A classification of one-dimensional local domains based on the invariant (c-δ)r-δ
Abstract
Let R be a one-dimensional, local, Noetherian domain, 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-δ)r- δ , where c, δ, r denote the conductor, the length of /R and the Cohen Macaulay type, respectively. In particular, the classification of the semigroups v(R) for rings having b≤ 2(r-1) is realized. This method of classification might be successfully utilized with similar arguments but more boring computations in the cases b≤ 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.