Sequence of valuation ideals of prime divisors of the second kind in 2- dimensional regular local rings
Let \((R,m,k)\) be a 2-dimensional regular local ring, and let \(w\) be a valuation of the quotient field \(K\) of \(R\) that dominates \(R\) and such that the transcendence degree of the field extension \(k(w)/k\) is one (these valuations are also called prime divisors of the second kind). The author studies the \(w\)-ideals \(I\) of \(R\), i.e. the ideals \(I\) satisfying \(IW\cap R=I\), where \(W\) is the valuation ring associated to \(w\). The main tool used is Lipman's theory of intersection multiplicity of integrally closed ideals [see \textit{J. Lipman} in ``Algebraic geometry and commutative algebra, Papers in Honor of M. Nagata'', Vol. I, 203-231 (1982; Zbl 0693.13011].
scientific article; zbMATH DE number 218568