Skew category algebras and modules on ringed finite sites (Q6169049)

If \(\mathfrak{R}\)\ is a sheaf of rings on a site \(\boldsymbol{C}=\left( \mathcal{C},\mathcal{J}\right) \), one can consider the right modules on the ringed site \(\left( \boldsymbol{C},\mathfrak{R}\right) \), which form an abelian category denoted by \(\mathfrak{M}od-\mathfrak{R}\). This is the major subject of this paper. The authors focus on finite categories, because all of their Grothendieck topologies, as well as the resulting topoi, are classified [\textit{M. Artin} (ed.) et al., Séminaire de géométrie algébrique du Bois-Marie 1963--1964. Théorie des topos et cohomologie étale des schémas. (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne, B. Saint-Donat. Tome 1: Théorie des topos. Exposés I à IV. 2e éd. Springer, Cham (1972; Zbl 0234.00007)]. The authors characterize module categories on ringed finite sites. Let \(k\)\ be a commutative ring with identity with \(k\)\ \(-Alg\)\ the category of unital associative \(k\)-algebras with identity and unital \(k\)-algebra\ homomorphisms. Let \(\mathfrak{R}\)\ be a presheaf of unital \(k\)-algebras on \(\mathcal{C}\). Motivated by the \textit{Grothendieck construction} \(Gr_{\mathcal{C}}\mathfrak{R}\)\ of \(\mathfrak{R}\)\ on \(\mathcal{C}\)\ [\textit{M. Raynaud} and \textit{A. Grothendieck} (ed.), Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1), dirigé par Alexander Grothendieck. Augmenté de deux exposés de M. Raynaud. Revêtements étales et groupe fondamental. Exposés I à XIII. (Seminar on algebraic geometry at Bois Marie 1960/61 (SGA 1), directed by Alexander Grothendieck. Enlarged by two reports of M. Raynaud. Ètale coverings and fundamental group). Springer, Cham (1971; Zbl 0234.14002), VI.8], the authors introduce the \textit{skew category algebra} \(\mathfrak{R}\left[ \mathcal{C}\right] \), as a linearization of \(Gr_{\mathcal{C}}\mathfrak{R}\). The right modules of \ form the module category \(\mathrm{Mod}-\mathfrak{R}\left[ \mathcal{C}\right] \). When \(\mathcal{C}\)\ is a poset, this construction has been investigated by \textit{M. Gerstenhaber} and \textit{S. D. Schack} [Trans. Am. Math. Soc. 310, No. 1, 135--165 (1988; Zbl 0706.16021)]. The synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] recalls basics on Grothendieck topologies, topoi, ringed sites and their modules, presenting some concrete examples. \item[\S 3] characterizes the modules on object-finite categories, establishing Theorem. Let \(\mathcal{C}\)\ be a small category and let \(\mathfrak{R}:\mathcal{C}^{\mathrm{op}}\rightarrow k\)\ \(-Alg\) be a presheaf of unital \(k\)-algebras on \(\mathcal{C}\). If \(\mathrm{Ob\,}\mathcal{C}\)\ is finite, then we have a category equivalence \[ \mathfrak{M}od-\mathfrak{R}\simeq\mathrm{Mod}-\mathfrak{R}\left[ \mathcal{C}\right] \] \item[\S 4] recalls Grothendieck and Verdier's classification of finite topoi, which allows of reducing the characterizations of general module categories to the situation in the preceding section, enabling the authors to establish categorical equivalences as follows. Theorem. Let \(\boldsymbol{C}=\left( \mathcal{C},\mathcal{J}\right) \) be a finite site and \(\mathfrak{R}:\mathcal{C}^{\mathrm{op}}\rightarrow k\)\ be a sheaf of unital \(k\)-algebras on \(\boldsymbol{C}\). Then \(\mathcal{J}=\mathcal{J}^{\mathcal{D}}\)\ is determined by a strictly full subcategory \(\mathcal{D}\), and there is a category equivalence \[ \mathfrak{M}od-\mathfrak{R}\simeq\mathrm{Mod}-\mathfrak{R}\mid_{\mathcal{D} }\left[ \mathcal{D}\right] \] where \(\mathfrak{R}\mid_{\mathcal{D}}\)\ is the restriction of \(\mathfrak{R}\)\ to \(\mathcal{D}\). Theorem. Let \(\boldsymbol{C}=\left( \mathcal{C},\mathcal{J}^{\mathcal{D}}\right) \) be a finite site and \(\mathfrak{R}:\mathcal{C}^{\mathrm{op}}\rightarrow k\)\ be a sheaf of \(k\)-algebras on \(\boldsymbol{C}\). If for each \(x\in\mathrm{Ob\,} \mathcal{D}\), \(\mathfrak{R}\left( x\right) \)\ is Noetherian, then there is a category equivalence \[ \mathrm{coh}-\mathfrak{R}\simeq\mathrm{mod}-\mathfrak{R}\mid_{\mathcal{D} }\left[ \mathcal{D}\right] \] \item[\S 5] gives several remarks. \end{itemize}