Algebras and Representation Theory

, Volume 20, Issue 2, pp 487–529 | Cite as

Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings

Article

Abstract

In this paper we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with monomorphism categories. In the second part, we investigate monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.

Keywords

Monomorphism categories Morita rings Homological embeddings Gorenstein artin algebras Gorenstein-projective modules Gorenstein (sub)categories Coherent functors 

Mathematics Subject Classification (2010)

16E10 16E65 16G 16G50 16S50 

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References

  1. 1.
    Amiot, C., Iyama, O., Reiten, I.: Stable categories of Cohen-Macaulay modules and cluster categories. Am. J. Math. 137(3), 813–857 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Auslander, M.: Coherent functors. In: Proceeding Conference Categorical Algebra (La Jolla, Calif., 1965), pp 189–231. Springer, New York (1966)Google Scholar
  3. 3.
    Auslander, M. Representation dimension of artin algebras, Queen Mary College Notes (1971)Google Scholar
  4. 4.
    Auslander, M., Reiten, I.: Cohen-Macaulay and Gorenstein Algebras. Progress in Math 95, 221–245 (1991)MathSciNetMATHGoogle Scholar
  5. 5.
    Auslander, M., Reiten, I., Smalø, S.: Representation Theory of Artin Algebras. Cambridge University Press (1995)Google Scholar
  6. 6.
    Bass, H. The Morita theorems, mimeographed notes, University of Oregon (1962)Google Scholar
  7. 7.
    Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux Pervers, (French) [Perverse sheaves], Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Asterisque 100 Soc. Math. France, Paris (1982)Google Scholar
  8. 8.
    Beligiannis, A.: The Homological Theory of Contravariantly Finite Subcategories: Gorenstein Categories, Auslander-Buchweitz Contexts and (Co-)Stabilization. Comm. Algebra 28, 4547–4596 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Beligiannis, A.: On the Relative Homology of Cleft Extensions of Rings and Abelian Categories. J. Pure Appl. Algebra 150, 237–299 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Beligiannis, A.: Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras. J. Algebra 288(1), 137–211 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Beligiannis, A.: On algebras of finite Cohen-Macaulay type. Adv. Math. 226(2), 1973–2019 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc. 188(883), viii+207 (2007)MathSciNetMATHGoogle Scholar
  13. 13.
    Bennis, D., Mahdou, N.: Strongly Gorenstein projective, injective, and flat modules. J. Pure Appl. Algebra 210, 437–445 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Buchweitz, R.-O.: Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein rings, unpublished manuscript, p. 155 (1987)Google Scholar
  15. 15.
    Buchweitz, R.-O.: Morita contexts, idempotents, and Hochschild cohomology - with applications to invariant rings, Contemp. Math. Amer. Math, Soc., Providence, RI 331, 25–53 (2003)MathSciNetMATHGoogle Scholar
  16. 16.
    Bühler, T.: Exact categories. Expo. Math. 28(1), 1–69 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Chen, X.-W.: The stable monomorphism category of a Frobenius category. Math. Res. Lett. 18(1), 125–137 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Chen, X.-W.: Three results on Frobenius categories. Math. Z. 270(1-2), 43–58 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Christensen, L.W.: Gorenstein dimensions. In: Lecture Notes in math, vol. 1747. Springer, Berlin (2000)Google Scholar
  20. 20.
    Cline, E., Parshall, B., Scott, L.: Stratifying endomorphisms algebras. Mem. Amer. Math. Soc. 124(591), viii+119 (1996)MathSciNetMATHGoogle Scholar
  21. 21.
    Cohn, P.M. Morita equivalence and duality, Qeen Mary College Math. Notes (1966)Google Scholar
  22. 22.
    Fossum, R., Griffith, P., Reiten, I.: Trivial Extensions of Abelian Categories with Applications to Ring Theory, vol. 456. Springer L.N.M. (1975)Google Scholar
  23. 23.
    Franjou, V., Pirashvili, T.: Comparison of abelian categories recollements. Documenta Math. 9, 41–56 (2004)MathSciNetMATHGoogle Scholar
  24. 24.
    Green, E.L.: On the representation theory of rings in matrix form. Pacific. J. Math. 100(1), 138–152 (1982)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Green, E.L., Psaroudakis, C.: On Artin algebras arising from Morita contexts. Algebr. Represent. Theory 17(5), 1485–1525 (2014)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Happel, D.: Triangulated categories in the representation theory of finite dimensional algebras, London Math. Soc. Lecture Notes Ser., vol. 119. Cambridge University Press, Cambridge (1988)Google Scholar
  27. 27.
    Happel, D.: On Gorenstein algebras. In: Representation theory of finite groups and finite-dimensional algebras, Prog. Math., vol. 95, pp 389–404 (1991)Google Scholar
  28. 28.
    Keller, B.: Chain complexes and stable categories. Manuscripta Math. 67, 379–417 (1990)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Koenig, S., Nagase, H.: Hochschild cohomology and stratifying ideals. J. Pure Appl. Algebra 213(5), 886–891 (2009)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kussin, D., Lenzing, H., Meltzer, H.: Nilpotent operators and weighted projective lines. J. Reine Angew. Math. 685, 33–71 (2013)MathSciNetMATHGoogle Scholar
  31. 31.
    Li, Z.-W., Zhang, P.: A construction of Gorenstein-projective modules. J. Algebra 323(6), 1802–1812 (2010)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Matsui, H., Takahashi, R. Singularity categories and singular equivalences for resolving subcategories, arXiv:1412.8061, Math. Z. (to appear)
  33. 33.
    McConnell, J.C., Robson, J.C.: Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication. Wiley, Chichester (1987)Google Scholar
  34. 34.
    Orlov, D.: Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Tr. Mat. Inst. Steklova 246, 240–262 (2004). English transl.: Proc. Steklov. Inst. Math. 246 (2204), no. 3, 227–248MathSciNetMATHGoogle Scholar
  35. 35.
    Psaroudakis, C.: Representation Dimension, Cohen-Macaulay Modules and Triangulated Categories, Ph.D. thesis, p 201. University of Ioannina, Greece (2013)Google Scholar
  36. 36.
    Psaroudakis, C.: Homological Theory of Recollements of Abelian Categories. J. Algebra 398, 63–110 (2014)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Psaroudakis, C., Vitória, J.: Recollements of Module Categories. Appl. Categ. Structures 22(4), 579–593 (2014)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Psaroudakis, C., Vitória, J. Realisation functors in tilting theory, arXiv:1511.02677
  39. 39.
    Quillen, D.: Higher algebraic K-theory. I, in Algebraic K-theory, I: higher K-theories, Seattle, WA, 1972. Lecture Notes in Mathematics, vol. 341, pp 85–147. Springer, Berlin (1973)Google Scholar
  40. 40.
    Ringel, C.M., Schmidmeier, M.: Submodule categories of wild representation type. J. Pure Appl. Algebra 205(2), 412–422 (2006)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Ringel, C.M., Schmidmeier, M.: The Auslander-Reiten translation in submodule categories. Trans. Amer. Math. Soc. 360(2), 691–716 (2008)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Ringel, C.M., Schmidmeier, M.: Invariant subspaces of nilpotent linear operators. I, J. Reine Angew. Math. 614, 1–52 (2008)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Rowen, L.H.: Ring Theory. Student edition. Academic Press, Inc., Doston (1991)Google Scholar
  44. 44.
    Xi, C.C.: Adjoint functors and representation dimensions. Acta. Math. Sin. 22 (2), 625–640 (2006)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Xiong, B.L., Zhang, P., Zhang, Y.H.: Auslander-Reiten translations in monomorphism categories. Forum Math. 26, 863–912 (2014)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Xiong, B.L., Zhang, P.: Gorenstein-projective modules over triangular matrix Artin algebras. J. Algebra Appl. 11(4), 14 (2012)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Zhang, P.: Gorenstein-projective modules and symmetric recollements. J. Algebra 388, 65–80 (2013)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Zhang, P.: Monomorphism categories, cotilting theory, and Gorenstein-projective modules. J. Algebra 339, 181–202 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiPeople’s Republic of China
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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