Resolving subcategories whose finitely presented module categories are abelian
Categories of additive contravariant functors from an essentially small (pre)additive category \(\mathcal{X}\) to abelian groups form a nice abelian category \(\operatorname{Mod}\mathcal{X}\) which has been studied in a systematic way, together with its full subcategory \(\operatorname{mod}\mathcal{X}\) of finitely presented functors, since the 1960's, in particular by Auslander -- see for example his seminal work on \textit{coherent functors} [\textit{M. Auslander}, Proc. Conf. Categor. Algebra, La Jolla 1965, 189--231 (1966; Zbl 0192.10902)]. In an article with \textit{S. O. Smalø} [J. Algebra 66, 61--122 (1980; Zbl 0477.16013)], \textit{M. Auslander} introduced the notion of \textit{contravariantly finite} subcategory of a nice abelian category. If an additive and essentially small category \(\mathcal{X}\) is contravariantly finite, then the category \(\operatorname{mod}\mathcal{X}\) is abelian. The article under review studies the converse for some additive full subcategories of finitely generated modules over a commutative noetherian ring. It shows that the category \(\operatorname{mod}\mathcal{X}\) may be abelian without \(\mathcal{X}\) being contravariantly finite and gives additional conditions which ensure that \(\mathcal{X}\) is contravariantly finite if \(\operatorname{mod}\mathcal{X}\) is abelian.
modules over commutative noetherian rings
scientific article; zbMATH DE number 7371717
scientific article