Tilting objects in periodic triangulated categories

Abstract

A triangulated category $\mathcal{T}$ whose suspension functor $\Sigma$ satisfies ${\Sigma }^m\simeq {\mathrm{Id}}_{\mathcal{T}}$ as additive functors is called an $m$-periodic triangulated category. Such a category does not have a tilting object by the periodicity. In this paper, we introduce the notion of an $m$-periodic tilting object in an $m$-periodic triangulated category, which is a periodic analogue of a tilting object in a triangulated category, and prove that an $m$-periodic triangulated category having an $m$-periodic tilting object is triangulated equivalent to the $m$-periodic derived category of an algebra under some homological assumptions. As an application, we construct a triangulated equivalence between the stable category of a self-injective algebra and the $m$-periodic derived category of a hereditary algebra.

23 pages, Added new and corrected an error in the previous version, comments welcome

References in corpus (1)

• A note on Grothendieck groups of periodic derived categories

Cited by in corpus (2)

• The Donovan--Wemyss Conjecture via the Derived Auslander--Iyama Correspondence
• Formality and strongly unique enhancements