标题： 卡拉比-丘对偶微分分次双模的根完成 显示英文标题

标题： Calabi-Yau completions for roots of dualizing dg bimodules

摘要： 移位Serre函子的根在表示论和代数几何中自然出现。 我们给出了Keller的Calabi-Yau完成在移位反自对偶双模的根上的类似物。 给定一个正整数$a$，我们引入了光滑dg范畴上的$a$次根对的概念，并定义其Calabi-Yau完成。 我们证明了当$a$次根对在阶数为$a$的循环群作用下具有某种不变性时，Calabi-Yau完成具有Calabi-Yau性质，并观察到一般情况下它只是扭曲的Calabi-Yau。 接下来，我们建立了Gorenstein参数为$a$的Adams分次Calabi-Yau dg范畴与具有循环不变性的dg范畴上的$a$次根对之间的双射。 应用这个双射，我们证明了对dg范畴的一种称为$a$-Segre乘积的操作，允许我们重现Calabi-Yau dg范畴。 此外，我们讨论这些卡拉比-丘完成的聚类范畴，并证明它是通常聚类范畴的$\mathbb{Z}/a\mathbb{Z}$-商，从而建立了聚类范畴的$a$-次根版本。 在附录中，我们将关于射影空间上倾斜丛的贝林森定理推广到阿德姆分次微分分次范畴的设置中。 显示英文摘要

摘要： Roots of shifted Serre functors appear naturally in representation theory and algebraic geometry. We give an analogue of Keller's Calabi-Yau completion for roots of shifted inverse dualizing bimodules over dg categories. Given a positive integer $a$, we introduce the notion of the $a$-th root pair on smooth dg categories and define its Calabi-Yau completion. We prove that the Calabi-Yau completion has the Calabi-Yau property when the $a$-th root pair has certain invariance under an action of the cyclic group of order $a$, and observe that it is only twisted Calabi-Yau in general. Next, we establish a bijection between Adams graded Calabi-Yau dg categories of Gorenstein parameter $a$ and $a$-th root pairs on a dg category with the cyclic invariance. Applying this bijection, we prove that a certain operation on dg categories, called the $a$-Segre product, allows us to reproduce Calabi-Yau dg categories. Furthermore, we discuss the cluster category of these Calabi-Yau completions, and prove that it is a $\mathbb{Z}/a\mathbb{Z}$-quotient of the usual cluster category, which thereby establishes the $a$-th root versions of cluster categories. In the appendix, we give a generalization of Beilinson's theorem on tilting bundles on projective spaces to the setting of Adams graded dg categories.