Every finitely generated abelian group is the class group of a generalized cluster algebra
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908343506829312 |
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| author | Pompili, Mara |
| author_facet | Pompili, Mara |
| contents | We determine the class group of those generalized cluster algebras that are Krull domains. In particular, this provides a criterion for determining whether or not a generalized cluster algebra is a UFD. In fact, any finitely generated abelian group can be realized as the class group of a generalized cluster algebra. Additionally, we show that generalized cluster algebras are FF-domains and that their cluster variables are strong atoms. Finally, we examine the factorization and ring-theoretic properties of Laurent phenomenon algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14963 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Every finitely generated abelian group is the class group of a generalized cluster algebra Pompili, Mara Commutative Algebra 13F60, 13F05, 13F15 We determine the class group of those generalized cluster algebras that are Krull domains. In particular, this provides a criterion for determining whether or not a generalized cluster algebra is a UFD. In fact, any finitely generated abelian group can be realized as the class group of a generalized cluster algebra. Additionally, we show that generalized cluster algebras are FF-domains and that their cluster variables are strong atoms. Finally, we examine the factorization and ring-theoretic properties of Laurent phenomenon algebras. |
| title | Every finitely generated abelian group is the class group of a generalized cluster algebra |
| topic | Commutative Algebra 13F60, 13F05, 13F15 |
| url | https://arxiv.org/abs/2411.14963 |