A characterization of class groups via sets of lengths II

Alfred Geroldinger; Qinghai Zhong

doi:10.5802/jtnb.983

Alfred Geroldinger ¹ ; Qinghai Zhong ¹

¹ University of Graz, NAWI Graz Institute for Mathematics and Scientific Computing Heinrichstraße 36 8010 Graz, Austria

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

Résumé
Abstract

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 $a = u_{1} \dots u_{k}$ en éléments irréductibles $u_{1}, ..., u_{k} \in H$ , alors nous appelons $k$ la longueur de la factorisation et l’ensemble $L (a)$ de toutes les longueurs de factorisation possibles l’ensemble des longueurs de $a$ . C’est bien connu que le système $ℒ (H) = {L (a) ∣ a \in H}$ 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 $ℒ (H)$ caractérise le groupe des classes. Nous vérifions la conjecture si le groupe des classes est isomorphe à $C_{n}^{r}$ avec $r, n \geq 2$ et $r \leq max {2, (n + 2) / 6}$ .

En effet, soit $H^{'}$ un autre monoïde de Krull avec un groupe des classes $G^{'}$ tel que chaque classe contient un diviseur premier, et supposons que $ℒ (H) = ℒ (H^{'})$ . Nous montrons que, si l’un des groupes $G$ et $G^{'}$ est isomorphe à $C_{n}^{r}$ avec $r, n$ donnés comme ci-dessus, alors $G$ et $G^{'}$ 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 $a = u_{1} \cdot ... \cdot u_{k}$ into irreducible elements $u_{1}, ..., u_{k} \in H$ , then $k$ is called the length of the factorization and the set $L (a)$ of all possible factorization lengths is the set of lengths of $a$ . It is classical that the system $ℒ (H) = {L (a) ∣ a \in H}$ of all sets of lengths depends only on the class group $G$ , and a standing conjecture states that conversely the system $ℒ (H)$ is characteristic for the class group. We verify the conjecture if the class group is isomorphic to $C_{n}^{r}$ with $r, n \geq 2$ and $r \leq max {2, (n + 2) / 6}$ . Indeed, let $H^{'}$ be a further Krull monoid with class group $G^{'}$ such that every class contains a prime divisor and suppose that $ℒ (H) = ℒ (H^{'})$ . We prove that, if one of the groups $G$ and $G^{'}$ is isomorphic to $C_{n}^{r}$ with $r, n$ as above, then $G$ and $G^{'}$ are isomorphic (apart from two well-known pairings).

Reçu le : 2015-06-16
Accepté le : 2016-02-26
Publié le : 2017-06-12

DOI : 10.5802/jtnb.983

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

Affiliations des auteurs :

Alfred Geroldinger ¹ ; Qinghai Zhong ¹

¹ University of Graz, NAWI Graz Institute for Mathematics and Scientific Computing Heinrichstraße 36 8010 Graz, Austria

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     journal = {Journal de th\'eorie des nombres de Bordeaux},
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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, Tome 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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