Gorenstein categories relative to G-admissible triples

TL;DR

This paper introduces Gorenstein categories relative to G-admissible triples, establishing a framework for hereditary abelian model structures with Gorenstein objects, and explores their connections with tilting theory and homological dimensions.

Contribution

It develops the concept of Gorenstein categories relative to G-admissible triples, expanding the framework for Gorenstein homological algebra and model structures.

Findings

Established hereditary abelian model structures with Gorenstein objects.

Linked relative Gorenstein categories with tilting theory.

Provided examples and applications of the new structures.

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.