One-Sided Gorenstein Subcategories

TL;DR

This paper introduces and studies the properties of one-sided Gorenstein subcategories in abelian categories, providing characterizations and structural results that extend the understanding of Gorenstein homological algebra.

Contribution

It defines right and left Gorenstein subcategories relative to an additive subcategory and proves their closure properties and characterizations, extending Gorenstein theory.

Findings

Right Gorenstein subcategory is closed under extensions, kernels of epimorphisms, direct summands, and finite direct sums.

Objects with finite Gorenstein projective dimension relate to kernels and cokernels of morphisms involving objects with finite projective dimension.

Application to weak Auslander-Buchweitz context and cotorsion pairs in abelian categories.

Abstract

We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $\C$ of an abelian category $\A$, and prove that the right Gorenstein subcategory $r\mathcal{G}(\mathscr{C})$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $\C$ is self-orthogonal, we give a characterization for objects in $r\mathcal{G}(\mathscr{C})$, and prove that any object in $\A$ with finite $r\mathcal{G}(\C)$-projective dimension is isomorphic to a kernel (resp. a cokernel) of a morphism from an object in $\A$ with finite $\C$-projective dimension to an object in $r\mathcal{G}(\C)$. As an application, we obtain a weak Auslander-Buchweitz context related to the kernel of a hereditary cotorsion pair in $\A$ having enough injectives.