The image of the derived category in the cluster category

TL;DR

This paper investigates the relationship between derived categories and cluster categories for non-hereditary algebras, showing that orbit categories often do not coincide with cluster categories, highlighting limitations in their triangulated structure.

Contribution

It analyzes the extent to which orbit categories approximate cluster categories in non-hereditary cases, introducing new insights into their structural differences.

Findings

Orbit categories rarely equal cluster categories for non-piecewise hereditary algebras

The natural functor's image often does not cover the entire cluster category

Highlights limitations of orbit categories in non-hereditary algebra contexts

Abstract

Cluster categories of hereditary algebras have been introduced as orbit categories of their derived categories. Keller has pointed out that for non-hereditary algebras orbit categories need not be triangulated, and he introduced the notion of triangulated hull to overcome this problem. In this paper we study the image if the natural functor from the bounded derived category to the cluster category, that is we investigate how far the orbit category is from being the cluster category. We show that for wide classes of non-piecewise hereditary algebras the orbit category is never equal to the cluster category.