Relative Gorenstein objects in abelian categories

TL;DR

This paper introduces and studies relative Gorenstein objects in abelian categories, generalizing classical Gorenstein projective modules and related concepts, and develops a relative Gorenstein homological dimension theory.

Contribution

It defines weak and relative Gorenstein objects in abelian categories, extending existing notions, and establishes foundational properties and a relative Gorenstein dimension theory.

Findings

Generalization of Gorenstein projective modules to relative settings

Extension of Gorenstein homological dimension concepts

Introduction of $ ext{W}$-cotilting pairs and their relation to cotorsion pairs

Abstract

Let $\mathcal{A}$ be an abelian category. For a pair $(\mathcal{X},\mathcal{Y}$ of classes of objects in $\mathcal{A},$ we define the weak and the $(\mathcal{X},\mathcal{Y})$-Gorenstein relative projective objects in $\mathcal{A}$. We point out that such objects generalize the usual Gorenstein projective objects and others generalizations appearing in the literature as Ding-projective, Ding-injective, $\mathcal{X}$-Gorenstein projective, Gorenstein AC-projective and $G_C$-projective modules and Cohen-Macaulay objects in abelian categories. We show that the principal results on Gorenstein projective modules remains true for the weak and the $(\mathcal{X},\mathcal{Y}$-Gorenstein relative objects. Furthermore, by using Auslander-Buchweitz approximation theory, a relative version of Gorenstein homological dimension is developed. Finally, we introduce the notion of $\mathcal{W}$-cotilting pair in the abelian category $\mathcal{A}$, which is very strong connected with the cotorsion pairs related with relative Gorenstein objects in $\mathcal{A}$. It is worth mentioning that the $\mathcal{W}$-cotilting pairs generalize the notion of cotilting objects in the sense of L. Angeleri H\"ugel and F. Coelho.