On the stability question of Gorenstein categories (Q1701179)

For a subcategory \(\mathcal{X}\) of an abelian category \(\mathcal{A}\), the Gorenstein subcategory of \(\mathcal{A}\) associated to \(\mathcal{X}\) is defined as: \(\mathcal{G}(\mathcal{X})=\) prop-coprop-res(\(\mathcal{X}\)) \(\cap\) prop-coprop-cores(\(\mathcal{X}\)). Set \(\mathcal{G}^{0}(\mathcal{X})=\mathcal{X}\), \(\mathcal{G}^{1}(\mathcal{X})=\mathcal{G}(\mathcal{X})\) and \(\mathcal{G}^{n+1}(\mathcal{X})=\mathcal{G}^{n}(\mathcal{G}(\mathcal{X}))\) for each \(n\in\mathbb{N}\). \textit{S. Sather-Wagstaff} et al. [Algebr. Represent. Theory 14, No. 3, 403--428 (2011; Zbl 1317.13029)] proved that if \(\mathcal{X}\) is self-orthogonal and closed under finite direct sums then one has \(\mathcal{G}^{n}(\mathcal{X})=\mathcal{G}(\mathcal{X})\) for each \(n\geq1\). \textit{Z. Huang}'s result [J. Algebra 393, 142--169 (2013; Zbl 1291.18022)] shows that the above equality holds if \(\mathcal{X}\) is an additive subcategory of \(\mathcal{A}\) (not necessarily self-orthogonal). In the paper under review, the authors further generalize Huang's result by studying the subcategories of an abelian category \(\mathcal{A}\) constituted of all objects that admit (coproper) proper resolutions (resp. (proper) coproper coresolutions).