Arithmetic of Weakly Krull Domains#

Abstract

A noetherian domain R is called weakly Krull if , where 𝔛(R) denotes the set of hight one prime ideals of R. We study arithmetical properties and in particular the structure of sets of lengths of a noetherian weakly Krull domain R. This structure is well investigated if the integral closure R¯ of R is a finitely generated R-module and if the class group of R is finite. The main emphasis of the paper lies on the study of the case when R¯ fails to be a finite R-module. In the first part of the article we study the structure of v-ideals of R and derive an exact sequence which connects the class groups of R and R¯. This sequence is the well known Mayer-Vietoris sequence if R is one-dimensional with finite integral closure. In the second part we introduce the concept of local monoids and study factorization properties of R locally.