Special Precovered Categories of Gorenstein Categories

TL;DR

This paper investigates special precovering categories related to Gorenstein categories within an abelian category, establishing their closure properties and resolving structures under certain conditions.

Contribution

It introduces and characterizes special Gorenstein precovering subcategories, showing their closure under extensions and their resolving properties in abelian categories.

Findings

The right 1-orthogonal of Gorenstein subcategories is projectively resolving.

The subcategory of objects with special Gorenstein precovers is extension-closed.

Under certain conditions, this subcategory is minimal and resolving with a proper generator.

Abstract

Let $\mathscr{A}$ be an abelian category and $\mathscr{P}(\mathscr{A})$ the subcategory of $\mathscr{A}$ consisting of projective objects. Let $\mathscr{C}$ be a full, additive and self-orthogonal subcategory of $\mathscr{A}$ with $\mathscr{P}(\mathscr{A})$ a generator, and let $\mathcal{G}(\mathscr{C})$ be the Gorenstein subcategory of $\mathscr{A}$. Then the right 1-orthogonal category ${\mathcal{G}(\mathscr{C})^{\bot_1}}$ of $\mathcal{G}(\mathscr{C})$ is both projectively resolving and injectively coresolving in $\mathscr{A}$. We also get that the subcategory $\spc(\mathcal{G}(\mathscr{C}))$ of $\mathscr{A}$ consisting of objects admitting special $\mathcal{G}(\mathscr{C})$-precovers is closed under extensions and $\mathscr{C}$-stable direct summands (*). Furthermore, if $\mathscr{C}$ is a generator for $\mathcal{G}(\mathscr{C})^{\perp_1}$, then we have that $\spc(\mathcal{G}(\mathscr{C}))$ is the minimal subcategory of $\mathscr{A}$ containing $\mathcal{G}(\mathscr{C})^{\perp_1}\cup \mathcal{G}(\mathscr{C})$ with respect to the property (*), and that $\spc(\mathcal{G}(\mathscr{C}))$ is $\mathscr{C}$-resolving in $\mathscr{A}$ with a $\mathscr{C}$-proper generator $\mathscr{C}$.