Valuation ideals of order two in 2-dimensional regular local rings

✍ Scribed by Sunsook Noh

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 233 KB
• Volume: 261-262
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310116

✦ Synopsis

Abstract

Let K be the quotient field of a 2‐dimensional regular local ring (R, m) and let v be a prime divisor of R, i.e., a valuation of K birationally dominating R which is residually transcendental over R. Zariski showed that: such prime divisor v is uniquely associated to a simple__m__‐primary integrally closed ideal I of R, there are only finitely many simple v‐ideals including I, and all the other v‐ideals can be uniquely factored into products of simple v‐ideals. The number of nonmaximal simple v‐ideals is called the rank of v or the rank of I as well. It is also known that such an__m__‐primary ideal I is minimally generated by o(I)+1 elements, where o(I) denotes the m‐adic order of I. Given a simple valuation ideal of order two associated to a prime divisor v of arbitrary rank, in this paper we find minimal generating sets of all the simple v‐ideals and the value semigroup v(R) in terms of its rank and the v‐value difference of two elements in a regular system of parameters of R. We also obtain unique factorizations of all the composite v‐ideals and describe the complete sequence of v‐ideals as explicitly as possible. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)