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Exact Krull–Schmidt categories with finitely many indecomposables

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More than 30 years ago, M. Auslander proved, for Cohen–Macaulay orders over a complete regular local ring of dimension d, that existence of almost split sequences is equivalent to the presence of an isolated singularity. CM-finite Cohen–Macaulay orders form an important special class. They have been studied intensively for \(d\le 2\), with scattered results for \(d>2\). More recently, the question of CM-finiteness in its widest sense has become relevant for exact categories arising in commutative algebra, non-commutative singularity theory, Gorenstein homological algebra, and related topics. In the paper, two types of criteria for CM-finiteness are established which extend previously known results to arbitrary dimension. The first type of criteria deals with Krull–Schmidt categories with almost split sequences. It is shown that finite CM-type is closely related, but not equivalent to finiteness with respect to L-functors. The second type of criteria appeals to non-commutative crepant resolutions.

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References

  1. Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Springer, New York (1974)

    Book  MATH  Google Scholar 

  2. Angeleri Hugel, L., Coelho, F.U.: Infinitely generated tilting modules of finite projective dimension. Forum Math. 13(2), 239–250 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras, vol. 1, London Mathematical Society Student Texts, p. 65. Cambridge University Press, Cambridge (2006)

  4. Auslander, M.: Anneaux de Gorenstein, et torsion en algèbre commutative. In: Sém. Algèbre Commutative P. Samuel, 1966/67, Paris (1967)

  5. Auslander, M.: The Representation Dimension of Artin Algebras, Queen Mary College Mathematics Notes (1971), Republished in Selected works of Maurice Auslander. Amer. Math. Soc., Providence (1999)

    Google Scholar 

  6. Auslander, M.: Isolated Singularities and Existence of Almost Split Sequences. Springer Lecture Notes in Mathematics, vol. 1178 (1986)

  7. M. Auslander, M. Bridger: Stable Module Theory, Memoirs of the American Mathematical Society, vol. 94, p. 146. American Mathematical Society, Providence (1969)

  8. Auslander, M., Buchweitz, R.: The homological theory of maximal Cohen–Macaulay approximations. Mém. Soc. Math. France 38, 5–37 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Auslander, M., Smalø, S.O.: Preprojective modules over artin algebras. J. Algebra 66, 61–122 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  11. Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras, Corrected reprint of the 1995 original, Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  12. Bazzoni, S.: A characterization of \(n\)-cotilting and \(n\)-tilting modules. J. Algebra 273(1), 359–372 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Beligiannis, A.: On algebras of finite Cohen–Macaulay type. Adv. Math. 226(2), 1973–2019 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc., vol. 188, no. 883 (2007)

  15. Brown, K.A., Lenagan, T.H.: A note on Jacobson’s conjecture for right Noetherian rings. Glasg. Math. J. 23, 7–8 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Buchweitz, R.-O.: Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings, p. 155 (1987)

  17. 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 

  18. Bühler, T.: Exact categories. Expo. Math. 28(1), 1–69 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cauchon, G.: Sur l’intersection des puissances du radical d’un T-anneau noethérie. C. R. Acad. Sci. Paris Sér. A 279, 91–93 (1974)

    MathSciNet  MATH  Google Scholar 

  20. Chen, X.-W.: Three results on Frobenius categories. Math. Z. 270(1–2), 43–58 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Christensen, L.W., Piepmeyer, G., Striuli, J., Takahashi, R.: Finite Gorenstein representation type implies simple singularity. Adv. Math. 218(4), 1012–1026 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dao, H., Iyama, O., Takahashi, R., Vial, C.: Non-commutative resolutions and Grothendieck groups. J. Noncommut. Geom. 9, 21–34 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Enochs, E.E.: Injective and flat covers, envelopes and resolvents. Isr. J. Math. 39(3), 189–209 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  24. Enochs, E.E., Jenda, O.M.G.: Gorenstein injective and projective modules. Math. Z. 220(4), 611–633 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  26. Enomoto, H.: Classifying exact categories via Wakamatsu tilting. J. Algebra 485, 1–44 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Freyd, P.: Representations in Abelian Categories. In: Proc. Conf. Categorial Algebra, La Jolla, pp. 95–120. Springer, New York (1966)

  28. Gabriel, P., Roiter, A.V.: Representations of Finite-Dimensional Algebras. Springer, Berlin (1997)

    Book  Google Scholar 

  29. Holm, H.: Gorenstein homological dimensions. J. Pure Appl. Algebra 189(1–3), 167–193 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Holm, H.: Gorenstein derived functors. Proc. Am. Math. Soc. 132(7), 1913–1923 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hovey, M.: Cotorsion pairs, model category structures, and representation theory. Math. Z. 241(3), 553–592 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Igusa, K., Todorov, G.: Radical Layers of Representable Functors. J. Algebra 89, 105–147 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  33. Igusa, K., Todorov, G.: A characterization of finite Auslander–Reiten quivers. J. Algebra 89, 148–177 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  34. Igusa, K., Todorov, G.: A numerical characterization of finite Auslander–Reiten quivers. Representation Theory I, Finite Dimensional Algebras, Proceedings, Springer LNM, vol. 1177, pp. 181–198 (1984)

  35. Iyama, O.: \(\tau \)-categories, I, Ladders. Algebras Represent. Theory 8(3), 297–321 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Iyama, O.: \(\tau \)-categories II: Nakayama pairs and rejective subcategories, algebras and representation theory. Algebras Represent. Theory 8(4), 449–477 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Iyama, O.: \(\tau \)-categories III: Auslander orders and Auslander–Reiten quivers. Algebras Represent. Theory 8(5), 601–619 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Iyama, O., Reiten, I.: Fomin–Zelevinsky mutation and tilting modules over Calabi–Yau algebras. Am. J. Math. 130(4), 1087–1149 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Jategaonkar, A.V.: Jacobson’s conjecture and modules over fully bounded Noetherian rings. J. Algebra 30, 103–121 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  41. Keller, B.: Chain complexes and stable categories. Manuscr. Math. 67, 379–417 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kalck, M., Iyama, O., Wemyss, M., Yang, D.: Frobenius categories, Gorenstein algebras and rational surface singularities. Compos. Math. 151(3), 502–534 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kerner, O., Skowroński, A.: On module categories with nilpotent infinite radical. Compos. Math. 77(3), 313–333 (1991)

    MathSciNet  MATH  Google Scholar 

  44. Knörrer, H.: Cohen-Macaulay modules on hypersurface singularities. I. Invent. Math. 88(1), 153–164 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  45. Krause, H.: A short proof for Auslander’s defect formula, Special issue on linear algebra methods in representation theory. Linear Algebra Appl. 365, 267–270 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Li, Z.-W., Zhang, P.: Gorenstein algebras of finite Cohen–Macaulay type. Adv. Math. 223(2), 728–734 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  47. S. Mac Lane: Categories for the Working Mathematician, New York, Heidelberg, Berlin (1971)

  48. Miyashita, Y.: Tilting modules of finite projective dimension. Math. Z. 193(1), 113–146 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  49. Miyachi, J.: Duality for derived categories and cotilting bimodules. J. Algebra 185(2), 583–603 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  50. Orlov, D.O.: Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004). Algebr. Geom. Metody, Svyazi i Prilozh., pp. 240–262; Engl. translation: Proc. Steklov Inst. Math. 246, 227–248 (2004)

  51. Reiner, I.: Maximal Orders, Corrected reprint of the 1975 original, With a foreword by M. J. Taylor, London Mathematical Society Monographs, New Series, vol. 28. The Clarendon Press, Oxford University Press, Oxford (2003)

  52. Rump, W.: Almost Abelian Categories. Cahiers de topologie et géométrie différentielle catégoriques XLI I, pp. 163–225 (2001)

  53. Rump, W.: Triads. J. Algebra 280, 435–462 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  54. Rump, W.: Large lattices over orders. Proc. Lond. Math. Soc. 91(1), 105–128 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  55. Rump, W.: Large indecomposables over representation-infinite orders and algebras. Arch. Math. 84(1), 11–21 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  56. Rump, W.: The category of lattices over a lattice-finite ring. Algebras Represent. Theory 8(3), 323–345 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  57. Rump, W.: L-functors and almost split sequences. Commun. Algebra 33(1), 73–95 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  58. Rump, W.: Triadic categories without localization. J. Algebra 322(4), 995–1012 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  59. Salce, L.: Cotorsion theories for abelian groups, Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977), pp. 11–32. Academic Press, London-New York (1979)

  60. Šaroch, J., Trlifaj, J.: Completeness of cotorsion pairs. Forum Math. 19(4), 749–760 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  61. Takahashi, R.: Contravariantly finite resolving subcategories over commutative rings. Am. J. Math. 133(2), 417–436 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  62. Takahashi, R.: Classifying resolving subcategories over a Cohen–Macaulay local ring. Math. Z. 273(1–2), 569–587 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  63. Thomason, R.W., Trobaugh, T.: Higher algebraic \(K\)-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III. Progr. Math., vol. 88, pp. 247–435. Birkhäuser Boston, Boston (1990)

  64. van den Bergh, M.: Three-dimensional flops and noncommutative rings. Duke Math. J. 122(3), 423–455 (2004)

    MathSciNet  MATH  Google Scholar 

  65. van den Bergh, M.: Non-Commutative Crepant Resolutions, The Legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  66. Wakamatsu, T.: Tilting modules and Auslander’s Gorenstein property. J. Algebra 275(1), 3–39 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author wishes to thank an anonymous referee for a very careful reading of the manuscript and many thoughtful comments which led to an improved presentation of the paper and a strengthening of a previous version of Theorem 5.6.

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  1. Institute for Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70550, Stuttgart, Germany

    Wolfgang Rump

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Rump, W. Exact Krull–Schmidt categories with finitely many indecomposables. Math. Z. 300, 761–789 (2022). https://doi.org/10.1007/s00209-021-02710-0

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