Special Precovered Categories of Gorenstein Categories

Abstract

Let $\mathcal{A}$ be an abelian category and $\mathcal{P}(\mathcal{A})$ the subcategory of $\mathcal{A}$ consisting of projective objects. Let $\mathcal{C}$ be a full, additive and self-orthogonal subcategory of $\mathcal{A}$ with $\mathcal{P}(\mathcal{A})$ a generator, and let $\mathcal{G}(\mathcal{C})$ be the Gorenstein subcategory of $\mathcal{A}$. Then the right 1-orthogonal category $\mathcal{G}(\mathcal{C}{)}^{{\perp }_1}$ of $\mathcal{G}(\mathcal{C})$ is both projectively resolving and injectively coresolving in $\mathcal{A}$. We also get that the subcategory $\spc(\mathcal{G}(\mathscr{C}))$ of $\mathcal{A}$ consisting of objects admitting special $\mathcal{G}(\mathcal{C})$-precovers is closed under extensions and $\mathcal{C}$-stable direct summands (*). Furthermore, if $\mathcal{C}$ is a generator for $\mathcal{G}(\mathcal{C}{)}^{{\perp }_1}$, then we have that $\spc(\mathcal{G}(\mathscr{C}))$ is the minimal subcategory of $\mathcal{A}$ containing $\mathcal{G}(\mathcal{C}{)}^{{\perp }_1}\cup \mathcal{G}(\mathcal{C})$ with respect to the property (*), and that $\spc(\mathcal{G}(\mathscr{C}))$ is $\mathcal{C}$-resolving in $\mathcal{A}$ with a $\mathcal{C}$-proper generator $\mathcal{C}$.

19 pages, accepted for publication in Science China Mathematics