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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, 29 no. 2 (2017), p. 327-346, doi: 10.5802/jtnb.983
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Class. Math.: 11B30, 11R27, 13A05, 13F05, 20M13
Keywords: Krull monoids, maximal orders, seminormal orders, class groups, arithmetical characterizations, sets of lengths, zero-sum sequences, Davenport constant

Résumé - Abstract

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

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