A classification of one-dimensional local domains based on the invariant $(c-\delta)r-\delta$

TL;DR

This paper classifies one-dimensional local domains using an invariant derived from the conductor, Cohen-Macaulay type, and integral closure, providing a structured understanding of their value semigroups.

Contribution

It introduces a classification method for such domains based on the invariant $(c- ext{delta})r- ext{delta}$, especially for cases where this invariant is less than or equal to twice the Cohen-Macaulay type.

Findings

Classified semigroups for rings with invariant $b \\leq 2(r-1)$.

Developed a method using type sequences and the invariant $k$ for classification.

Potential extension of the method to cases with higher invariant values.

Abstract

Let $R$ be a one-dimensional, local, Noetherian domain, $\R$ the integral closure of $R$ in its quotient field and $v(R)$ the value set defined by the usual valuation. The aim of the paper is to study the non-negative invariant $b:=(c-\delta)r- \delta $, where $c, \delta, r$ denote the conductor, the length of $\R/R$ and the Cohen Macaulay type, respectively. In particular, the classification of the semigroups $v(R)$ for rings having $b\leq 2(r-1)$ is realized. This method of classification might be successfully utilized with similar arguments but more boring computations in the cases $b\leq q(r-1), $ for reasonably low values of $q$. The main tools are type sequences and the invariant $k$ which estimates the number of elements in $v(R)$ belonging to the interval $[c-e,c), e$ being the multiplicity of $R$.