Simple valuation ideals of order two in$2$-dimensional regular local rings
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Abstract
Let (R;m) be a 2-dimensional regular local ring with algebraically closed residue ¯eld R=m. Let K be the quotient ¯eld 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 ¯nitely many simple v-ideals m = P0 ¾ P1 ¾ ¢ ¢ ¢ ¾ Pt = 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 dierence jv(x) ¡ v(y)j by nv. In this paper, if the m-adic order of P is 2, we show that o(Pi) = 1 for 1 · i · db+1 2 e and o(Pi) = 2 for db+3 2 e · i · t, where b = nv. We also show that nw = nv when w is the prime divisor associated to a simple v-ideal Q ¾ P of order 2 and that w(R) = v(R) as well.
참고문헌 (13)
- "On the valuations centered in a local domain", 78, pp.70~99 (1956)
- "Zero-dimensional valuation ideals associated with plane curvebranches", 6(3), pp.70~99 (1956)
- "Integrally closed ideals in two-dimensional regular local rings", 15, pp.325~337 (1989), Springer-Verlag
- "Birational extensions in dimension two and integrallyclosed ideals", pp.481~500 (1988)
- "Adjoints and polars of simple complete ideals in two-dimensionalregular local rings", pp.223~244 (1993)
- "Proximity inequalities for complete ideals in two-dimensional regularlocal rings", 159, pp.293~306 (1994)
- "The value semigroups of prime divisors of the second kind on 2", pp.607~619 (1993)
- "Sequence of valuation ideals of prime divisors of the second kind in 2-dimensional regular local rings" (1993)
- "Adjacent integrally closed ideals in dimension two", pp.163~184 (1993)
- "Valuation ideals of order one in two-dimensional regular local rings", 28(2), pp.613~624 (2000)
- "Valuation ideals of order two in 2-dimensional regular local rings", 261-262, pp.123~140 (2003)
- "The theory of classical valuations" (1999), Springer-Verlag
- "Polynomial ideals dened by innitely near base points", 60, pp.151~204 (1938)