论文标题
莫里塔定理的遗传性卡拉比豪类别
Morita theorem for hereditary Calabi-Yau categories
论文作者
论文摘要
我们通过遗传簇倾斜对象为Calabi-yau三角剖分类别提供了一个结构定理。 We prove that an algebraic $d$-Calabi-Yau triangulated category with a $d$-cluster tilting object $T$ such that its shifted sum $T\oplus\cdots\oplus T[-(d-2)]$ has hereditary endomorphism algebra $H$ is triangle equivalent to the orbit category $\mathscr{D}^b(\mathrm{\mathop{mod}}\, H)/τ^{-1/(d-1)}[1]$ of the derived category of $H$ for a naturally defined $(d-1)$-st root $τ^{1/(d-1)}$ of the AR translation, provided $H$ is of non-Dynkin type.我们还表明，$ h $的遗传性来自$ t $的$ d = 3 $，当$ t \ oplus t [-1] $当$ d = 4 $时，同样是从较小的内态代数来实现较小的尺寸，以消失在某些负面自我t $ $ t $的负面自我范围内。因此，我们的结果概括了凯勒（Reiten）和凯勒（Keller） - 梅尔菲特（Murfet） - van den bergh的既定定理。此外，我们表明这种三角类别的增强是独一无二的。最后，我们将结果应用于有限维代数的较高群集类别的Calabi-yau降低和不变子次数的奇异性类别的较高类别。
We give a structure theorem for Calabi-Yau triangulated category with a hereditary cluster tilting object. We prove that an algebraic $d$-Calabi-Yau triangulated category with a $d$-cluster tilting object $T$ such that its shifted sum $T\oplus\cdots\oplus T[-(d-2)]$ has hereditary endomorphism algebra $H$ is triangle equivalent to the orbit category $\mathscr{D}^b(\mathrm{\mathop{mod}}\, H)/τ^{-1/(d-1)}[1]$ of the derived category of $H$ for a naturally defined $(d-1)$-st root $τ^{1/(d-1)}$ of the AR translation, provided $H$ is of non-Dynkin type. We also show that hereditaryness of $H$ follows from that of $T$ is when $d=3$, that of $T\oplus T[-1]$ when $d=4$, and similarly from a smaller endomorphism algebra for higher dimensions under vanishing of some negative self-extensions of $T$. Our result therefore generalizes the established theorems by Keller--Reiten and Keller--Murfet--Van den Bergh. Furthermore, we show that enhancements of such triangulated categories are unique. Finally we apply our results to Calabi-Yau reductions of a higher cluster category of a finite dimensional algebra and of the singularity category of an invariant subring.
