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 generalized Cohen-Macaulay modules


Authors: Mihai Cipu, Jürgen Herzog and Dorin Popescu
Journal: Trans. Amer. Math. Soc. 342 (1994), 107-136
MSC: Primary 13C14; Secondary 13C05, 13D02
DOI: https://doi.org/10.1090/S0002-9947-1994-1104198-3
MathSciNet review: 1104198
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this paper is to study the indecomposable modules which are Cohen-Macaulay on the punctured spectrum of a local ring, and to describe some of their invariants such as their local cohomology groups and ranks. One of our main concerns is to find indecomposable quasi-Buchsbaum modules of high rank with prescribed cohomology over a regular local ring.


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

  • [1] M. André, Localisation de la lissité formelle, Manuscripta Math. 13 (1974), 297-307. MR 0357403 (50:9871)
  • [2] M. Auslander, Isolated singularities and existence of almost split sequences, Proc. ICRA IV, Lecture Notes in Math., vol. 1178, Springer-Verlag, Berlin and New York, 1986, pp. 194-241. MR 842486 (87j:13029)
  • [3] M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. No. 94, 1969. MR 0269685 (42:4580)
  • [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 877011 (88d:14005)
  • [5] A. Brezuleanu and N. Radu, Sur la localisation de la lissité formelle, C.R. Acad. Sci. Paris Sér. I Math. 276 (1973), 439-441. MR 0318142 (47:6691)
  • [6] -, Excellent rings and good separation of the module of differentials, Rev. Roumaine Math. Pures Appl. 23 (1978), 1455-1740. MR 530677 (80h:13029)
  • [7] W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Math., vol. 1327, Springer-Verlag, Berlin, Heidelberg and New York, 1988. MR 953963 (89i:13001)
  • [8] N. T. Cuong, P. Schenzel, and N. V. Trung, Verallgemeinerte Cohn-Macaulay Moduln, Math. Nachr. 85 (1978), 57-73. MR 517641 (80i:13008)
  • [9] E. Dieterich, Reduction of isolated singularities, Comment. Math. Helv. 62 (1987), 654-676. MR 920064 (90b:14003)
  • [10] R. Elkik, Solutions d'équations à coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup. (4) 6 (1973), 533-604. MR 0345966 (49:10692)
  • [11] E. G. Evans and P. Griffith, Syzygies, London Math. Soc. Lecture Note Ser., vol. 106, Cambridge Univ. Press, Cambridge and New York, 1985. MR 811636 (87b:13001)
  • [12] D. Eisenbud and J. Herzog, The classification of homogeneous Cohen-Macaulay rings of finite representation type, Math. Ann. 280 (1988), 347-352. MR 929541 (89c:13012)
  • [13] S. Goto, Maximal Buchsbaum modules over regular local rings and a structure theorem for generalized Cohen-Macaulay modules, Commutative Algebra and Combinatorics, Advanced Studies (M. Nagata and H. Matsumura, eds.), Kinokuniya, Tokyo, 1987, pp. 39-64. MR 951196 (89h:13028)
  • [14] -, Surface singularities of finite Buchsbaum-representation type, Commutative Algebra (M. Hochster, C. Huneke, and J. D. Sally, eds.), Springer-Verlag, Berlin, and New York, 1989, pp. 247-263. MR 1015521 (90h:13021)
  • [15] S. Goto and K. Nishida, Rings with only finitely many isomorphisms classes of indecomposable maximal Buchsbaum modules, J. Math. Soc. Japan 40 (1988), 501-518. MR 945349 (89e:13022)
  • [16] A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique, I. Inst. Hautes Etudes Sci. Publ. Math. No. 4, 1964. MR 2334323 (2008g:46001)
  • [17] M. Harada and Y. Sai, On categories of indecomposable modules. I, Osaka J. Math. 8 (1971), 309-321. MR 0304431 (46:3566)
  • [18] J. Herzog and E. Kunz, Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Math., vol. 238, Springer-Verlag, Berlin, Heidelberg, and New York, 1971. MR 0412177 (54:304)
  • [19] H. Knörrer, Cohen-Macaulay modules on hypersurface singularities, I, Invent. Math. 88 (1987), 153-164. MR 877010 (88d:14004)
  • [20] H. Matsumura, Commutative algebra, 2nd ed., Benjamin-Cummings, New York, 1980. MR 575344 (82i:13003)
  • [21] -, Commutative ring theory, Cambridge Univ. Press, Cambridge and New York, 1986.
  • [22] L. A. Nazarova and A. V. Rojter, Categorical matrix problems and the Brauer-Thrall conjecture, Inst. Math. Acad. Sci., Kiev, 1973. MR 0469832 (57:9613)
  • [23] R. S. Pierce, Associative algebras, Springer-Verlag, Berlin, Heidelberg, and New York, 1982. MR 674652 (84c:16001)
  • [24] D. Popescu, Indecomposable Cohen-Macaulay modules and their multiplicities, Trans. Amer. Math. Soc. 326 (1991), 369-387. MR 979959 (91c:13019)
  • [25] D. Popescu and M. Roczen, Indecomposable Cohen-Macaulay modules and irreducible maps, Compositio Math. 76 (1990), 277-294. MR 1078867 (91j:13008)
  • [26] -, The second Brauer-Thrall conjecture for isolated singularities of excellent hypersurface, preprint, Bucharest, 1990.
  • [27] I. Reiten, Finite dimensional algebras and singularities, Singularities, Representation of Algebras and Vector Bundles, Lecture Notes in Math., vol. 1273, Springer-Verlag, Berlin and New York, 1987, pp. 35-57. MR 915168 (89a:16031)
  • [28] C. M. Ringel, Report on the Brauer-Thrall conjectures: Rojter's theorem and the theorem of Nazarova and Rojter (Proc. ICRA I, Ottawa), Lecture Notes in Math., vol. 831, Berlin and New York, 1980, pp. 104-136. MR 607142 (82j:16055)
  • [29] J. Stückrad and W. Vogel, Toward a theory of Buchsbaum singularities, Amer. J. Math. 100 (1978), 727-746. MR 509072 (80d:14004)
  • [30] -, Buchsbaum rings and applicatioms, VEB, Deutscher Verlag Wiss., Berlin, 1986.
  • [31] N. V. Trung, Toward a theory of generalized Cohen-Macaulay modules, Nagoya Math. J. 102 (1986), 1-49. MR 846128 (87h:13018)
  • [32] Y. Yoshino, Brauer-Thrall type theorem for maximal Cohen-Macaulay modules, J. Math. Soc. Japan 39 (1987), 719-739. MR 905636 (88k:13006)
  • [33] -, Cohen-Macaulay modules over Cohen-Macaulay rings, London Math. Soc. Lecture Note Ser., vol. 146, Cambridge Univ. Press, Cambridge and New York, 1990. MR 1079937 (92b:13016)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 13C14, 13C05, 13D02

Retrieve articles in all journals with MSC: 13C14, 13C05, 13D02


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1104198-3
Keywords: Maximal generalized Cohen-Macaulay modules, quasi-Buchsbaum modules, indecomposable modules, local cohomology
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society