Integrally Closed Ideals and Type Sequences in One-Dimensional Local Rings

Abstract

0. Introduction. Let (R,m) be a one-dimensional, local, Noetherian domain. LetR be the integral closure of R in its quotient field. The conductor of R in R will be denoted by C, and the length function on R-modules by λ(−). We also assume that R is analytically irreducible, that is, R is a domain, or equivalently R is a DVR and is a finite Rmodule. If n is the maximal ideal of R, we assume that R/m R/n. To any such ring we can associate a numerical semigroup as follows. Let v denote the valuation of the quotient field K of R, v(K) = Z ∪ {∞}, with valuation ring R and set v(R) = {v(x) | x ∈ R, x = 0}. As R is a DVR and a finite R-module, C = rR, where rR = n. Therefore, v(R) is a numerical semigroup such that |N − v(R)| < ∞. We have v(R) = {0 = s0, s1, . . . , Sn−1, sn = g + 1,→}, where 0 = s0 < s1 < · · · < sn−1 < sn = g + 1, and the arrow indicates that any integer strictly greater than g is in v(R). The integer g is the greatest integer not in v(R) and is called the Frobenius number of R. Matsuoka [7] defines a chain of ideals Ui as follows