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Mathematische Zeitschrift

, Volume 287, Issue 3–4, pp 1109–1155 | Cite as

Gorenstein homological algebra and universal coefficient theorems

  • Ivo Dell’Ambrogio
  • Greg Stevenson
  • Jan Šťovíček
Article

Abstract

We study criteria for a ring—or more generally, for a small category—to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman’s Brown–Adams representability theorem for compactly generated categories.

Keywords

Gorenstein homological algebra Triangulated category Universal coefficient theorem Kasparov’s KK-theory 

Mathematics Subject Classification

Primary 16E65 Secondary 18E30 19K35 46L80 

Notes

Acknowledgements

We are very grateful to Ralf Meyer for asking us the interesting questions that have prompted our collaboration. Thanks are also due to him for first noticing that the Gorenstein closed condition of Theorem 8.6 is not only sufficient but also necessary. Another source of inspiration was a talk given in Bielefeld by Claus Ringel where he presented the results of [61] (see Example 9.21). The paper was finished during the research program IRTATCA (Interactions between Representation Theory, Algebraic Topology and Commutative Algebra) at Centre de Recerca Matemàtica in Barcelona. Our thanks also belong to the organizers for the hospitality and the creative environment there.

References

  1. 1.
    Auslander, M.: Representation theory of Artin algebras. I. Commun. Algebra 1, 177–268 (1974)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Auslander, M., Bridger, M.: Stable Module Theory. Memoirs of the American Mathematical Society. No. 94. American Mathematical Society, Providence (1969)MATHGoogle Scholar
  3. 3.
    Auslander, M., Buchweitz, R.-O.: The homological theory of maximal Cohen-Macaulay approximations. Mém. Soc. Math. France (N.S.) 38, 5–37 (1989) (English, with French summary). Colloque en l’honneur de Pierre Samuel (Orsay, 1987)Google Scholar
  4. 4.
    Becker, H.: Models for singularity categories. Adv. Math. 254, 187–232 (2014)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Balmer, P., Schlichting, M.: Idempotent completion of triangulated categories. J. Algebra 236(2), 819–834 (2001)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Beligiannis, A.: Relative homological algebra and purity in triangulated categories. J. Algebra 227(1), 268–361 (2000)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Beligiannis, A.: Some Ghost Lemmas, available at http://www.mathematik.uni-bielefeld.de/~sek/2008/ghosts.pdf. Survey for “The Representation Dimension of Artin Algebras”, Bielefeld (2008)
  9. 9.
    Bondarko, M.V.: On torsion pairs, (well generated) weight structures, adjacent t-structures, and related (co)homological functors. preprint (2016), available at arXiv:1611.00754v3
  10. 10.
    Bravo, D., Gillespie, J., Hovey, M.: The stable module category of a general ring. preprint (2014), available at arXiv:1405.5768v1
  11. 11.
    Bruns, W., Herzog, J.: Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  12. 12.
    Buchweitz, R.O.: Maximal Cohen-Macauley modules and Tate-cohomology over Gorenstein rings. unpublished manuscript (1986), available at https://tspace.library.utoronto.ca/bitstream/1807/16682/1/maximal_cohen-macaulay_modules_1986.pdf
  13. 13.
    Bühler, T.: Exact categories. Expos. Math. 28(1), 1–69 (2010)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Christensen, J.D.: Ideals in triangulated categories: phantoms, ghosts and skeleta. Adv. Math. 136(2), 284–339 (1998)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Crawley-Boevey, W.: Locally finitely presented additive categories. Commun. Algebra 22(5), 1641–1674 (1994)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Dadarlat, M., Loring, T.A.: A universal multicoefficient theorem for the Kasparov groups. Duke Math. J. 84(2), 355–377 (1996)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Dell’Ambrogio, I.: Localizing subcategories in the bootstrap category of separable \(C^\ast \)-algebras. J. K-Theory 8(3), 493–505 (2011)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Dell’Ambrogio, I.: Equivariant Kasparov theory of finite groups via mackey functors. J. Noncommut. Geom. 8(3), 837–871 (2014)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Dell’Ambrogio, I., Emerson, H., Meyer, R.: An equivariant Lefschetz fixed-point formula for correspondences. Doc. Math. 19, 141–194 (2014)MATHMathSciNetGoogle Scholar
  20. 20.
    Ding, N.Q., Chen, J.L.: The flat dimensions of injective modules. Manuscripta Math. 78(2), 165–177 (1993)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Ding, N., Chen, J.: Coherent rings with finite self-\(FP\)-injective dimension. Commun. Algebra 24(9), 2963–2980 (1996)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Enochs, E.E., Jenda, O.M.G.: Relative Homological Algebra, de Gruyter Expositions in Mathematics, vol. 30. Walter de Gruyter & Co., Berlin (2000)CrossRefGoogle Scholar
  23. 23.
    Enochs, E.E., García Rozas, J.R.: Gorenstein injective and projective complexes. Commun. Algebra 26(5), 1657–1674 (1998)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Eilenberg, S., Nakayama, T.: On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings. Nagoya Math. J. 9, 1–16 (1955)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc. 260(1), 35–64 (1980)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Enochs, E., Estrada, S., García-Rozas, J.R.: Gorenstein categories and Tate cohomology on projective schemes. Math. Nachr. 281(4), 525–540 (2008)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Enochs, E.E., Jenda, O.M.G.: Gorenstein balance of Hom and tensor. Tsukuba J. Math. 19(1), 1–13 (1995)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Happel, D.: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Mathematical Society Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988)Google Scholar
  29. 29.
    Hirschhorn, P.S.: Model Categories and Their Localizations, Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, Providence (2003)MATHGoogle Scholar
  30. 30.
    Hovey, M.: Cotorsion pairs, model category structures, and representation theory. Math. Z. 241(3), 553–592 (2002)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Hovey, M.: Model Categories, Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence (1999)Google Scholar
  32. 32.
    Iwanaga, Y.: On rings with finite self-injective dimension. II. Tsukuba J. Math. 4(1), 107–113 (1980)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Jategaonkar, A.V.: A counter-example in ring theory and homological algebra. J. Algebra 12, 418–440 (1969)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Kasch, F.: Projektive Frobenius-Erweiterungen. S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1960/61 (1960/1961) (German)Google Scholar
  35. 35.
    Kashiwara, M., Schapira, P.: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332. Springer, Berlin (2006)Google Scholar
  36. 36.
    Kasparov, G.G.: The operator \(K\)-functor and extensions of \(C^{\ast }\)-algebras. Izv. Akad. Nauk SSSR Ser. Mat. 44(3), 571–636, 719 (1980) (Russian)Google Scholar
  37. 37.
    Kasparov, G.G.: Equivariant \(KK\)-theory and the Novikov conjecture. Invent. Math. 91(1), 147–201 (1988)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Köhler, M.: Universal Coefficient Theorems in Equivariant \(KK\)-theory. PhD thesis (2011), available at http://hdl.handle.net/11858/00-1735-0000-0006-B6A9-9
  39. 39.
    Krause, H.: Report on locally finite triangulated categories. J. K-Theory 9(3), 421–458 (2012)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Krause, H.: The stable derived category of a Noetherian scheme. Compos. Math. 141(5), 1128–1162 (2005)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Krause, H.: Smashing subcategories and the telescope conjecture-an algebraic approach. Invent. Math. 139(1), 99–133 (2000)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Krause, H.: Derived categories, resolutions, and Brown representability, Interactions between homotopy theory and algebra, Contemp. Math., vol. 436, Am. Math. Soc., Providence, RI, pp. 101–139 (2007)Google Scholar
  43. 43.
    Krause, H.: Deriving auslander’s formula. Doc. Math. 20, 669–688 (2015)MATHMathSciNetGoogle Scholar
  44. 44.
    Le Gall, P.-Y.: Théorie de Kasparov équivariante et groupoïdes. I. K-Theory 16(4), 361–390 (1999). (French, with English and French summaries)CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Mac Lane, S.: Categories for the Working Mathematician, vol. 5, 2nd edn. Graduate Texts in Mathematics. Springer, New York (1998)MATHGoogle Scholar
  46. 46.
    Margolis, H.R.: Spectra and the Steenrod algebra, North-Holland Mathematical Library, vol. 29. North-Holland Publishing Co., Modules over the Steenrod algebra and the stable homotopy category (1983)Google Scholar
  47. 47.
    Meyer, R.: Categorical aspects of bivariant \(K\)-theory, \(K\)-theory and noncommutative geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, pp. 1-39 (2008)Google Scholar
  48. 48.
    Meyer, R.: Homological algebra in bivariant \(K\)-theory and other triangulated categories. II. Tbil. Math. J. 1, 165–210 (2008)MATHMathSciNetGoogle Scholar
  49. 49.
    Meyer, R., Nest, R.: The Baum–Connes conjecture via localisation of categories. Topology 45(2), 209–259 (2006)CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Meyer, R., Nest, R.: Homological algebra in bivariant \(K\)-theory and other triangulated categories. I. Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375. Cambridge Univ. Press, Cambridge, pp. 236–289 (2010)Google Scholar
  51. 51.
    Meyer, R., Nest, R.: \(C^*\)-algebras over topological spaces: the bootstrap class. Münster J. Math. 2, 215–252 (2009)MATHMathSciNetGoogle Scholar
  52. 52.
    Meyer, R., Nest, R.: \({\rm C}^*\)-algebras over topological spaces: filtrated \(K\)-theory. Can. J. Math. 64(2), 368–408 (2012)CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    Morita, K.: Adjoint pairs of functors and Frobenius extensions. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 9, 40–71 (1965)MATHMathSciNetGoogle Scholar
  54. 54.
    Neeman, A.: Some new axioms for triangulated categories. J. Algebra 139(1), 221–255 (1991)CrossRefMATHMathSciNetGoogle Scholar
  55. 55.
    Neeman, A.: The connection between the \(K\)-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. École Norm. Sup. (4) 25(5), 547–566 (1992)CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    Neeman, A.: The Brown representability theorem and phantomless triangulated categories. J. Algebra 151(1), 118–155 (1992)CrossRefMATHMathSciNetGoogle Scholar
  57. 57.
    Neeman, A.: On a theorem of Brown and Adams. Topology 36(3), 619–645 (1997)CrossRefMATHMathSciNetGoogle Scholar
  58. 58.
    Oberst, U., Röhrl, H.: Flat and coherent functors. J. Algebra 14, 91–105 (1970)CrossRefMATHMathSciNetGoogle Scholar
  59. 59.
    Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progr. Math., vol. 270. Birkhäuser Boston Inc., Boston, MA, pp. 503–531 (2009)Google Scholar
  60. 60.
    Patchkoria, I.: On the algebraic classification of module spectra. Algebraic Geom. Topol. 12(4), 2329–2388 (2012)CrossRefMATHMathSciNetGoogle Scholar
  61. 61.
    Ringel, C.M., Zhang, P.: Representations of quivers over the algebra of dual numbers. J. Algebra (2016). doi: 10.1016/j.jalgebra.2016.12.001
  62. 62.
    Rosenberg, J., Schochet, C.: The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized \(K\)-functor. Duke Math. J. 55(2), 431–474 (1987)CrossRefMATHMathSciNetGoogle Scholar
  63. 63.
    Schochet, C.L.: The UCT, the Milnor sequence, and a canonical decomposition of the Kasparov groups. K-Theory 10(1), 49–72 (1996)CrossRefMATHMathSciNetGoogle Scholar
  64. 64.
    Schochet, C.L.: Correction to: “the UCT, the Milnor sequence, and a canonical decomposition of the Kasparov groups” [\(K\)-Theory 10(1), 49–72 (1996)] \(K\)-Theory 14(2), 197–199 (1998)Google Scholar
  65. 65.
    Simson, D.: On pure global dimension of locally finitely presented Grothendieck categories. Fund. Math. 96(2), 91–116 (1977)CrossRefMATHMathSciNetGoogle Scholar
  66. 66.
    Šťovíček, J.: Derived equivalences induced by big cotilting modules. Adv. Math. 263, 45–87 (2014)CrossRefMATHMathSciNetGoogle Scholar
  67. 67.
    Šťovíček, J.: Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves. Advances in Representation Theory of Algebras (ICRA Bielefeld, Germany, 2012), EMS Series of Congress Reports, EMS Publishing House, Zürich, pp. 297–367 (2014)Google Scholar
  68. 68.
    Šťovíček, J., Pospíšil, D.: On compactly generated torsion pairs and the classification of co-\(t\)-structures for commutative noetherian rings. Trans. Am. Math. Soc. 368(9), 6325–6361 (2016)MATHMathSciNetGoogle Scholar
  69. 69.
    Šťovíček, J.: On purity and applications to coderived and singularity categories. preprint (2014), available at arXiv:1412.1615
  70. 70.
    Wang, R.: Gorenstein triangular matrix rings and category algebras. J. Pure Appl. Algebra 220(2), 666–682 (2016)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Ivo Dell’Ambrogio
    • 2
  • Greg Stevenson
    • 1
  • Jan Šťovíček
    • 3
  1. 1.Fakultät für Mathematik, BIREP GruppeUniversität BielefeldBielefeldGermany
  2. 2.Laboratoire de Mathématiques Paul PainlevéUniversité de Lille 1Villeneuve-d’Ascq CedexFrance
  3. 3.Department of AlgebraCharles University in PraguePraha 8Czech Republic

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