One-Sided Gorenstein Subcategories

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 rG($$\mathcal{G}(\mathscr{C})$$) 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 rG($$\mathcal{G}(\mathscr{C})$$), and prove that any object in $$\mathscr{A}$$ with finite rG($$\mathcal{G}(\mathscr{C})$$)-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 rG($$\mathcal{G}(\mathscr{C})$$). 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.