A CLASSIFICATION OF ONE-DIMENSIONAL LOCAL DOMAINS BASED ON THE INVARIANT (c-delta)r-delta

作者：Oneto Anna^*; Zatini Elsa

来源：Journal of Commutative Algebra, 2011, 3(1): 89-115.

DOI：10.1216/JCA-2011-3-1-89

Let (R, m) be a one-dimensional, local, Noetherian domain, and let (R) over bar be the integral closure of R in its quotient field K. We assume that R is not regular, analytically irreducible and residually rational. The usual valuation v : K -> U infinity associated to (R) over bar defines the numerical semi-group v(R) = {v(a), a is an element of R, a not equal 0} subset of N. 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) over bar /R and the Cohen-Macaulay type of R, 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.

• 出版日期2011