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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.07252 |
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| _version_ | 1866909864234582016 |
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| author | Han, Zhe He, Ping |
| author_facet | Han, Zhe He, Ping |
| contents | Given a torsion pair $(\mathcal{T},\mathcal{F})$ in an abelian category $\mathcal{A}$ and its Happel-Reiten-Smalø tilt $\mathcal{B}$, the equivalence of the realization functor $D^b({\mathcal B})\to D^b({\mathcal A})$ is determined by some properties of the torsion pair [9]. We call $(\mathcal{T},\mathcal{F})$ satisfying such a property effaceable. If $\mathcal{A}$ is an Ext-finite abelian category with Serre duality, we prove that $(\mathcal{T},\mathcal{F})$ is effaceable implies that $\mathcal{U}_{\mathcal T}$ is closed under Serre functor. Conversely, when $\mathcal A$ is the module category of a finite-dimensional hereditary algebra, we prove that the torsion pair $(\mathcal{T},\mathcal{F})$ is effaceable if and only if $\mathcal{U}_\mathcal{T}$ is closed under the Serre functor via a recollement of $D^b({\mathcal A})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07252 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Serre functor and torsion pairs Han, Zhe He, Ping Representation Theory Given a torsion pair $(\mathcal{T},\mathcal{F})$ in an abelian category $\mathcal{A}$ and its Happel-Reiten-Smalø tilt $\mathcal{B}$, the equivalence of the realization functor $D^b({\mathcal B})\to D^b({\mathcal A})$ is determined by some properties of the torsion pair [9]. We call $(\mathcal{T},\mathcal{F})$ satisfying such a property effaceable. If $\mathcal{A}$ is an Ext-finite abelian category with Serre duality, we prove that $(\mathcal{T},\mathcal{F})$ is effaceable implies that $\mathcal{U}_{\mathcal T}$ is closed under Serre functor. Conversely, when $\mathcal A$ is the module category of a finite-dimensional hereditary algebra, we prove that the torsion pair $(\mathcal{T},\mathcal{F})$ is effaceable if and only if $\mathcal{U}_\mathcal{T}$ is closed under the Serre functor via a recollement of $D^b({\mathcal A})$. |
| title | Serre functor and torsion pairs |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2403.07252 |