Gorenstein categories relative to G-admissible triples
Sergio Estrada, Octavio Mendoza, Marco A. Pérez
TL;DR
The paper develops relative Gorenstein categories through G-admissible triples, generalizing classical Gorenstein categories to a setting that supports homotopical methods even when enough projectives or injectives are absent. It introduces relative Gorenstein objects (GP-/GI-) and shows how to construct hereditary abelian model structures via compatible cotorsion pairs, yielding both GP- and GI-relative model frameworks. A central contribution is the notion of G-admissible triples and the notion of strongly relative Gorenstein categories, with the main results giving characterizations, cotorsion-theoretic structures, and model structures that connect to tilting theory and various examples (modules, quiver representations, and schemes). The work unifies and extends prior approaches (Beligiannis-Reiten, Tate, Xu) and provides a cohesive framework to study relative homological dimensions and stable categories with tilting-theoretic interpretations. Overall, it offers a versatile toolkit for relative Gorenstein homological algebra and its homotopical manifestations, with broad applications in representation theory and beyond.
Abstract
We present the notion of Gorenstein categories relative to G-admissible triples. This is a relativization of the concept of Gorenstein category (an abelian category with enough projective and injective objects, in which the suprema of the sets $\{ {\rm pd}(I) \ \text{:} \ I \text{ is injective} \}$ and $\{ {\rm id}(P) \ \text{:} \ P \text{ is projective} \}$ are finite). Such categories turn out to be a suitable setting on which it is possible to obtain hereditary abelian model structures where the (co)fibrant objects are Gorenstein injective (resp., Gorenstein projective) objects relative to GI-admissible (resp., GP-admissible) pairs. Applications and examples of these structures are given. Moreover, we link relative Gorenstein categories with tilting theory and obtain relations between different relative homological dimensions.