A characterization of class groups via sets of lengths - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2015

A characterization of class groups via sets of lengths

Résumé

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u_1 \cdot \ldots \cdot u_k$, with irreducibles $u_1, \ldots u_k \in H$, then $k$ is called the length of the factorization and the set $\mathsf L (a)$ of all possible $k$ is called the set of lengths of $a$. It is well-known that the system $\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}$ depends only on the class group $G$. In the present paper we study the inverse question asking whether or not the system $\mathcal L (H)$ is characteristic for the class group. Consider a further Krull monoid $H'$ with class group $G'$ such that every class contains a prime divisor and suppose that $\mathcal L (H) = \mathcal L (H')$. We show that, if one of the groups $G$ and $G'$ is finite and has rank at most two, then $G$ and $G'$ are isomorphic (apart from two well-known pairings).
Fichier principal
Vignette du fichier
systems-rank-two-groups_arxiv.pdf (479.02 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-01131955 , version 1 (16-03-2015)
hal-01131955 , version 2 (11-02-2019)
hal-01131955 , version 3 (14-03-2019)

Identifiants

Citer

Alfred Geroldinger, Wolfgang Schmid. A characterization of class groups via sets of lengths. 2015. ⟨hal-01131955v1⟩
182 Consultations
147 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More