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

Title & Authors
SIMPLE VALUATION IDEALS OF ORDER 3 IN TWO-DIMENSIONAL REGULAR LOCAL RINGS
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 $\small{\upsilon}$ 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 $\small{\upsilon}$-ideals $\small{m\;=\;P_0\;{\supset}\;P_1\;{\supset}\;{\cdots}\;{\supset}\;P_t\;=\;P}$ and all the other $\small{\upsilon}$-ideals are uniquely factored into a product of those simple ones [17]. Lipman further showed that the predecessor of the smallest simple $\small{\upsilon}$-ideal P is either simple or the product of two simple $\small{\upsilon}$-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 $\small{\upsilon}$-ideals when P is satellite of order 3 in terms of the invariant $\small{b_{\upsilon}\;=\;|\upsilon(x)\;-\;\upsilon(y)|}$, where $\small{\upsilon}$ is the prime divisor associated to P and m = (x, y). Denote $\small{b_{\upsilon}}$ by b and let b = 3k + 1 for k = 0, 1, 2. Let $\small{n_i}$ be the number of nonmaximal simple $\small{\upsilon}$-ideals of order i for i = 1, 2, 3. We show that the numbers $\small{n_{\upsilon}}$ = ($\small{n_1}$, $\small{n_2}$, $\small{n_3}$) = ($\small{{\lceil}\frac{b+1}{3}{\rceil}}$, 1, 1) and that the rank of P is $\small{{\lceil}\frac{b+7}{3}{\rceil}}$ = k + 3. We then describe all the $\small{\upsilon}$-ideals from m to P as products of those simple $\small{\upsilon}$-ideals. In particular, we find the conductor ideal and the $\small{\upsilon}$-predecessor of the given ideal P in cases of b = 1, 2 and for b = 3k + 1, 3k + 2, 3k for $\small{k\;{\geq}\;1}$. We also find the value semigroup $\small{\upsilon(R)}$ of a satellite simple valuation ideal P of order 3 in terms of $\small{b_{\upsilon}}$.
Keywords
simple valuation ideal;order of an ideal;prime divisor;proximity of simple integrally closed ideal;regular local ring;
Language
English
Cited by
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