Integrally closed ideals and type sequences in one-dimensional local rings (Q1265151)

To any one-dimensional local noetherian analytically irreducible domain \(R\) one can associate a numerical semigroup \(v(R)\) by using the valuation \(v\) of the quotient field of \(R\). Let \(v(R) = \{0=s_0,s_1,\ldots,s_n,s_n+1,s_n+2,\ldots\}\), where \(s_n\) is the smallest integer such that all integers \(\geq s_n\) belong to \(v(R)\). Put \(I_j = \{x \in R\mid v(x)\geq s_j\}\) for \(j \leq n\). The sequence \(t_j(R) = \ell(I_j^{-1}/I_{j-1}^{-1})\) is called the type sequence of \(R\). The authors characterize the integrally closed ideals of \(R\) as the ideals of the form \(\{x \in R\mid v(x) \geq r\}\) for some \(r\) and give a criterion to check when the ideals \(I_j\) are stable. Moreover, they use the type sequence to study the difference \(t\ell(R/C)-\ell(\overline R/R)\), where \(t\) is the type of \(R\), \(\bar R\) is the integral closure of \(R\) in its quotient field and \(C\) denotes the conductor of \(R\) in \(\overline R\).