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

Title & Authors
SIMPLE VALUATION IDEALS OF ORDER TWO IN 2-DIMENSIONAL REGULAR LOCAL RINGS
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 $\small{m=P_0\;{\supset}\;P_1\;{\supset}\;{\cdotS}\;{\supset}\;P_t=P}$ 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 $\small{n_v}$. In this paper, if the m-adic order of P is 2, we show that $\small{O(P_i)\;=\;1\;for\;1\;{\leq}\;i\; {\leq}\;{\lceil}\;{\frac{b+1}{2}}{\rceil}\;and\;O(P_i)\;=2\;for\;{\lceil}\;\frac{b+3}{2}\rceil\;{\leq}\;i\;\leq\;t,\;where\;b=n_v}$. We also show that $\small{n_w\;=\;n_v}$ when w is the prime divisor associated to a simple v-ideal $\small{Q\;{\supset}\;P}$ 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
1.
SIMPLE VALUATION IDEALS OF ORDER 3 IN TWO-DIMENSIONAL REGULAR LOCAL RINGS,;

대한수학회논문집, 2008. vol.23. 4, pp.511-528
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