Proper classes and Gorensteinness in extriangulated categories

By Jiangsheng Hu, Dongdong Zhang, Panyue Zhou. Journal of Algebra, 2019

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Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. A notion of proper class in an extriangulated category is defined in this paper. Let $\mathcal{C}$ be an extriangulated category and $\xi$ a proper class in $\mathcal{C}$. We prove that $\mathcal{C}$ admits a new extriangulated structure. This construction gives extriangulated categories which are neither exact categories nor triangulated categories. Moreover, we introduce and study $\xi$-Gorenstein projective objects in $\mathcal{C}$ and demonstrate that $\xi$-Gorenstein projective objects share some basic properties with Gorenstein projective objects in module categories or in triangulated categories. In particular, we refine a result of Asadollahi and Salarian [Gorenstein objects in triangulated categories, J. Algebra 281(2004), 264-286]. As an application, the $\xi$-$\mathcal{G}$projective model structures on extriangulated categories are obtained.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c8232849f3cb7798282467ed27bc0694efba98d6