Gorenstein Projective Objects in Comma Categories

TL;DR

This paper characterizes Gorenstein projective objects in comma categories using properties from the component categories and explores their stable categories through recollements, advancing the understanding of Gorenstein homological algebra.

Contribution

It provides an equivalent characterization of Gorenstein projective objects in comma categories and establishes a recollement framework for their stable categories.

Findings

Characterization of Gorenstein projective objects in comma categories.

Existence of a left recollement of stable categories.

Recollement can be extended to a full recollement under certain conditions.

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories and $\mathbf{F}:\mathcal{A}\to \mathcal{B}$ an additive and right exact functor which is perfect, and let $(\mathbf{F},\mathcal{B})$ be the left comma category. We give an equivalent characterization of Gorenstein projective objects in $(\mathbf{F},\mathcal{B})$ in terms of Gorenstein projective objects in $\mathcal{B}$ and $\mathcal{A}$. We prove that there exists a left recollement of the stable category of the subcategory of $(\mathbf{F},\mathcal{B})$ consisting of Gorenstein projective objects modulo projectives relative to the same kind of stable categories in $\mathcal{B}$ and $\mathcal{A}$. Moreover, this left recollement can be filled into a recollement when $\mathcal{B}$ is Gorenstein and $\mathbf{F}$ preserves projectives.