2025/08/18 by Hanihara, Norihiro, Iyama, Osamu
#FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2508.12836
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 Π, we show that the induction functor gives an embedding from the poset siltdA of d-silting objects of A to the poset siltΠ of silting objects of Π. Moreover, when H0Π is finite dimensional, this functor identifies the Hasse quiver of siltdA as a full subquiver of the Hasse quiver of siltΠ. In this case, we also prove that each d-silting object P of A gives a d-cluster tilting subcategory of per A as the ν[-d]-orbit of P. Secondly, for a connective Calabi-Yau dg algebra Π, we study the map from siltΠ to the set d-ctiltC(Π) of d-cluster tilting objects in the cluster category C(Π). We call Π F-liftable if the induced map siltΠ\capF→ d-ctiltC(Π) is bijective, where F is the fundamental domain in perΠ. We prove that F-liftable Calabi-Yau dg algebras Π such that H0Π 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 siltΠ→ d-ctiltC(Π) is surjective, which we call liftable. We explain our results by polynomial dg algebras and Calabi-Yau completions of type A2.