Applied Categorical Structures

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

Locally Stable Grothendieck Categories: Applications

  • Florencio Castaño-Iglesias
  • Constantin Năstăsescu
  • Laura Năstăsescu


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.


Grothendieck category Localizing subcategory V-category Coalgebra Graded ring 

Mathematics Subject Classifications (2010)

18E15 18E40 16W30 16W50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dăuş, L., Năstăsescu, C., Van Oystaeyen, F.: V-categories. Applications to graded rings. Commun. Algebra 37, 3248–3258 (2009)zbMATHCrossRefGoogle Scholar
  2. 2.
    Cahen, P.-J.: Torsion Theories and Commutative Algebras. Ph.D. Thesis, Queen’s University, Kingston (1973)Google Scholar
  3. 3.
    Gabriel, P.: Des catégories abeliennes. Bull. Soc. Math. France 90, 323–448 (1962)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Gómez-Torrecillas, J., Năstăsescu, C., Torrecillas, B.: Localization in coalgebras. Applications to finiteness conditions. J. Algebra Appl. 6(2), 233–243 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Lin, B. I.-P.: Morita’s theorem for coalgebras. Commun. Algebra 1(4), 311–344 (1974)zbMATHCrossRefGoogle Scholar
  6. 6.
    Năstăsescu, C., Popescu, N.: Sur la structure des objects de certaines catégories abéliennes. C.R. Math. Acad. Sci. Paris, Sér. A–B 262, 1295–1297 (1966)zbMATHGoogle Scholar
  7. 7.
    Năstăsescu, C., Van Oystaeyen, F.: Dimensions of Ring Theory, Mathematics and its Applications. Reidel, Dordrecht (1987)CrossRefGoogle Scholar
  8. 8.
    Năstăsescu, C., Van Oystaeyen, F.: Methods of Graded Rings. Lecture Notes, vol. 1836. Springer, Berlin (2004)zbMATHCrossRefGoogle Scholar
  9. 9.
    Năstăsescu, C., Torrecillas, B.: Atomical Grothendieck categories. Int. J. Math. Math. Sci. 71, 4501–4509 (2003)CrossRefGoogle Scholar
  10. 10.
    Stenström, B.: Rings of Quotients. Springer, Berlin (1975)zbMATHCrossRefGoogle Scholar
  11. 11.
    Sweedler, M.E.: Hopf Algebras. Benjamin, New York (1969)Google Scholar
  12. 12.
    Takeuchi, M.: Morita theorems for categories of comodules. J. Fac. Sci. Univ. Tokyo 24, 629–644 (1977)zbMATHGoogle Scholar

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

Personalised recommendations