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

37 pages