \mathbb {N}_0$$ , such that $${{{\,\mathrm{Ext}\,}}}^{i}_R(R/{\mathfrak {a}},M)$$ is weakly Laskerian for all i, then $${{\,\mathrm{H}\,}}_{{\mathfrak {a}}}^{i}(M)$$ is $${\mathfrak {a}}$$ -weakly cofinite for all i if (and only if) $${{\,\mathrm{Hom}\,}}_R(R/{\mathfrak {a}}, {{\,\mathrm{H}\,}}_{{\mathfrak {a}}}^{i}(M))$$ is weakly Laskerian for all i.