Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T20:42:28.907Z Has data issue: false hasContentIssue false
  • English
  • Français

An Uncountably Infinite Number of Indecomposable Totally Reflexive Modules

Published online by Cambridge University Press:  11 January 2016

Ryo Takahashi
Ryo Takahashi*
Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki 214-8571, Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Several years ago, Huneke and Leuschke proved a theorem solving a conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring of countable Cohen-Macaulay type which is complete or has uncountable residue field has at most a one-dimensional singular locus. In this paper, it is verified that the assumption of the excellent property can be removed, and the theorem is considered over an arbitrary local ring. The main purpose of this paper is to prove that the existence of a certain prime ideal and a certain totally reflexive module implies the existence of an uncountably infinite number of isomorphism classes of indecomposable totally reflexive modules.

Keywords

13C1416G60
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 187 , September 2007 , pp. 35 - 48
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

References

[1] Auslander, M., Anneaux de Gorenstein, et torsion en algèbre commutative, Séeminaire d’Algèbre Commutative dirigée par Pierre Samuel, 1966/67, Texte réedigée, d’après des exposées de Maurice Auslander, Marquerite Mangeney, Christian Peskine et Lucien é Szpiro, Ecole Normale Supéerieure de Jeunes Filles, Secréetariat mathéematique, Paris, 1967.Google Scholar
[2] Auslander, M., Isolated singularities and existence of almost split sequences, Representation theory, II (Ottawa, Ont., 1984), Lecture Notes in Math., vol. 1178, Springer, Berlin, 1986, pp. 194–242.Google Scholar
[3] Auslander, M. and Bridger, M. Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969.Google Scholar
[4] Araya, T., Takahashi, R. and Yoshino, Y. Homological invariants associated to semi-dualizing bimodules, J. Math. Kyoto Univ., 45 (2005), no. 2, 287–306.Google Scholar
[5] Avramov, L. L. and Martsinkovsky, A. Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3), 85 (2002), no. 2, 393–440.CrossRefGoogle Scholar
[6] Bruns, W. and Herzog, J., Cohen-Macaulay rings, revised edition, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1998.Google Scholar
[7] Buchweitz, R.-O., Greuel, G.-M. and Schreyer, F.-O., Cohen-Macaulay modules on hypersurface singularities, II, Invent. Math., 88 (1987), no. 1, 165–182.Google Scholar
[8] Burch, L., Codimension and analytic spread, Proc. Cambridge Philos. Soc., 72 (1972), 369–373.Google Scholar
[9] Christensen, L. W., Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000.Google Scholar
[10] Greuel, G.-M. and Knörrer, H., Einfache Kurvensingularitäten und torsionsfreie Moduln, Math. Ann., 270 (1985), no. 3, 417–425.Google Scholar
[11] Herzog, J., Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerleg-baren Cohen-Macaulay-Moduln, Math. Ann., 233 (1978), no. 1, 21–34.CrossRefGoogle Scholar
[12] Huneke, C. and Leuschke, G. J., Two theorems about maximal Cohen-Macaulay modules, Math. Ann., 324 (2002), no. 2, 391–404.CrossRefGoogle Scholar
[13] Huneke, C. and Leuschke, G. J., Local rings of countable Cohen-Macaulay type, Proc. Amer. Math. Soc., 131 (2003), no. 10, 3003–3007.Google Scholar
[14] Knörrer, H., Cohen-Macaulay modules on hypersurface singularities, I, Invent. Math., 88 (1987), no. 1, 153–164.CrossRefGoogle Scholar
[15] Leuschke, G. and Wiegand, R., Ascent of finite Cohen-Macaulay type, J. Algebra, 228 (2000), no. 2, 674–681.Google Scholar
[16] Schreyer, F.-O., Finite and countable CM-representation type, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 9–34.CrossRefGoogle Scholar
[17] Sharp, R. Y. and Váamos, P., Baire’s category theorem and prime avoidance in complete local rings, Arch. Math. (Basel), 44 (1985), no. 3, 243–248.Google Scholar
[18] Takahashi, R., On the category of Gorenstein dimension zero, Math. Z., 251 (2005), no. 2, 249–256.Google Scholar
[19] Takahashi, R., On the category of modules of Gorenstein dimension zero, II, J. Algebra, 278 (2004), no. 1, 402–410.CrossRefGoogle Scholar
[20] Takahashi, R., Modules of G-dimension zero over local rings of depth two, Illinois J. Math., 48 (2004), no. 3, 945–952.Google Scholar
[21] Takahashi, R., On the number of indecomposable totally reflexive modules, Bull. London Math. Soc., 39 (2007), 487–492.CrossRefGoogle Scholar
[22] Yoshino, Y., Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990.Google Scholar
[23] Yoshino, Y., Modules of G-dimension zero over local rings with the cube of maximal ideal being zero, Commutative algebra, singularities and computer algebra (Sinaia, 2002), NATO Sci. Ser. II Math. Phys. Chem., vol. 115, Kluwer Acad. Publ., Dordrecht, 2003, pp. 255–273.Google Scholar