Simple valuation ideals of order two\\ in $2$-dimensional regular local rings
Commun. Korean Math. Soc. 2005 Vol. 20, No. 3, 427-436
Printed September 1, 2005
Jooyoun Hong, Heisook Lee, Sunsook Noh
Purdue University, Ewha Womans University, 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. 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 $\vert v(x)-v(y) \vert $ by $n_v$. In this paper, if the $m$-adic order of $P$ is $2$, we show that $o(P_i)=1$ for $1 \le i \le \lceil \frac {b+1}{2} \rceil$ and $o(P_i)=2$ for $ \lceil \frac {b+3}{2} \rceil \le i \le t$, where $b=n_v$. We also show that $n_w = n_v$ when $w$ is the prime divisor associated to a simple $v$-ideal $Q \supset P$ of order $2$ and that $w(R)=v(R)$ as well.
Keywords : simple valuation ideal, order of an ideal, prime divisor
MSC numbers : 13A18, 13H05, 13B02, 13B22
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