The catenary degree of Krull monoids I

Geroldinger, Alfred; Grynkiewicz, David J.; Schmid, Wolfgang A.

Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 137-169.

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}\left(H\right)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each two factorizations $z,z^{\prime }$ of $a$, there exist factorizations $z=z_0,...,z_k=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. Under a very mild condition on the Davenport constant of $G$, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between $\mathsf{c}\left(H\right)$ and the set of distances of $H$ and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on $\mathsf{c}\left(H\right)$ and characterize when $\mathsf{c}\left(H\right)\le 4$.

Soit $H$ un monoïde de Krull de groupe de classes $G$ fini. On suppose que chaque classe contient un diviseur premier (par exemple, l’anneau des entiers d’un corps de nombres ou l’anneau d’holomorphie d’un corps de fonctions). Le degré de chaînage $\mathsf{c}\left(H\right)$ de $H$ est le plus petit entier $N$ ayant la propriété suivante : pour tout $a\in H$ et toute paire de factorisations $z,z^{\prime }$ de l’élément $a$, il existe des factorisations $z=z_0,...,z_k=z^{\prime }$ de $a$ telles que, pour chaque $i\in [1,k]$, on puisse obtenir $z_i$ à partir de $z_{i-1}$ en modifiant au plus $N$ atomes. Dans cet article, nous obtenons une nouvelle caractérisation du degré de chaînage pour les $H$ dont la constante de Davenport du groupe de classes vérifie une certaine hypothèse très peu restrictive. Cette caractérisation offre un nouveau point de vue, plus structurel, sur la notion de degré de chaînage. En particulier, elle clarifie la relation entre $\mathsf{c}\left(H\right)$ et l’ensemble des distances de $H$ et permet d’envisager l’obtention de résultats plus précis sur le degré de chaînage. Nous illustrons ce phénomène en donnant deux applications : une nouvelle borne supérieure pour $\mathsf{c}\left(H\right)$ et la caractérisation des $H$ tels que $\mathsf{c}\left(H\right)\le 4$.

Classification: 11R27, 13F05, 20M13
Keywords: non-unique factorizations, Krull monoids, catenary degree, zero-sum sequence

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