Zoran Skoda MR4328537

Mart'in Ortiz-Morales, Martha Lizbeth Shaid Sandoval-Miranda, Valente Santiago-Vargas,

Gabriel localization in functor categories.

Comm. Algebra 49 (2021), no. 12, 5273–5296

doi

18A25 (16D90 16G10 18E05 18E35)

Any unital ring $R$ defines a one object category $\Sigma R$ enriched over the category of abelian groups $\mathbf{Ab}$. Morphisms of $\Sigma R$ are the elements of $R$ and the composition is given by the multiplication of $R$. A left $R$-module is then an additive functor $\Sigma R\to\mathbf{Ab}$. More generally, for any small preadditive category $\mathcal{C}$, the category $\mathrm{Mod}(\mathcal{C})$ of additive functors $\mathcal{C}\to\mathbf{Ab}$ shares many properties with the category of modules over a ring. $\mathrm{Mod}(\mathcal{C})$ is a Grothendieck abelian category and the corepresentable functors form a set of finitely generated projective generators. In particular, $\mathrm{Mod}(\mathcal{C})$ is locally finitely presentable. Categories of this form are additive analogues of presheaf categories. In particular, the exact localizations functors having right adjoint are analogues of sheafification functors for presheaves on Grothendieck sites.

In his thesis $[$Des cat'egories ab'eliennes. (French) Bull. Soc. Math. France 90 (1962), 323–448 MR232821$]$, P.\ Gabriel developed general theory of localization of abelian categories, in terms of closed, thick and localizing subcategories, and related notions. Localizing subcategories of well-powered abelian categories correspond to exact localization functors having right adjoint. In the case of full categories of modules over a ring $R$, Gabriel defined topologizing (synonyms: linear, uniform) filters of ideals of $R$ and a subclass of topologizing filters which are idempotent with respect to certain associative operation (Gabriel composition of filters). He constructed a bijection between topologizing filters of $R$ and closed subcategories of $\mathrm{Mod}(R)$, under which the idempotent topologizing filters (called Gabriel filters in the work under review) correspond to localizing subcategories, hence also to exact localization functors having right adjoint.

The work under review is a sequel of the work $[$M.\ Ortiz-Morales, S.\ Diaz-Alvarado, Linear filters and hereditary torsion theories in functor categories. International Journal of Algebra 9:25–41 (2015)$]$ which generalizes Gabriel filters to small preadditive categories. The two works together develop localization theory of $\mathrm{Mod}(\mathcal{C})$ in terms of filters. Main result of the present paper is the bijection between Gabriel filters of any small preadditive category $\mathcal{C}$ and exact localization functors of $\mathrm{Mod}(\mathcal{C})$ having right adjoint. However, all these results are known. Popescu’s monograph $[$Abelian categories with applications to rings and modules, London Mathematical Society Monographs 3, Acad. Press 1973. xii+467 pp. MR340375$]$, cited by the authors, defines Gabriel filter on a small preadditive category $\mathcal{C}$ under the name of a left localizing system in Section 4.9 and proves the bijection with localizing sucategories (Theorem 4.9.1). Correspondence of the latter with hereditary torsion theories is standard and can be read from the rest of the book.

The present article is readable, detailed and explicit. Unfortunately, the complete analogy between Gabriel filters on $\mathcal{C}^{\mathrm{op}}$ and Grothendieck topologies is not even mentioned. For example, endofunctor $\mathbb{L}$ on $\mathrm{Mod}(\mathcal{C})$ studied in this work is an additive analogue of the $+$-construction in sheaf theory as apparent e.g. in [D. Murfet, Localization of ringoids. 2006].

Gabriel filters (either for $R$ or $\mathcal{C}$) are sometimes called additive Grothendieck topologies, e.g.\ in $[$W.Lowen, A generalization of the Gabriel–Popescu theorem. J. Pure Appl. Alg. 190:1–3 (2004) 197–211$]$ where Theorem 2.6 also states the main result of the present article (for the dual of $\mathcal{C}$). More generally, let $\mathcal{V}$ be a locally finitely presentable symmetrical monoidal closed category and $\mathcal{C}$ a small $\mathcal{V}$-enriched category. Then the category $[\mathcal{C}^\mathrm{op},\mathcal{V}]$ of $\mathcal{V}$-valued $\mathcal{V}$-enriched functors on the dual of $\mathcal{C}$ may be considered as a category of enriched presheaves. Axioms for $\mathcal{V}$-enriched Grothendieck topologies may be stated in terms of $\mathcal{V}$-subfunctors of representable functors. A bijection between $\mathcal{V}$-localizations of $[\mathcal{C}^\mathrm{op},\mathcal{V}]$ having right adjoint and preserving finite limits and $\mathcal{V}$-enriched Grothendieck topologies on $\mathcal{C}$ has been exhibited in [F. Borceux, C. Quinteiro, A theory of enriched sheaves. Cahiers Topologie Geom. Differentielle Catég. 37 (1996), no. 2, 145162 MR1394507].