Let R be a commutative Noetherian ring, a\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {a}}$$\end{document} an ideal of R, and M an R-module. We prove that the category of a\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {a}}$$\end{document}-weakly cofinite modules is a Melkersson subcategory of R-modules whenever dimR≤1\documentclass[12pt]{minimal}
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\begin{document}$$\dim R\le 1$$\end{document} and is an Abelian subcategory whenever dimR≤2\documentclass[12pt]{minimal}
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\begin{document}$$\dim R\le 2$$\end{document}. We also prove that if (R,m)\documentclass[12pt]{minimal}
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\begin{document}$$(R,{\mathfrak {m}})$$\end{document} is a local ring with dimR/a≤2\documentclass[12pt]{minimal}
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\begin{document}$$\dim R/{\mathfrak {a}}\le 2$$\end{document} and SuppR(M)⊆V(a)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Supp}\,}}_R(M)\subseteq {{\,\mathrm{V}\,}}({\mathfrak {a}})$$\end{document}, then M is a\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {a}}$$\end{document}-weakly cofinite if (and only if) HomR(R/a,M)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Hom}\,}}_R(R/{\mathfrak {a}}, M)$$\end{document}, ExtR1(R/a,M)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Ext}\,}}_{R}^{1}(R/{\mathfrak {a}}, M)$$\end{document} and ExtR2(R/a,M)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Ext}\,}}_{R}^{2}(R/{\mathfrak {a}}, M)$$\end{document} are weakly Laskerian. In addition, we prove that if (R,m)\documentclass[12pt]{minimal}
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\begin{document}$$(R,{\mathfrak {m}})$$\end{document} is a local ring with dimR/a≤2\documentclass[12pt]{minimal}
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\begin{document}$$\dim R/{\mathfrak {a}}\le 2$$\end{document} and n∈N0\documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}_0$$\end{document}, such that ExtRi(R/a,M)\documentclass[12pt]{minimal}
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\begin{document}$${{{\,\mathrm{Ext}\,}}}^{i}_R(R/{\mathfrak {a}},M)$$\end{document} is weakly Laskerian for all i, then Hai(M)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{H}\,}}_{{\mathfrak {a}}}^{i}(M)$$\end{document} is a\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {a}}$$\end{document}-weakly cofinite for all i if (and only if) HomR(R/a,Hai(M))\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Hom}\,}}_R(R/{\mathfrak {a}}, {{\,\mathrm{H}\,}}_{{\mathfrak {a}}}^{i}(M))$$\end{document} is weakly Laskerian for all i.
