Applied Categorical Structures

, Volume 21, Issue 2, pp 105–118 | Cite as

Locally Stable Grothendieck Categories: Applications

  • Florencio Castaño-Iglesias
  • Constantin Năstăsescu
  • Laura Năstăsescu
Article
  • 149 Downloads

Abstract

We introduce for any Grothendieck category the notion of stable localizing subcategory, as a localizing subcategory that can be written as an intersection of localizing subcategories defined by indecomposable injectives. A Grothendieck category in which every localizing subcategory is stable is called a locally stable category. As a main result we give a characterization of these categories in terms of the local stability of a localizing subcategory and its quotient category. The locally coirreducible categories (in particular, the categories with Gabriel dimension) and the locally noetherian categories are examples of locally stable categories. We also present some applications to the category of modules over a left fully bounded noetherian ring, to the category of comodules over a coalgebra and to the category of modules over graded rings.

Keywords

Grothendieck category Localizing subcategory V-category Coalgebra Graded ring 

Mathematics Subject Classifications (2010)

18E15 18E40 16W30 16W50 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Florencio Castaño-Iglesias
    • 1
  • Constantin Năstăsescu
    • 2
  • Laura Năstăsescu
    • 2
  1. 1.Departamento de Estadística y Matemática AplicadaUniversidad de AlmeríaAlmeríaSpain
  2. 2.Faculty of Mathematics and InformaticsUniversity of BucharestBucharest 1Romania

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