Calabi-Yau completions for roots of dualizing dg bimodules

Abstract

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.