Integrally closed ideals and type sequences in one-dimensional local rings
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 \(\overline R\).
local noetherian analytically irreducible domain
scientific article; zbMATH DE number 1206686