Simple valuation ideals of order 3 in two-dimensional regular local rings

Commun. Korean Math. Soc. 2008 Vol. 23, No. 4, 511-528 Printed December 1, 2008

Sunsook Noh Ewha Womans University

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 $m=P_0 \supset P_1 \supset \cdots \supset P_t=P$ and all the other $v$-ideals are uniquely factored into a product of those simple ones \cite{ZS}. Lipman further showed that the predecessor of the smallest simple $v$-ideal $P$ is either simple or the product of two simple $v$-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 $v$-ideals when $P$ is satellite of order 3 in terms of the invariant $b_v = \vert v(x) - v(y) \vert $, where $v$ is the prime divisor associated to $P$ and $\mathfrak m =(x, y)$. Denote $b_v$ by $b$ and let $b=3k+1$ for $k=0, 1, 2$. Let $n_i$ be the number of nonmaximal simple $v$-ideals of order $i$ for $i=1,2,3$. We show that the numbers $n_v=(n_1, n_2, n_3)=(\lceil \frac{b+1}{3} \rceil, 1, 1)$ and that the rank of $P$ is $ \lceil \frac { b + 7}{3} \rceil = k+3$. We then describe all the $v$-ideals from $\mathfrak m$ to $P$ as products of those simple $v$-ideals. In particular, we find the conductor ideal and the $v$-predecessor of the given ideal $P$ in cases of $b=1, 2$ and for $b=3k+1, 3k+2, 3k$ for $k\ge 1$. We also find the value semigroup $v(R)$ of a satellite simple valuation ideal $P$ of order $3$ in terms of $b_v$.

Keywords : simple valuation ideal, order of an ideal, prime divisor, proximity of simple integrally closed ideal, regular local ring