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Local rings of countable Cohen-Macaulay type


Authors: Craig Huneke and Graham J. Leuschke
Journal: Proc. Amer. Math. Soc. 131 (2003), 3003-3007
MSC (2000): Primary 13C14; Secondary 13H10, 13C05
DOI: https://doi.org/10.1090/S0002-9939-03-07167-3
Published electronically: May 9, 2003
MathSciNet review: 1993205
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Abstract: We prove (the excellent case of) Schreyer's conjecture that a local ring with countable CM type has at most a one-dimensional singular locus. Furthermore, we prove that the localization of a Cohen-Macaulay local ring of countable CM type is again of countable CM type.


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  • 1. M. Auslander, Isolated singularities and the existence of almost split sequences, Proc. ICRA IV, Lecture Notes in Mathematics, vol. 1178, Springer-Verlag, New York-Berlin, 1986, pp. 194-241. MR 87j:13029
  • 2. -, Rational singularities and almost split sequences, Trans. Amer. Math. Soc. 293 (1986), no. 2, 511-531. MR 87e:16073
  • 3. M. Auslander and R.-O. Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.) (1989), no. 38, 5-37, Colloque en l'honneur de Pierre Samuel (Orsay, 1987). MR 91h:13010
  • 4. R.-O. Buchweitz, G.-M. Greuel, and F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities II, Invent. Math. 88 (1987), 165-182. MR 88d:14005
  • 5. L. Burch, Codimension and analytic spread, Proc. Camb. Phil. Soc. 72 (1972), 369-373. MR 46:3512
  • 6. G.-M. Greuel and H. Knörrer, Einfache Kurvensingularitäten und torsionfreie Moduln, Math. Ann. 270 (1985), 417-425. MR 86d:14025
  • 7. A. Grothendieck, Éléments de Géometrie Algébrique Chapter IV, 2nd partie, Publ. Math. I.H.E.S. 24 (1965). MR 33:7330
  • 8. C. Huneke and G. Leuschke, Two theorems about maximal Cohen-Macaulay modules, Math. Ann. 324 (2002), 391-404.
  • 9. H. Knörrer, Cohen-Macaulay modules on hypersurface singularities I, Invent. Math. 88 (1987), 153-164. MR 88d:14004
  • 10. G. Leuschke and R. Wiegand, Ascent of finite Cohen-Macaulay type, J. Algebra 228 (2000), 674-681. MR 2001k:13035
  • 11. P. Roberts, Homological invariants of modules over commutative rings, Seminaire de Mathematiques Superieures 72 (1980), University of Montreal Press. MR 82j:13020
  • 12. F.-O. Schreyer, Finite and countable CM-representation type, Singularities, Representation of Algebras, and Vector Bundles: Proceedings Lambrecht 1985 (G.-M. Greuel and G. Trautmann, eds.), Lecture Notes in Mathematics, vol. 1273, Springer-Verlag, New York-Berlin, 1987, pp. 9-34. MR 88j:14005
  • 13. R. Y. Sharp and P. Vámos, Baire's category theorem and prime avoidance in complete local rings, Arch. Math. (Basel) 44 (1985), no. 3, 243-248. MR 86h:13004
  • 14. R. Wiegand, Local rings of finite Cohen-Macaulay type, J. Algebra 203 (1998), 158-168. MR 99c:13025
  • 15. Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Math. Soc. Lect. Notes Ser., vol. 146, Cambridge University Press, 1990. MR 92b:13016

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Additional Information

Craig Huneke
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email: huneke@math.ukans.edu

Graham J. Leuschke
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email: gleuschke@math.ukans.edu

DOI: https://doi.org/10.1090/S0002-9939-03-07167-3
Keywords: Maximal Cohen--Macaulay modules, CM representation type, countable CM type
Received by editor(s): May 10, 2002
Published electronically: May 9, 2003
Additional Notes: Both authors were supported by the National Science Foundation
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2003 American Mathematical Society

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