Torsion classes of extended Dynkin quivers over commutative rings
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866918005554806784 |
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| author | Iyama, Osamu Kimura, Yuta |
| author_facet | Iyama, Osamu Kimura, Yuta |
| contents | For a Noetherian $R$-algebra $Λ$, there is a canonical inclusion $\mathsf{tors}Λ\to\prod_{\mathfrak{p}\in \mathrm{Spec} R}\mathsf{tors}(κ(\mathfrak{p})Λ)$, and each element in the image satisfies a certain compatibility condition. We call $Λ$ compatible if the image coincides with the set of all compatible elements. For example, for a Dynkin quiver $Q$ and a commutative Noetherian ring $R$ containing a field, the path algebra $RQ$ is compatible. In this paper, we prove that $RQ$ is compatible when $Q$ is an extended Dynkin quiver and $R$ is either a Dedekind domain or a Noetherian semilocal normal ring of dimension two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_02928 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Torsion classes of extended Dynkin quivers over commutative rings Iyama, Osamu Kimura, Yuta Representation Theory Commutative Algebra Rings and Algebras For a Noetherian $R$-algebra $Λ$, there is a canonical inclusion $\mathsf{tors}Λ\to\prod_{\mathfrak{p}\in \mathrm{Spec} R}\mathsf{tors}(κ(\mathfrak{p})Λ)$, and each element in the image satisfies a certain compatibility condition. We call $Λ$ compatible if the image coincides with the set of all compatible elements. For example, for a Dynkin quiver $Q$ and a commutative Noetherian ring $R$ containing a field, the path algebra $RQ$ is compatible. In this paper, we prove that $RQ$ is compatible when $Q$ is an extended Dynkin quiver and $R$ is either a Dedekind domain or a Noetherian semilocal normal ring of dimension two. |
| title | Torsion classes of extended Dynkin quivers over commutative rings |
| topic | Representation Theory Commutative Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2303.02928 |