Special Precovered Categories of Gorenstein Categories

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 $\mathscr{C}$.