Schur roots and tilting modules of acyclic quivers over commutative rings
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917975691362304 |
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| author | Iyama, Osamu Kimura, Yuta |
| author_facet | Iyama, Osamu Kimura, Yuta |
| contents | Let $Q$ be a finite acyclic quiver and $A_Q$ the cluster algebra of $Q$. It is well-known that for each field $k$, the additive equivalence classes of support tilting $kQ$-modules correspond bijectively with the clusters of $A_Q$. The aim of this paper is to generalize this result to any ring indecomposable commutative Noetherian ring $R$, that is, the additive equivalence classes of 2-term silting complexes of $RQ$ correspond bijectively with the clusters of $A_Q$. As an application, for a Dynkin quiver $Q$, we prove that the torsion classes of $\mathrm{mod} RQ$ corresponds bijectively with the order preserving maps from $\mathrm{Spec} R$ to the set of clusters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schur roots and tilting modules of acyclic quivers over commutative rings Iyama, Osamu Kimura, Yuta Representation Theory Commutative Algebra Rings and Algebras Let $Q$ be a finite acyclic quiver and $A_Q$ the cluster algebra of $Q$. It is well-known that for each field $k$, the additive equivalence classes of support tilting $kQ$-modules correspond bijectively with the clusters of $A_Q$. The aim of this paper is to generalize this result to any ring indecomposable commutative Noetherian ring $R$, that is, the additive equivalence classes of 2-term silting complexes of $RQ$ correspond bijectively with the clusters of $A_Q$. As an application, for a Dynkin quiver $Q$, we prove that the torsion classes of $\mathrm{mod} RQ$ corresponds bijectively with the order preserving maps from $\mathrm{Spec} R$ to the set of clusters. |
| title | Schur roots and tilting modules of acyclic quivers over commutative rings |
| topic | Representation Theory Commutative Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2504.02371 |