Silting correspondences and Calabi-Yau dg algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918256550346752 |
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| author | Hanihara, Norihiro Iyama, Osamu |
| author_facet | Hanihara, Norihiro Iyama, Osamu |
| contents | This paper is devoted to studying two important classes of objects in triangulated categories; silting objects and $d$-cluster tilting objects, and their correspondences. First, we introduce the notion of $d$-silting objects as a generalization tilting objects whose endomorphism algebras have global dimension at most $d$. For a smooth dg algebra $A$ and its $(d+1)$-Calabi-Yau completion $Π$, we show that the induction functor gives an embedding from the poset $\operatorname{silt}^dA$ of $d$-silting objects of $A$ to the poset $\operatorname{silt}Π$ of silting objects of $Π$. Moreover, when $H^0Π$ is finite dimensional, this functor identifies the Hasse quiver of $\operatorname{silt}^dA$ as a full subquiver of the Hasse quiver of $\operatorname{silt}Π$. In this case, we also prove that each $d$-silting object $P$ of $A$ gives a $d$-cluster tilting subcategory of $\operatorname{per} A$ as the $ν[-d]$-orbit of $P$. Secondly, for a connective Calabi-Yau dg algebra $Π$, we study the map from $\operatorname{silt}Π$ to the set $d\text{-}\operatorname{ctilt}\mathcal{C}(Π)$ of $d$-cluster tilting objects in the cluster category $\mathcal{C}(Π)$. We call $Π$ $\mathcal{F}$-liftable if the induced map $\operatorname{silt}Π\cap\mathcal{F}\to d\text{-}\operatorname{ctilt}\mathcal{C}(Π)$ is bijective, where $\mathcal{F}$ is the fundamental domain in $\operatorname{per}Π$. We prove that $\mathcal{F}$-liftable Calabi-Yau dg algebras $Π$ such that $H^0Π$ is hereditary are precisely the Calabi-Yau completions of hereditary algebras. As an application, we obtain counter-examples to an open question posed in [IYa1]. We also study Calabi-Yau dg algebras such that the map $\operatorname{silt}Π\to d\text{-}\operatorname{ctilt}\mathcal{C}(Π)$ is surjective, which we call liftable. We explain our results by polynomial dg algebras and Calabi-Yau completions of type $A_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12836 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Silting correspondences and Calabi-Yau dg algebras Hanihara, Norihiro Iyama, Osamu Representation Theory Rings and Algebras This paper is devoted to studying two important classes of objects in triangulated categories; silting objects and $d$-cluster tilting objects, and their correspondences. First, we introduce the notion of $d$-silting objects as a generalization tilting objects whose endomorphism algebras have global dimension at most $d$. For a smooth dg algebra $A$ and its $(d+1)$-Calabi-Yau completion $Π$, we show that the induction functor gives an embedding from the poset $\operatorname{silt}^dA$ of $d$-silting objects of $A$ to the poset $\operatorname{silt}Π$ of silting objects of $Π$. Moreover, when $H^0Π$ is finite dimensional, this functor identifies the Hasse quiver of $\operatorname{silt}^dA$ as a full subquiver of the Hasse quiver of $\operatorname{silt}Π$. In this case, we also prove that each $d$-silting object $P$ of $A$ gives a $d$-cluster tilting subcategory of $\operatorname{per} A$ as the $ν[-d]$-orbit of $P$. Secondly, for a connective Calabi-Yau dg algebra $Π$, we study the map from $\operatorname{silt}Π$ to the set $d\text{-}\operatorname{ctilt}\mathcal{C}(Π)$ of $d$-cluster tilting objects in the cluster category $\mathcal{C}(Π)$. We call $Π$ $\mathcal{F}$-liftable if the induced map $\operatorname{silt}Π\cap\mathcal{F}\to d\text{-}\operatorname{ctilt}\mathcal{C}(Π)$ is bijective, where $\mathcal{F}$ is the fundamental domain in $\operatorname{per}Π$. We prove that $\mathcal{F}$-liftable Calabi-Yau dg algebras $Π$ such that $H^0Π$ is hereditary are precisely the Calabi-Yau completions of hereditary algebras. As an application, we obtain counter-examples to an open question posed in [IYa1]. We also study Calabi-Yau dg algebras such that the map $\operatorname{silt}Π\to d\text{-}\operatorname{ctilt}\mathcal{C}(Π)$ is surjective, which we call liftable. We explain our results by polynomial dg algebras and Calabi-Yau completions of type $A_2$. |
| title | Silting correspondences and Calabi-Yau dg algebras |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2508.12836 |