Proper resolutions and Gorensteinness in triangulated categories

Abstract

Let $\mathcal {T}$ be a triangulated category with triangulation $\Delta $, $\xi \subseteq \Delta $ a proper class of triangles and $\mathcal {C}$ an additive full subcategory of $\mathcal {T}$. We provide a method for constructing a proper $\mathcal {C}(\xi )$-resolution (respectively, coproper $\mathcal {C}(\xi )$-coresolution) of one term in a triangle in $\xi $ from those of the other two terms. By using this construction, we show the stability of the Gorenstein category $\mathcal {GC}(\xi )$ in triangulated categories. Some applications are given.