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Integrally closed modules over two-dimensional regular local rings


Author: Vijay Kodiyalam
Journal: Trans. Amer. Math. Soc. 347 (1995), 3551-3573
MSC: Primary 13H05; Secondary 13C13
DOI: https://doi.org/10.1090/S0002-9947-1995-1308016-0
MathSciNet review: 1308016
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Abstract: This paper is based on work of Rees on integral closures of modules and initiates the study of integrally closed modules over two-dimensional regular local rings in analogy with the classical theory of complete ideals of Zariski. The main results can be regarded as generalizations of Zariski's product theorem. They assert that the tensor product mod torsion of integrally closed modules is integrally closed, that the symmetric algebra mod torsion of an integrally closed module is a normal domain and that the first Fitting ideal of an integrally closed module is an integrally closed ideal. A construction of indecomposable integrally closed modules is also given. The primary technical tool is a study of the Buchsbaum-Rim multiplicity.


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  • [BrnHrz] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Stud. Adv. Math., 39, Cambridge University Press, 1993. MR 1251956 (95h:13020)
  • [BchSnb] D. A. Buchsbaum and D. Eisenbud, Some structure theorems for finite free resolutions, Adv. Math. 12 (1974), 84-139. MR 0340240 (49:4995)
  • [BchRm] D. A. Buchsbaum and D. S. Rim, A generalized Koszul complex II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1964), 197-224. MR 0159860 (28:3076)
  • [Ctk] S. D. Cutkosky, Factorization of complete ideals, J. Algebra 115 (1988), 151-204. MR 937605 (89d:13006)
  • [Ctk2] -, Complete ideals in algebra and geometry, Commutative Algebra: Syzygies, Multiplicities and Birational Algebra, Contemporary Math., vol. 159, Amer. Math. Soc., Providence, RI, 1993, pp. 27-39. MR 1266177 (95g:13023)
  • [Ghn] H. Gohner, Semifactoriality and Muhly's condition $ ({\text{N}})$ in two-dimensional local rings, J. Algebra 34 (1975), 403-429. MR 0379489 (52:394)
  • [Hnk] C. Huneke, Complete ideals in two dimensional regular local rings, Commutative Algebra, Proceedings of a Microprogram, MSRI publ. no. 15, Springer-Verlag, 1989, pp. 417-436. MR 1015525 (90i:13020)
  • [Hnk2] -, The primary components of and integral closures of ideals in three-dimensional regular local rings, Math. Ann. 275 (1986), 617-635. MR 859334 (87k:13038)
  • [HnkSll] C. Huneke and J. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), 481-500. MR 943272 (89e:13025)
  • [JhnVrm] B. Johnston and J. K. Verma, On the length formula of Hoskin and Deligne and associated graded rings of two-dimensional regular local rings, Math. Proc. Cambridge Philos. Soc. 111 (1992), 423-432. MR 1151321 (93e:13004)
  • [Ktz] D. Katz, Reduction criteria for modules, Preprint. MR 1352554 (96j:13022)
  • [KlmThr] S. Kleiman and A. Thorup, A geometric theory of the Buchsbaum-Rim multiplicity, J. Algebra 167 (1994), 168-231. MR 1282823 (96a:14007)
  • [Krb] D. Kirby, On the Buchsbaum-Rim multiplicity associated with a matrix, J. London Math. Soc. (2) 32 (1985), 57-61. MR 813385 (87d:13025)
  • [KrbRs] D. Kirby and D. Rees, Hilbert functions of multigraded modules and the Buchsbaum-Rim multiplicity.
  • [Lpm] J. Lipman, On complete ideals in regular local rings, Algebraic Geometry and Commutative Algebra in honor of Masayoshi Nagata, Academic Press, 1987, pp. 203-231. MR 977761 (90g:14003)
  • [Lpm2] -, Rational singularities with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Hautes Etudes SCi. 36 (1969), 195-279. [Rs] D. Rees, Reduction of modules, Math. Proc. Cambridge Philos. Soc. 101 (1987), 431-449. MR 0276239 (43:1986)
  • [ZrsSml] O. Zariski and P. Samuel, Commutative algebra, Vol. II, Van Nostrand Reinhold, New York, 1960. MR 0120249 (22:11006)

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DOI: https://doi.org/10.1090/S0002-9947-1995-1308016-0
Article copyright: © Copyright 1995 American Mathematical Society

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