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

Balint Rago

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.

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

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.

Paper Structure

This paper contains 6 sections, 11 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Monoids and factorization theory
  4. Dedekind domains and orders
  5. Regular elements of transfer Krull orders
  6. Localizations of transfer Krull orders

Key Result

Theorem 1

Let $R$ be a Dedekind domain with torsion class group and let $\mathcal{O} \subseteq R$ be an order with conductor $\mathfrak f$ such that $R/\mathfrak f$ is finite.

Theorems & Definitions (22)

  • Theorem 1
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 12 more