Triangulated categories of periodic complexes and orbit categories

TL;DR

This paper explores the structure of orbit categories related to derived categories of rings, revealing their connection to periodic complexes and establishing new equivalences and dualities in derived algebraic geometry.

Contribution

It introduces the triangulated hull of orbit categories of derived categories, linking them to categories of periodic complexes and extending known results to new algebraic contexts.

Findings

Triangulated hull corresponds to compact objects in categories of periodic complexes.

Derived equivalences of flat algebras induce equivalences of periodic derived categories.

Established a periodic version of Koszul duality.

Abstract

We investigate the triangulated hull of the orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull will correspond to the full subcategory of compact objects of certain triangulated categories of periodic complexes. This specializes to Stai and Zhao's result when the ring is a finite dimensional algebra with finite global dimension over a field. As the first application, if $A,B$ are flat algebras over a commutative ring and they are derived equivalent, then the corresponding derived categories of $n$-periodic complexes are triangle equivalent. As the second application, we get the periodic version of the Koszul duality.