Proper Resolutions and Gorenstein Categories

Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ an additive full subcategory of $\mathcal{A}$. We provide a method to construct a proper $\mathcal{C}$-resolution (resp. coproper $\mathcal{C}$-coresolution) of one term in a short exact sequence in $\mathcal{A}$ from that of the other two terms. By using these constructions, we answer affirmatively an open question on the stability of the Gorenstein category $\mathcal{G}(\mathcal{C})$ posed by Sather-Wagstaff, Sharif and White; and also prove that $\mathcal{G}(\mathcal{C})$ is closed under direct summands. In addition, we obtain some criteria for computing the $\mathcal{C}$-dimension and the $\mathcal{G}(\mathcal{C})$-dimension of an object in $\mathcal{A}$.