A characterization of transfer Krull orders in Dedekind domains with torsion class group

TL;DR

This paper characterizes certain orders in Dedekind domains that admit transfer homomorphisms to zero-sum sequence monoids, linking their arithmetic properties to those of the Dedekind domain.

Contribution

It provides a natural condition-based characterization of transfer Krull orders in Dedekind domains, clarifying when their arithmetic aligns with the domain's.

Findings

Inclusion map to Dedekind domain is a transfer homomorphism except in specific cases.

Transfer homomorphisms imply shared arithmetic properties between orders and Dedekind domains.

Not all orders in Dedekind domains have transfer homomorphisms, highlighting differences in their arithmetic structures.

Abstract

We establish a characterization (under some natural conditions) of those orders in Dedekind domains which allow a transfer homomorphism to a monoid of zero-sum sequences. As a consequence, the inclusion map to the Dedekind domain is a transfer homomorphism, with the exception of a particular case. The arithmetic of Krull and Dedekind domains is well understood, and the existence of a transfer homomorphism implies that the order and the associated Dedekind domain share the same arithmetic properties. This is not the case for arbitrary orders in Dedekind domains.