The triangulated hull of periodic complexes

TL;DR

This paper explicitly describes how the orbit category of an algebra's derived category embeds into its triangulated hull, revealing new insights into the structure and automorphisms of these categories.

Contribution

It provides an explicit description of the embedding of orbit categories into their triangulated hulls for $n$-periodic derived categories, highlighting cases where orbit categories are strictly smaller.

Findings

Orbit categories can be strictly smaller than their triangulated hulls.

Automorphisms may not induce the identity functor on orbit categories.

Explicit embeddings are constructed for $n$-periodic derived categories.

Abstract

In the terms of an `$n$-periodic derived category', we describe explicitly how the orbit category of the bounded derived category of an algebra with respect to powers of the shift functor embeds in its triangulated hull. We obtain a large class of algebras whose orbit categories are strictly smaller than their triangulated hulls and a realization of the phenomenon that an automorphism need not induce the identity functor on the associated orbit category.