Locally finitely presented categories of sheaves
type: Item
instance of: scholarly article
title: Locally finitely presented categories of sheaves
zbMATH Open document ID: 1187.18004
author: Wolfgang Rump
published in: Journal of Pure and Applied Algebra
publication date: 2009-11-23T00:00:00Z
review text: 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).
reviewed by: Simion Sorin Breaz
Mathematics Subject Classification ID: 18C35
Mathematics Subject Classification ID: 18E05
Mathematics Subject Classification ID: 18A35
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 18E35
zbMATH DE Number: 5637054
zbMATH Keywords: Grothendieck category
zbMATH Keywords: finitely presented object
zbMATH Keywords: Bousfield localization
zbMATH Keywords: torsion theory
zbMATH Keywords: presheaf
zbMATH Keywords: ringed space
MaRDI profile type: Publication
full work available at URL: j.jpaa.2009.05.014
OpenAlex ID: W2005925509
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DOI: 10.1016/J.JPAA.2009.05.014
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description: scientific article; zbMATH DE number 5637054