Cluster tilting objects in generalized higher cluster categories

TL;DR

This paper proves the existence of m-cluster tilting objects in generalized m-cluster categories derived from Calabi-Yau dg algebras, extending previous results and applicable to various algebraic structures.

Contribution

It generalizes the existence of m-cluster tilting objects to broader classes of Calabi-Yau categories, including those from Ginzburg dg algebras and finite-dimensional algebras.

Findings

Existence of m-cluster tilting objects in generalized m-cluster categories.

Application to categories from finite-dimensional algebras of finite global dimension.

Application to categories from Ginzburg dg algebras with graded quivers.

Abstract

We prove the existence of an $m$-cluster tilting object in a generalized $m$-cluster category which is $(m+1)$-Calabi-Yau and Hom-finite, arising from an $(m+2)$-Calabi-Yau dg algebra. This is a generalization of the result for the ${m = 1}$ case in Amiot's Ph.~D.~thesis. Our results apply in particular to higher cluster categories associated to suitable finite-dimensional algebras of finite global dimension, and higher cluster categories associated to Ginzburg dg categories coming from suitable graded quivers with superpotential.