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On homological dimensions in some functor categories

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Abstract

In this paper, we investigate the homological properties of the functor categories (mod−R, Ab) and ((mod−R)op, Ab). Some new homological dimensions in these functor categories such as FP-projecive dimensions and cotorsion dimensions for functors and functor categories are introduced and studied. We also characterize functor categories of homological dimensions zero and explore the connections among some different homological dimensions.

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References

  1. M. Auslander, “Coherent Functors,” in Proceedings of the Conference on Categorical Algebra, La Jolla, CA, 1965 (Springer, New York, 1966), pp. 189–231.

    Google Scholar 

  2. M. Auslander, Representation Dimension of Artin Algebras, in Queen Mary Coll. Math. Notes (Queen Mary College, London, 1971).

    Google Scholar 

  3. M. Auslander and I. Reiten, “Stable equivalence of Artin algebras,” in Lecture Notes in Math., Vol. 353: Proceedings of the Conference on Orders, Group Rings and Related Topics, Columbus, OH, 1972 (Springer-Verlag, Berlin, 1973), pp. 8–71.

    Google Scholar 

  4. M. Auslander, “Functors and morphisms determined by objects,” in Lecture Notes in Pure and Appl. Math. (Marcel Dekker, New York, 1978), Vol. 37, pp. 1–244.

    MathSciNet  MATH  Google Scholar 

  5. M. Auslander, “A functorial approach to representation theory,” in Lecture Notes in Math., Vol. 944: Representations of Algebras (Springer-Verlag, New York, 1982), pp. 105–179.

    Google Scholar 

  6. A. Beligiannis, “Homotopy theory of modules and Gorenstein rings,” Math. Scand. 89, 5–45 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Brune, “On the global dimension of the functor category ((mod R)op, Ab) and a theorem of Kulikov,” J. Pure Appl. Algebra 28, 31–39 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  8. W. Crawley-Boevey, “Modules of finite length over their endomorphism rings,” in LondonMath. Soc. Lecture Note Ser., Vol. 168: Representations ofAlgebras and Related Topics (Cambridge Univ. Press, Cambridge, 1992), pp. 127–184.

    Chapter  Google Scholar 

  9. W. Crawley-Boevey, “Locally finitely presented additive categories,” Comm. Algebra 22 (5), 1641–1674 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Gruson and C. U. Jensen, “Dimensions cohomologique reliées aux foncteurs lim i,” in Lecture Notes in Math., Vol: 867: Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, Paris 1980 (Springer-Verlag, 1981), pp. 234–294.

    Chapter  Google Scholar 

  11. I. Herzog, “The phantom cover of a module,” Adv. Math. 215 (1), 220–249 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  12. I. Herzog, “Contravariant functors on the category of finitely presented modules,” Israel J. Math. 167, 347–410 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  13. C. U. Jensen, “On the global dimension of the functor category (mod R, Ab),” J. Pure Appl. Algebra 11, 45–51 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  14. L. X. Mao, “On covers and envelopes in some functor categories,” Comm. Algebra 41(5), 1655–1684 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  15. L. X. Mao, “Notes on several orthogonal classes of flat and FP-injective functors,” Math. Notes 95 (1–2), 78–90 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Prest, Model Theory and Modules, in London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 1988), Vol.130.

  17. M. Saorin and A. Del Valle, “Covers and envelopes in functor categories,” in Lecture Notes in Pure and Appl. Math., Vol. 210: Interactions between Ring Theory and Representations of Algebras (Marcel Dekker, New York, 2000).

    Google Scholar 

  18. D. Simson, “On pure global dimension of locally finitely presented Grothendieck categories,” Fund. Math. 96, 91–116 (1977).

    MathSciNet  MATH  Google Scholar 

  19. B. Stenström, Rings of Quotients: An Introduction to Methods of Ring Theory, in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 1975), Vol.217.

    Google Scholar 

  20. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, in de Gruyter Exp. Math. (Walter de Gruyter, Berlin, 2000), Vol. 30.

  21. E. E. Enochs and L. Oyonarte, Covers, Envelopes and Cotorsion Theories (Nova Sci. Publ., New York, 2002).

    MATH  Google Scholar 

  22. T. Y. Lam, Lectures on Modules and Rings (Springer-Verlag, New York–Heidelberg–Berlin, 1999).

    Book  MATH  Google Scholar 

  23. J. J. Rotman, An Introduction to Homological Algebra, in Pure Appl.Math. (Academic Press, New York, 1979), Vol.85.

    Google Scholar 

  24. R. S. Pierce, “The global dimension of Boolean rings,” J. Algebra 7, 91–99 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Kielpinski and D. Simson, “On pure homological dimension,” Bull. Acad. Polon. Sci. XXIII (1), 1–6 (1975).

    MathSciNet  MATH  Google Scholar 

  26. N. Q. Ding and J. L. Chen, “The flat dimensions of injective modules,” Manuscripta Math. 78, 165–177 (1993).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Nanjing Institute of Technology, Nanjing, China

    L. X. Mao

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  1. L. X. Mao
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Correspondence to L. X. Mao.

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Mao, L.X. On homological dimensions in some functor categories. Math Notes 101, 631–644 (2017). https://doi.org/10.1134/S000143461703021X

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  • DOI: https://doi.org/10.1134/S000143461703021X

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