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The Category of Maximal Cohen–Macaulay Modules as a Ring with Several Objects

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Abstract

Over a commutative local Cohen–Macaulay ring, we view and study the category of maximal Cohen–Macaulay modules as a ring with several objects. We compute the global dimension of this category and thereby extend some results of Iyama and Leuschke.

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References

  1. Auslander, M.: Coherent functors. In: Proceedings of Conference on Categorical Algebra (La Jolla, California, 1965), pp. 189–231. Springer, New York (1966)

  2. Auslander, M.: Isolated singularities and existence of almost split sequences. In: Representation Theory, II (Ottawa, Ontario, 1984), Lecture Notes in Mathematics, vol. 1178, pp. 194–242. Springer, Berlin (1986)

  3. Auslander, M., Buchweitz, R.-O.: The homological theory of maximal Cohen-Macaulay approximations. In: Mém. Soc. Math. France (N.S.), Colloque en l’honneur de Pierre Samuel (Orsay, 1987), no. 38, pp. 5–37 (1989)

  4. Auslander M., Reiten I.: Grothendieck groups of algebras and orders. J. Pure Appl. Algebra 39(1–2), 1–51 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)

  6. Buchweitz R.-O., Greuel G.-M., Schreyer F.-O.: Cohen–Macaulay modules on hypersurface singularities II. Invent. Math. 88(1), 165–182 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Christensen, L.W.: Gorenstein Dimensions, Lecture Notes in Mathematics, vol. 1747. Springer, Berlin (2000)

  8. Enochs, E.E., Jenda, Overtoun M.G.: Relative Homological Algebra, de Gruyter Expositions in Mathematics, vol. 30. Walter de Gruyter & Co., Berlin (2000)

  9. Freyd, P.: Abelian Categories. An Introduction to the Theory of Functors, Harper’s Series in Modern Mathematics. Harper & Row Publishers, New York (1964)

  10. Freyd, P.: Representations in abelian categories. In: Proceedings of Conference on Categorical Algebra (La Jolla, California, 1965), pp. 95–120. Springer, New York (1966)

  11. Ischebeck F.: Eine Dualität zwischen den Funktoren Ext und Tor. J. Algebra 11, 510–531 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  12. Iyama, O.: Rejective subcategories of artin algebras and orders (preprint) (2003). arXiv:math/0311281v1 [math.RT]

  13. Iyama O.: Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories. Adv. Math. 210(1), 22–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Leuschke G.J.: Endomorphism rings of finite global dimension. Can. J. Math. 59(2), 332–342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mitchell B.: Rings with several objects. Adv. Math. 8, 1–161 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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  1. University of Copenhagen, 2100, Copenhagen Ø, Denmark

    Henrik Holm

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  1. Henrik Holm
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Correspondence to Henrik Holm.

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Holm, H. The Category of Maximal Cohen–Macaulay Modules as a Ring with Several Objects. Mediterr. J. Math. 13, 885–898 (2016). https://doi.org/10.1007/s00009-015-0557-8

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  • DOI: https://doi.org/10.1007/s00009-015-0557-8

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