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

MathSciNet review:
1308016

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[BrnHrz]**Winfried Bruns and Jürgen Herzog,*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR**1251956****[BchSnb]**David A. Buchsbaum and David Eisenbud,*Some structure theorems for finite free resolutions*, Advances in Math.**12**(1974), 84–139. MR**0340240****[BchRm]**David A. Buchsbaum and Dock S. Rim,*A generalized Koszul complex. II. Depth and multiplicity*, Trans. Amer. Math. Soc.**111**(1964), 197–224. MR**0159860**, 10.1090/S0002-9947-1964-0159860-7**[Ctk]**Steven D. Cutkosky,*Factorization of complete ideals*, J. Algebra**115**(1988), no. 1, 144–149. MR**937605**, 10.1016/0021-8693(88)90286-4**[Ctk2]**Steven Dale Cutkosky,*Complete ideals in algebra and geometry*, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 27–39. MR**1266177**, 10.1090/conm/159/01502**[Ghn]**Hartmut Göhner,*Semifactoriality and Muhly’s condition (𝑁) in two dimensional local rings*, J. Algebra**34**(1975), 403–429. MR**0379489****[Hnk]**Craig Huneke,*Complete ideals in two-dimensional regular local rings*, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 325–338. MR**1015525**, 10.1007/978-1-4612-3660-3_16**[Hnk2]**Craig Huneke,*The primary components of and integral closures of ideals in 3-dimensional regular local rings*, Math. Ann.**275**(1986), no. 4, 617–635. MR**859334**, 10.1007/BF01459141**[HnkSll]**Craig Huneke and Judith D. Sally,*Birational extensions in dimension two and integrally closed ideals*, J. Algebra**115**(1988), no. 2, 481–500. MR**943272**, 10.1016/0021-8693(88)90274-8**[JhnVrm]**Bernard L. Johnston and Jugal 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), no. 3, 423–432. MR**1151321**, 10.1017/S0305004100075526**[Ktz]**D. Katz,*Reduction criteria for modules*, Comm. Algebra**23**(1995), no. 12, 4543–4548. MR**1352554**, 10.1080/00927879508825485**[KlmThr]**Steven Kleiman and Anders Thorup,*A geometric theory of the Buchsbaum-Rim multiplicity*, J. Algebra**167**(1994), no. 1, 168–231. MR**1282823**, 10.1006/jabr.1994.1182**[Krb]**D. Kirby,*On the Buchsbaum-Rim multiplicity associated with a matrix*, J. London Math. Soc. (2)**32**(1985), no. 1, 57–61. MR**813385**, 10.1112/jlms/s2-32.1.57**[KrbRs]**D. Kirby and D. Rees,*Hilbert functions of multigraded modules and the Buchsbaum-Rim multiplicity*.**[Lpm]**Joseph Lipman,*On complete ideals in regular local rings*, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 203–231. MR**977761****[Lpm2]**Joseph Lipman,*Rational singularities, with applications to algebraic surfaces and unique factorization*, Inst. Hautes Études Sci. Publ. Math.**36**(1969), 195–279. MR**0276239****[ZrsSml]**Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. II*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. MR**0120249**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
13H05,
13C13

Retrieve articles in all journals with MSC: 13H05, 13C13

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1308016-0

Article copyright:
© Copyright 1995
American Mathematical Society