Locally finitely presented categories of sheaves (Q1039770)

Let \(\mathcal{A}\) be a locally finitely presented Grothendieck category and \(\mathrm{fp}(\mathcal{A})\) the full subcategory of finitely presented objects in \(\mathcal{A}\). In the main result of the paper the author shows that if a full subcategory \(\mathcal{B}\) of \(\mathcal{A}\) is of finite type (i.e., it is equivalent to a localization in the sense of Bousfield \(\mathcal{B}(\Sigma)\), where \(\Sigma\) is a set of morphisms in \(\mathrm{fp}(\mathcal{A})\)) then \(\mathcal{B}\) is locally finitely presented. This result is applied to hereditary torsion theories in \(\mathcal{A}\), sheaves and separated presheaves on a generalized ringed space. For torsion theories it is proved that there is a one-to-one correspondence between the Serre subcategories of \(\mathrm{fp}(\mathcal{A})\) and hereditary torsion theories of \(\mathcal{A}\) such that the class of torsion-free objects is closed under filtered colimits. Moreover, these torsion-free classes are locally finitely presented (Theorem 1 and Proposition 5). For (pre)sheaves it is proved that if \(\mathcal{C}\) is a presheaf of skeletally small left abelian categories on a topological space \(X\) such that the quasi-compact open sets form a basis for \(X\), then the full subcategories \(\mathrm{Sh}(\mathcal{C})\) of \(\mathcal{C}\)-sheaves and \(\mathrm{PScs}(\mathcal{C})\) of separated continuous \(\mathcal{C}\)-presheaves are of finite type, hence finitely presented (Proposition 8). This result is applied to (noetherian) ringed spaces (Corollary 1 and Corollary 2).