Anéis Gorenstein e semigrupos

Abstract

This work can be divided into two parts. In the first part we study algebraic function field of one variable and valuation. As an application of these concepts we investigate the relation between points of an irreducible algebraic plane curve C and the valuations in its rational function field, K(C). An example of an algebraic functions field of one variable is K(C). The second part is devoted to the study of Gorenstein rings and numeric semigroups. More specifically, an integrally closed one-dimensional local domain Noetherian ring R is discrete valuation domain. Therefore, there is a valuation v : K → Z ∪ {∞}, where K is the field fractions fiel of R, such that v(R) := {v(x); x ∈ R \ {0}} is a numeric semigroup. Some properties can be obtained by of a ring R can revealed by its semigroup v(R). For example, we have seen that an analytically irreducible local ring R which is residually rational is Gorenstein iff the value-semigroup v(R) is simetric. The integral closure of the local ring of a plane algebraic curve C at an unibranch point p, Op(C), is an example of a Gorenstein ring