2018/04/17 by Ohshita, Tatsuya
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1804.06292
In this article, we study the monoid of fractional ideals and the ideal class semigroup of an arbitrary given one dimensional normal domain O obtained by an infinite integral extension of a Dedekind domain. We introduce a notion of "upper semicontinuous functions" whose domain is the maximal spectrum of O equipped with a certain topology, and whose codomain is a certain totally ordered monoid containing the set of real numbers. We construct an isomorphism between a monoid consisting of such upper semicontinuous functions satisfying certain conditions and the monoid of fractional ideals of O. This result can be regarded as a generalization of the theory of prime ideal factorization for Dedekind domains. By using such isomorphism, we study the Galois-monoid structure of the ideal class semigroup of O.