On infinite extensions of Dedekind domains, upper semicontinuous functions and the ideal class semigroups

Abstract

We study the monoid of fractional ideals and the ideal class semigroup of an arbitrary given one dimensional normal domain $\mathfrak{𝔒}$ 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 $\mathfrak{𝔒}$ equipped with the inverse topology introduced by Hochster, and whose codomain is a certain totally ordered monoid containing $ℝ$. We construct an isomorphism between a monoid consisting of such upper semicontinuous functions satisfying certain conditions and the monoid of fractional ideals of $\mathfrak{𝔒}$. 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 $\mathfrak{𝔒}$.