Silting correspondences and Calabi-Yau dg algebras

Abstract

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 $\Pi$, we show that the induction functor gives an embedding from the poset $\operatorname{silt}^dA$ of $d$-silting objects of $A$ to the poset $\operatorname{silt}\Pi$ of silting objects of $\Pi$. Moreover, when $H^0\Pi$ is finite dimensional, this functor identifies the Hasse quiver of $\operatorname{silt}^dA$ as a full subquiver of the Hasse quiver of $\operatorname{silt}\Pi$. 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 $\nu[-d]$-orbit of $P$. Secondly, for a connective Calabi-Yau dg algebra $\Pi$, we study the map from $\operatorname{silt}\Pi$ to the set $d\text{-}\operatorname{ctilt}\mathcal{C}(\Pi)$ of $d$-cluster tilting objects in the cluster category $\mathcal{C}(\Pi)$. We call $\Pi$ $\mathcal{F}$-liftable if the induced map $\operatorname{silt}\Pi\cap\mathcal{F}\to d\text{-}\operatorname{ctilt}\mathcal{C}(\Pi)$ is bijective, where $\mathcal{F}$ is the fundamental domain in $\operatorname{per}\Pi$. We prove that $\mathcal{F}$-liftable Calabi-Yau dg algebras $\Pi$ such that $H^0\Pi$ 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 $\operatorname{silt}\Pi\to d\text{-}\operatorname{ctilt}\mathcal{C}(\Pi)$ is surjective, which we call liftable. We explain our results by polynomial dg algebras and Calabi-Yau completions of type $A_2$.