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

Tatsuya Ohshita

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arXiv · 1804.06292

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

Tatsuya Ohshita

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Abstract

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.

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BibTeX RIS

Tatsuya Ohshita. 2018-04-17. On infinite extensions of Dedekind domains, upper semicontinuous functions and the ideal class semigroups. https://arxiv.org/abs/1804.06292

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