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

Abstract

Let $R$ be a one-dimensional, local, Noetherian domain, $\mathbb{R}$ 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 $\mathbb{R}/R$ and the Cohen Macaulay type, respectively. In particular, the classification of the semigroups $v(R)$ for rings having $b\le 2(r-1)$ is realized. This method of classification might be successfully utilized with similar arguments but more boring computations in the cases $b\le q(r-1),$ 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 $R$.

Journal of Commutative Algebra (JCA), Rocky Mountain Consortium