Factoriality and class groups of cluster algebras
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866912815163375616 |
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| author | Elsener, Ana Garcia Lampe, Philipp Smertnig, Daniel |
| author_facet | Elsener, Ana Garcia Lampe, Philipp Smertnig, Daniel |
| contents | 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--Schröer. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1712_06512 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Factoriality and class groups of cluster algebras Elsener, Ana Garcia Lampe, Philipp Smertnig, Daniel Commutative Algebra Rings and Algebras 13F60 (Primary) 13A15, 13F05, 13F15 (Secondary) 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--Schröer. |
| title | Factoriality and class groups of cluster algebras |
| topic | Commutative Algebra Rings and Algebras 13F60 (Primary) 13A15, 13F05, 13F15 (Secondary) |
| url | https://arxiv.org/abs/1712.06512 |