Local and global tameness in Krull monoids

TL;DR

This paper investigates the relationship between local and global tameness in Krull monoids with finite class groups, establishing bounds for the tame degree and analyzing cases where these bounds are tight, with specific results for cyclic and elementary 2-groups.

Contribution

It provides new bounds for the tame degree in Krull monoids and analyzes the asymptotic behavior of these bounds, especially for specific types of class groups.

Findings

Global tame degree equals zero iff the monoid is factorial.

Bounds for tame degree in terms of the Davenport constant.

Asymptotic growth of tame degree matches bounds for large class group rank.

Abstract

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Then the global tame degree t (H) equals zero if and only if H is factorial (equivalently, |G|=1). If |G| > 1, then D (G) <= t (H) <= 1 + D (G) (D (G) -1) / 2, where D (G) is the Davenport constant of G. We analyze the case when t (H) equals the lower bound, and we show that t (H) grows asymptotically as the upper bound, when both terms are considered as functions of the rank of G. We provide more precise results if G is either cyclic or an elementary 2-group.