Lifting recollements of abelian categories and model structures

TL;DR

This paper develops a systematic method using Quillen model structures to lift recollements from hereditary abelian categories to their homotopy categories, enabling new insights into module and Gorenstein categories.

Contribution

It introduces a framework for lifting recollements via Quillen adjoint triples, extending the understanding of derived and stable categories in algebra.

Findings

Liftings of module category recollements to derived categories.

Applications to Gorenstein projective and injective modules.

Framework for transferring abelian model structures along adjoint pairs.

Abstract

We use Quillen model structures to show a systematic method to lift recollements of hereditary abelian model categories to recollements of their associated homotopy categories. To that end, we use the notion of Quillen adjoint triples and we investigate transfers of abelian model structures along adjoint pairs. Applications include liftings of recollements of module categories to their derived counterpart, liftings to homotopy categories that provide models for stable categories of Gorenstein projective and injective modules and liftings to homotopy categories of n-morphism categories over Iwanaga-Gorenstein rings.