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Pure Injective and Absolutely Pure Sheaves

Part of: Categories and theories Categories and geometry Modules, bimodules and ideals Abelian categories Algebraic geometry: Foundations Categories with structure

Published online by Cambridge University Press:  20 November 2015

Edgar Enochs ,
Sergio Estrada  and
Sinem Odabaşi
Edgar Enochs
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA (enochs@ms.uky.edu)
Sergio Estrada
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain (sestrada@um.es; sinem.odabasi1@um.es)
Sinem Odabaşi
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain (sestrada@um.es; sinem.odabasi1@um.es)
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Abstract

We study two notions of purity in categories of sheaves: the categorical and the geometric. It is shown that pure injective envelopes exist in both cases under very general assumptions on the scheme. Finally, we introduce the class of locally absolutely pure (quasi-coherent) sheaves with respect to the geometrical purity, and characterize locally Noetherian closed subschemes of a projective scheme in terms of the new class.

Keywords

local puritycategorical purityconcentrated schemelocally finitely presented categorypure injective sheafabsolutely pure sheaf

MSC classification

Secondary: 18E15: Grothendieck categories 18C35: Accessible and locally presentable categories 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 18F20: Presheaves and sheaves 14A15: Schemes and morphisms 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 623 - 640
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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