Gorenstein complexes and recollements from cotorsion pairs

TL;DR

This paper establishes a correspondence between recollements of triangulated categories and cotorsion pairs, applying it to Gorenstein modules over various rings, and introduces Gorenstein AC-modules for broader contexts.

Contribution

It provides a unified model category framework linking recollements and cotorsion pairs, extending results to Gorenstein AC-modules over general rings.

Findings

Recollements relate to cotorsion pairs in triangulated categories.

Gorenstein modules are used to construct specific recollements over Noetherian and coherent rings.

Gorenstein injective objects have a maximality property in abelian categories.

Abstract

We describe a general correspondence between injective (resp. projective) recollements of triangulated categories and injective (resp. projective) cotorsion pairs. This provides a model category description of these recollement situations. Our applications focus on displaying several recollements that glue together various full subcategories of K(R), the homotopy category of chain complexes of modules over a general ring R. When R is (left) Noetherian ring, these recollements involve complexes built from the Gorenstein injective modules. When R is a (left) coherent ring for which all flat modules have finite projective dimension we obtain the duals. These results extend to a general ring R by replacing the Gorenstein modules with the Gorenstein AC-modules introduced recently in the work of Bravo-Gillespie-Hovey. We also see that in any abelian category with enough injectives, the Gorenstein injective objects enjoy a maximality property in that they contain every other class making up the right half of an injective cotorsion pair.