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