SIMPLE VALUATION IDEALS OF ORDER 3 IN TWO-DIMENSIONAL REGULAR LOCAL RINGS

- Journal title : Communications of the Korean Mathematical Society
- Volume 23, Issue 4, 2008, pp.511-528
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2008.23.4.511

Title & Authors

SIMPLE VALUATION IDEALS OF ORDER 3 IN TWO-DIMENSIONAL REGULAR LOCAL RINGS

Noh, Sun-Sook;

Noh, Sun-Sook;

Abstract

Let (R, m) be a 2-dimensional regular local ring with algebraically closed residue field R/m. Let K be the quotient field of R and be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple -ideals and all the other -ideals are uniquely factored into a product of those simple ones [17]. Lipman further showed that the predecessor of the smallest simple -ideal P is either simple or the product of two simple -ideals. The simple integrally closed ideal P is said to be free for the former and satellite for the later. In this paper we describe the sequence of simple -ideals when P is satellite of order 3 in terms of the invariant , where is the prime divisor associated to P and m = (x, y). Denote by b and let b = 3k + 1 for k = 0, 1, 2. Let be the number of nonmaximal simple -ideals of order i for i = 1, 2, 3. We show that the numbers = (, , ) = (, 1, 1) and that the rank of P is = k + 3. We then describe all the -ideals from m to P as products of those simple -ideals. In particular, we find the conductor ideal and the -predecessor of the given ideal P in cases of b = 1, 2 and for b = 3k + 1, 3k + 2, 3k for . We also find the value semigroup of a satellite simple valuation ideal P of order 3 in terms of .

Keywords

simple valuation ideal;order of an ideal;prime divisor;proximity of simple integrally closed ideal;regular local ring;

Language

English

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