Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Indecomposable Cohen-Macaulay modules and their multiplicities


Author: Dorin Popescu
Journal: Trans. Amer. Math. Soc. 323 (1991), 369-387
MSC: Primary 13H10; Secondary 13C05, 13H15, 13J15
MathSciNet review: 979959
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main aim of this paper is to find a large class of rings for which there are indecomposable maximal Cohen-Macaulay modules of arbitrary high multiplicity (or rank in the case of domains).


References [Enhancements On Off] (What's this?)

  • [An] M. André, Localisation de la lissité formelle, Manuscripta Math. 13 (1974), 297-307. MR 0357403 (50:9871)
  • [Ar] M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23-58. MR 0268188 (42:3087)
  • [Au$ _{1}$] M. Auslander, Representation theory of Artin algebras. II, Comm. Algebra 1 (1974), 269-310. MR 0349747 (50:2240)
  • [Au$ _{2}$] -, Rational singularities and almost split sequences, Trans. Amer. Math. Soc. 293 (1986), 511-531. MR 816307 (87e:16073)
  • [Au$ _{3}$] -, Isolated singularities and existence of almost split sequences, Proc. ICRA IV, Lecture Notes in Math., vol. 1178, Springer-Verlag, 1986, pp. 194-241. MR 842486 (87j:13029)
  • [AR$ _{1}$] M. Auslander and I. Reiten, Representation theory of Artin algebras. III, Comm. Algebra 3 (1975), 239-294. MR 0379599 (52:504)
  • [AR$ _{2}$] -, Representation theory of Artin algebras. IV, Comm. Algebra 5 (1977), 443-518. MR 0439881 (55:12762)
  • [BR$ _{1}$] A. Brezuleanu and N. Radu, Sur la localisation de la lissité formelle, C. R. Acad. Sci. Paris 276 (1973), 439-441. MR 0318142 (47:6691)
  • [BR$ _{2}$] -, Excellent rings and good separation of the module of differentials, Rev. Roumaine Math. Pures Appl. 23 (1978), 1455-1470. MR 530677 (80h:13029)
  • [BGS] R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities. II, Invent. Math. 88 (1987), 165-183. MR 877011 (88d:14005)
  • [GK] G. M. Greuel and H. Knàrrer, Einfache Kurvensingularitaten und torsionsfreie Moduln, Math. Ann. 270 (1985), 417-425. MR 774367 (86d:14025)
  • [D] E. Dieterich, Reduction of isolated singularities, Comment. Math. Helv. 62 (1987), 654-676. MR 920064 (90b:14003)
  • [EGA] A. Grothendieck and J. Dieudonne, Éléments de géométrie algébrique. IV, part 1, Inst. Hautes Études Sci. Publ. Math. no. 20 (1964). MR 0173675 (30:3885)
  • [HS] M. Harada and Y. Sai, On categories of indecomposable modules. I, Osaka J. Math. 8 (1971), 309-321. MR 0304431 (46:3566)
  • [K] H. Knàrrer, Cohen Macaulay modules on hypersurface singularities. I, Invent. Math. 88 (1987), 153-165. MR 877010 (88d:14004)
  • [M$ _{1}$] H. Matsumura, Commutative algebra, Benjamin, New York, 1970. MR 0266911 (42:1813)
  • [M$ _{2}$] -, Commutative ring theory, Cambridge Univ. Press, 1986. MR 879273 (88h:13001)
  • [NP] V. Nica and D. Popescu, A structure theorem on formally smooth morphisms in positive characteristic, J. Algebra 100 (1986), 436-455. MR 840587 (87j:14007)
  • [P] R. S. Pierce, Associative algebras, Graduate Texts in Math., no. 88, Springer-Verlag, 1982. MR 674652 (84c:16001)
  • [Po] D. Popescu, General Néron desingularization and approximation, Nagoya Math. J. 104 (1986), 85-115. MR 868439 (88a:14007)
  • [R] A. V. Roiter, Unbounded dimensionality of indecomposable representation of an algebra with an infinite number of indecomposable representations, Izv. Akad. Nauk SSSR 32 (1968), 1275-1282. (Russian) MR 0238893 (39:253)
  • [S] G. Scheja and U. Storch, Differentielle Eigenschaften der Lokalisierungen analytischer Algebren, Math. Ann. 197 (1972), 137-170. MR 0306172 (46:5299)
  • [Ya] K. Yamagata, On Artin rings of finite representation type, J. Algebra 50 (1978), 276-283. MR 0469978 (57:9758)
  • [Y] Y. Yoshino, Brauer-Thrall type theorem for maximal Cohen-Macaulay modules, J. Math. Soc. Japan 39 (1987), 719-739. MR 905636 (88k:13006)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 13H10, 13C05, 13H15, 13J15

Retrieve articles in all journals with MSC: 13H10, 13C05, 13H15, 13J15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-0979959-9
Keywords: Maximally Cohen-Macaulay modules, indecomposable modules, multiplicity
Article copyright: © Copyright 1991 American Mathematical Society