On relative derived categories

TL;DR

This paper explores the properties and invariants of relative derived categories, extending existing results, establishing new equivalences, and analyzing their relations with Gorenstein modules and quiver reflections.

Contribution

It generalizes results of Happel, proves the existence of AR-triangles in Gorenstein derived categories, and studies relations between Gorenstein derived categories and quiver reflection functors.

Findings

Invariants of recollements involving relative derived categories are characterized.

AR-triangles are proven to exist in Gorenstein derived categories.

Relations between Gorenstein derived categories of quivers and their reflections are established.

Abstract

The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalise two results of Happel by proving the existence of AR-triangles in Gorenstein derived categories, provide situations for which relative derived categories with respect to Gorenstein projective and Gorenstein injective modules are equivalent and finally study relations between the Gorenstein derived category of a quiver and its image under a reflection functor. Some interesting applications are provided.