 A characterization of class groups via sets of lengths II
Journal de Théorie des Nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 327-346.

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 $ℒ\left(H\right)=\left\{\mathsf{L}\left(a\right)\mid a\in H\right\}$ 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 $r\le max\left\{2,\left(n+2\right)/6\right\}$. 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).

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 $ℒ\left(H\right)=\left\{\mathsf{L}\left(a\right)\mid a\in H\right\}$ 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 $r\le max\left\{2,\left(n+2\right)/6\right\}$.

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).

Accepted:
Published online:
DOI: 10.5802/jtnb.983
Classification: 11B30,  11R27,  13A05,  13F05,  20M13
Keywords: Krull monoids, maximal orders, seminormal orders, class groups, arithmetical characterizations, sets of lengths, zero-sum sequences, Davenport constant
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Alfred Geroldinger; Qinghai Zhong. A characterization of class groups via sets of lengths II. Journal de Théorie des Nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 327-346. doi : 10.5802/jtnb.983. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.983/

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