SIMPLE VALUATION IDEALS OF ORDER TWO IN 2-DIMENSIONAL REGULAR LOCAL RINGS

- Journal title : Communications of the Korean Mathematical Society
- Volume 20, Issue 3, 2005, pp.427-436
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2005.20.3.427

Title & Authors

SIMPLE VALUATION IDEALS OF ORDER TWO IN 2-DIMENSIONAL REGULAR LOCAL RINGS

Hong, Joo-Youn; Lee, Hei-Sook; Noh, Sun-Sook;

Hong, Joo-Youn; Lee, Hei-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 v 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 v-ideals and all the other v-ideals are uniquely factored into a product of those simple ones. It then was also shown by Lipman that the predecessor of the smallest simple v-ideal P is either simple (P is free) or the product of two simple v-ideals (P is satellite), that the sequence of v-ideals between the maximal ideal and the smallest simple v-ideal P is saturated, and that the v-value of the maximal ideal is the m-adic order of P. Let m = (x, y) and denote the v-value difference |v(x) - v(y)| by . In this paper, if the m-adic order of P is 2, we show that . We also show that when w is the prime divisor associated to a simple v-ideal of order 2 and that w(R) = v(R) as well.

Keywords

simple valuation ideal;order of an ideal;prime divisor;

Language

English

Cited by

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