A reduction approach to silting objects for derived categories of hereditary categories
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866917589189394432 |
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| author | Dai, Wei Fu, Changjian |
| author_facet | Dai, Wei Fu, Changjian |
| contents | Let $\mathcal{H}$ be a hereditary abelian category over a field $k$ with finite dimensional $\operatorname{Hom}$ and $\operatorname{Ext}$ spaces. It is proved that the bounded derived category $\mathcal{D}^b(\mathcal{H})$ has a silting object iff $\mathcal{H}$ has a tilting object iff $\mathcal{D}^b(\mathcal{H})$ has a simple-minded collection with acyclic $\operatorname{Ext}$-quiver. Along the way, we obtain a new proof for the fact that every presilting object of $\mathcal{D}^b(\mathcal{H})$ is a partial silting object. We also consider the question of complements for pre-simple-minded collections. In contrast to presilting objects, a pre-simple-minded collection $\mathcal{R}$ of $\mathcal{D}^b(\mathcal{H})$ can be completed into a simple-minded collection iff the $\operatorname{Ext}$-quiver of $\mathcal{R}$ is acyclic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_10728 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A reduction approach to silting objects for derived categories of hereditary categories Dai, Wei Fu, Changjian Rings and Algebras Let $\mathcal{H}$ be a hereditary abelian category over a field $k$ with finite dimensional $\operatorname{Hom}$ and $\operatorname{Ext}$ spaces. It is proved that the bounded derived category $\mathcal{D}^b(\mathcal{H})$ has a silting object iff $\mathcal{H}$ has a tilting object iff $\mathcal{D}^b(\mathcal{H})$ has a simple-minded collection with acyclic $\operatorname{Ext}$-quiver. Along the way, we obtain a new proof for the fact that every presilting object of $\mathcal{D}^b(\mathcal{H})$ is a partial silting object. We also consider the question of complements for pre-simple-minded collections. In contrast to presilting objects, a pre-simple-minded collection $\mathcal{R}$ of $\mathcal{D}^b(\mathcal{H})$ can be completed into a simple-minded collection iff the $\operatorname{Ext}$-quiver of $\mathcal{R}$ is acyclic. |
| title | A reduction approach to silting objects for derived categories of hereditary categories |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2011.10728 |