Factorization of an integrally closed ideal in two-dimensional regular local rings
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- by Mee-Kyoung Kim PDF
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Abstract:
Let $(R,m,k)$ be a two-dimensional regular local ring with algebraically closed residue field $k$ and $I$ be an $m$-primary integrally closed ideal in $R$. Let $T(I)$ be the set of Rees valuations of $I$ and $k(v)$ be the residue field of the valuation ring $V$ associated with $v\in T(I)$. Assume that $(a,b)$ is any minimal reduction of $I$. We show that if $I$ is the product of the distinct simple $m$-primary integrally closed ideals in $(R,m,k)$, then $k(v)$ is generated by the image of $a/b$ over $k$ for all $v\in T(I)$, and the converse of this is also true.References
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
- C. Huneke, Lecture note on complete ideals in two-dimensional regular local rings, Purdue University, 1987.
- 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
- Daniel Katz, Note on multiplicity, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1021–1026. MR 929434, DOI 10.1090/S0002-9939-1988-0929434-8
- M. K. Kim, Product of distinct simple integrally closed ideals in two dimensional regular local rings, Proc. Amer. Math. Soc. (to appear).
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
- 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
Additional Information
- Mee-Kyoung Kim
- Email: mkkim@yurim.skku.ac.kr
- Received by editor(s): July 16, 1993
- Received by editor(s) in revised form: July 12, 1996
- Communicated by: Wolmer V. Vasconcelos
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 3509-3513
- MSC (1991): Primary 13A18; Secondary 13B20, 13C05
- DOI: https://doi.org/10.1090/S0002-9939-97-04064-1
- MathSciNet review: 1415330