Gorenstein Homological Dimensions for Extriangulated Categories

• DOI: 10.1007/s40840-020-01057-9
• Publication Date: 2021-01-06T21:09:24Z
• Authors: Jiangsheng Hu; Dongdong Zhang; Panyue Zhou
• Keywords: 18E30, 18E10, 18G25, 55N20; FOS: Mathematics; Mathematics - Category Theory; Category Theory (math.CT); Representation Theory (math.RT); 0101 mathematics; 01 natural sciences; Mathematics - Representation Theory

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $��$ of $\mathbb{E}$-triangles. The authors introduced and studied $��$-$\mathcal{G}$projective and $��$-$\mathcal{G}$injective in \cite{HZZ}. In this paper, we discuss Gorenstein homological dimensions for extriangulated categories. More precisely, we first give some characterizations of $��$-$\mathcal{G}$projective dimension by using derived functors on $\mathcal{C}$. Second, let $\mathcal{P}(��)$ (resp. $\mathcal{I}(��)$) be a generating (resp. cogenerating) subcategory of $\mathcal{C}$. We show that the following equality holds under some assumptions: $$\sup\{��\textrm{-}\mathcal{G}{\rm pd}M \ | \ \textrm{for} \ \textrm{any} \ M\in{\mathcal{C}}\}=\sup\{��\textrm{-}\mathcal{G}{\rm id}M \ | \ \textrm{for} \ \textrm{any} \ M\in{\mathcal{C}}\},$$ where $��\textrm{-}\mathcal{G}{\rm pd}M$ (resp. $��\textrm{-}\mathcal{G}{\rm id}M$) denotes $��$-$\mathcal{G}$projective (resp. $��$-$\mathcal{G}$injective) dimension of $M$. As an application, our main results generalize their work by Bennis-Mahdou and Ren-Liu. Moreover, our proof is not far from the usual module or triangulated case.<br/>14 pages. arXiv admin note: text overlap with arXiv:1906.10989<br/>