Abstract
We study the monoid of fractional ideals and the ideal class semigroup of an arbitrary given one dimensional normal domain 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 equipped with the inverse topology introduced by Hochster, and whose codomain is a certain totally ordered monoid containing . We construct an isomorphism between a monoid consisting of such upper semicontinuous functions satisfying certain conditions and the monoid of fractional ideals of . 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 .
Citation
Tatsuya Ohshita. "On infinite extensions of Dedekind domains, upper semicontinuous functions and the ideal class semigroups." J. Commut. Algebra 13 (3) 407 - 434, Fall 2021. https://doi.org/10.1216/jca.2021.13.407
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