A characterization of class groups via sets of lengths II

Geroldinger, Alfred; Zhong, Qinghai

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 327-346.

Soit $H$ un monoïde de Krull avec un groupe des classes fini $G$, et supposons que chaque classe contient un diviseur premier. Si un élément $a\in H$ a une factorisation $a=u_1\cdots u_k$ en éléments irréductibles $u_1,...,u_k\in H$, alors nous appelons $k$ la longueur de la factorisation et l’ensemble $\mathsf{L}\left(a\right)$ de toutes les longueurs de factorisation possibles l’ensemble des longueurs de $a$. C’est bien connu que le système $ℒ\left(H\right)=\left\{\mathsf{L}\right(a)\mid a\in H\}$ de tous les ensembles de longueurs ne dépend que du groupe des classes $G$, et c’est bien une conjecture de longue date que, inversement, le système $ℒ\left(H\right)$ caractérise le groupe des classes. Nous vérifions la conjecture si le groupe des classes est isomorphe à $C_n^r$ avec $r,n\ge 2$ et $r\le max\{2,(n+2)/6\}$.

En effet, soit $H^{\text{'}}$ un autre monoïde de Krull avec un groupe des classes $G^{\text{'}}$ tel que chaque classe contient un diviseur premier, et supposons que $ℒ\left(H\right)=ℒ\left(H^{\text{'}}\right)$. Nous montrons que, si l’un des groupes $G$ et $G^{\text{'}}$ est isomorphe à $C_n^r$ avec $r,n$ donnés comme ci-dessus, alors $G$ et $G^{\text{'}}$ sont isomorphes (à part deux exceptions bien connues).

Let $H$ be a Krull monoid with finite class group $G$ and suppose that every class contains a prime divisor. If an element $a\in H$ has a factorization $a=u_1·...·u_k$ into irreducible elements $u_1,...,u_k\in H$, then $k$ is called the length of the factorization and the set $\mathsf{L}\left(a\right)$ of all possible factorization lengths is the set of lengths of $a$. It is classical that the system $ℒ\left(H\right)=\left\{\mathsf{L}\right(a)\mid a\in H\}$ of all sets of lengths depends only on the class group $G$, and a standing conjecture states that conversely the system $ℒ\left(H\right)$ is characteristic for the class group. We verify the conjecture if the class group is isomorphic to $C_n^r$ with $r,n\ge 2$ and $r\le max\{2,(n+2)/6\}$. Indeed, let $H^{\text{'}}$ be a further Krull monoid with class group $G^{\text{'}}$ such that every class contains a prime divisor and suppose that $ℒ\left(H\right)=ℒ\left(H^{\text{'}}\right)$. We prove that, if one of the groups $G$ and $G^{\text{'}}$ is isomorphic to $C_n^r$ with $r,n$ as above, then $G$ and $G^{\text{'}}$ are isomorphic (apart from two well-known pairings).

Classification : 11B30, 11R27, 13A05, 13F05, 20M13
Mots clés : Krull monoids, maximal orders, seminormal orders, class groups, arithmetical characterizations, sets of lengths, zero-sum sequences, Davenport constant

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