Integrally closed modules over two-dimensional regular local rings

Kodiyalam, Vijay

doi:10.1090/S0002-9947-1995-1308016-0

Integrally closed modules over two-dimensional regular local rings
HTML articles powered by AMS MathViewer

by Vijay Kodiyalam PDF

Trans. Amer. Math. Soc. 347 (1995), 3551-3573 Request permission

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.

References

Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
David A. Buchsbaum and David Eisenbud, Some structure theorems for finite free resolutions, Advances in Math. 12 (1974), 84–139. MR 340240, DOI 10.1016/S0001-8708(74)80019-8
David A. Buchsbaum and Dock S. Rim, A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1964), 197–224. MR 159860, DOI 10.1090/S0002-9947-1964-0159860-7
Steven D. Cutkosky, Factorization of complete ideals, J. Algebra 115 (1988), no. 1, 144–149. MR 937605, DOI 10.1016/0021-8693(88)90286-4
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, DOI 10.1090/conm/159/01502
Hartmut Göhner, Semifactoriality and Muhly’s condition $(\textrm {N})$ in two dimensional local rings, J. Algebra 34 (1975), 403–429. MR 379489, DOI 10.1016/0021-8693(75)90166-0
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, DOI 10.1007/978-1-4612-3660-3_{1}6
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, DOI 10.1007/BF01459141
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, DOI 10.1016/0021-8693(88)90274-8
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, DOI 10.1017/S0305004100075526
D. Katz, Reduction criteria for modules, Comm. Algebra 23 (1995), no. 12, 4543–4548. MR 1352554, DOI 10.1080/00927879508825485
Steven Kleiman and Anders Thorup, A geometric theory of the Buchsbaum-Rim multiplicity, J. Algebra 167 (1994), no. 1, 168–231. MR 1282823, DOI 10.1006/jabr.1994.1182
D. Kirby, On the Buchsbaum-Rim multiplicity associated with a matrix, J. London Math. Soc. (2) 32 (1985), no. 1, 57–61. MR 813385, DOI 10.1112/jlms/s2-32.1.57

Hilbert functions of multigraded modules and the Buchsbaum-Rim multiplicity

Joseph Lipman, On complete ideals in regular local rings, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 203–231. MR 977761
Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239
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