Yoneda algebras and their singularity categories
Yoneda algebras and their singularity categories
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DOI：
10.1112/plms.12441
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发表时间：
2019-02
影响因子：
1.8
通讯作者：
Norihiro Hanihara
中科院分区：
文献类型：
--
作者：
Norihiro Hanihara
For a finite dimensional algebra Λ$\Lambda$ of finite representation type and an additive generator M$M$ for modΛ$\operatorname{mod}\Lambda$ , we investigate the properties of the Yoneda algebra Γ=⨁i⩾0ExtΛi(M,M)$\Gamma =\bigoplus _{i \geqslant 0}\operatorname{Ext}_\Lambda ^i(M,M)$ . We show that Γ$\Gamma$ is graded coherent and Gorenstein of self‐injective dimension at most 1, and the graded singularity category DsgZ(Γ)$\mathrm{D_{sg}^\mathbb {Z}}(\Gamma )$ of Γ$\Gamma$ is triangle equivalent to the derived category of the stable Auslander algebra of Λ$\Lambda$ . These results remain valid for representation‐infinite algebras. For this we introduce the Yoneda category Y$\mathcal {Y}$ of Λ$\Lambda$ as the additive closure of the shifts of the Λ$\Lambda$ ‐modules in the derived category Db(modΛ)$\mathrm{D^b}(\operatorname{mod}\Lambda )$ . We show that Y$\mathcal {Y}$ is coherent and Gorenstein of self‐injective dimension at most 1, and the singularity category of Y$\mathcal {Y}$ is triangle equivalent to the derived category Db(mod(mod̲Λ))$\mathrm{D^b}(\operatorname{mod}(\operatorname{\underline{\operatorname{mod}}}\Lambda ))$ of the stable category mod̲Λ$\operatorname{\underline{\operatorname{mod}}}\Lambda$ . To give a triangle equivalence, we apply the theory of realization functors. We show that any algebraic triangulated category has an f‐category over itself by formulating the filtered derived category of a DG category, which assures the existence of a realization functor.
For a finite dimensional algebra Λ$\Lambda$ of finite representation type and an additive generator M$M$ for modΛ$\operatorname{mod}\Lambda$ , we investigate the properties of the Yoneda algebra Γ=⨁i⩾0ExtΛi(M,M)$\Gamma =\bigoplus _{i \geqslant 0}\operatorname{Ext}_\Lambda ^i(M,M)$ . We show that Γ$\Gamma$ is graded coherent and Gorenstein of self‐injective dimension at most 1, and the graded singularity category DsgZ(Γ)$\mathrm{D_{sg}^\mathbb {Z}}(\Gamma )$ of Γ$\Gamma$ is triangle equivalent to the derived category of the stable Auslander algebra of Λ$\Lambda$ . These results remain valid for representation‐infinite algebras. For this we introduce the Yoneda category Y$\mathcal {Y}$ of Λ$\Lambda$ as the additive closure of the shifts of the Λ$\Lambda$ ‐modules in the derived category Db(modΛ)$\mathrm{D^b}(\operatorname{mod}\Lambda )$ . We show that Y$\mathcal {Y}$ is coherent and Gorenstein of self‐injective dimension at most 1, and the singularity category of Y$\mathcal {Y}$ is triangle equivalent to the derived category Db(mod(mod̲Λ))$\mathrm{D^b}(\operatorname{mod}(\operatorname{\underline{\operatorname{mod}}}\Lambda ))$ of the stable category mod̲Λ$\operatorname{\underline{\operatorname{mod}}}\Lambda$ . To give a triangle equivalence, we apply the theory of realization functors. We show that any algebraic triangulated category has an f‐category over itself by formulating the filtered derived category of a DG category, which assures the existence of a realization functor.