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Factoriality and class groups of cluster algebras. (arXiv:1712.06512v1 [math.AC])
来源于:arXiv
Locally acyclic cluster algebras are Krull domains. Hence their factorization
theory is determined by their (divisor) class group and the set of classes
containing height-1 prime ideals. Motivated by this, we investigate class
groups of cluster algebras. We show that any cluster algebra that is a Krull
domain has a finitely generated free abelian class group, and that every class
contains infinitely many height-$1$ prime ideals. For a cluster algebra
associated to an acyclic seed, we give an explicit description of the class
group in terms of the initial exchange matrix. As a corollary, we reprove and
extend a classification of factoriality for cluster algebras of Dynkin type. In
the acyclic case, we prove the sufficiency of necessary conditions for
factoriality given by Geiss--Leclerc--Schr\"oer. 查看全文>>