$τ$-exceptional sequences for representations of quivers over local algebras
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915431299678208 |
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| author | Nonis, Iacopo |
| author_facet | Nonis, Iacopo |
| contents | Let $k$ be an algebraically closed field. Let $R$ be a finite dimensional commutative local $k$-algebra and let $Q$ be a quiver with no oriented cycles. In this paper, we study (signed) $τ$-exceptional sequences over the algebra $Λ= R\otimes kQ$, which is isomorphic to $RQ$. We show there is a bijection between the set of complete (signed) $τ$-exceptional sequences in $\text{mod }kQ$ and the set of complete (signed) $τ$-exceptional sequences in $\text{mod }Λ$. Moreover, we prove that every $τ$-perpendicular subcategory of $\text{mod }Λ$ is equivalent to the module category of $R\otimes kQ'$, for some quiver $Q'$. As a consequence, we prove that the $τ$-cluster morphism categories of $kQ$ and $Λ$ are equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $τ$-exceptional sequences for representations of quivers over local algebras Nonis, Iacopo Representation Theory 16G20, 16D90, 16G70, 18E40 Let $k$ be an algebraically closed field. Let $R$ be a finite dimensional commutative local $k$-algebra and let $Q$ be a quiver with no oriented cycles. In this paper, we study (signed) $τ$-exceptional sequences over the algebra $Λ= R\otimes kQ$, which is isomorphic to $RQ$. We show there is a bijection between the set of complete (signed) $τ$-exceptional sequences in $\text{mod }kQ$ and the set of complete (signed) $τ$-exceptional sequences in $\text{mod }Λ$. Moreover, we prove that every $τ$-perpendicular subcategory of $\text{mod }Λ$ is equivalent to the module category of $R\otimes kQ'$, for some quiver $Q'$. As a consequence, we prove that the $τ$-cluster morphism categories of $kQ$ and $Λ$ are equivalent. |
| title | $τ$-exceptional sequences for representations of quivers over local algebras |
| topic | Representation Theory 16G20, 16D90, 16G70, 18E40 |
| url | https://arxiv.org/abs/2502.15417 |