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- W4288272488 abstract "Let $(mathcal{C},mathbb{E},mathfrak{s})$ be an extriangulated category with a proper class $xi$ of $mathbb{E}$-triangles. The authors introduced and studied $xi$-$mathcal{G}$projective and $xi$-$mathcal{G}$injective in cite{HZZ}. In this paper, we discuss Gorenstein homological dimensions for extriangulated categories. More precisely, we first give some characterizations of $xi$-$mathcal{G}$projective dimension by using derived functors on $mathcal{C}$. Second, let $mathcal{P}(xi)$ (resp. $mathcal{I}(xi)$) be a generating (resp. cogenerating) subcategory of $mathcal{C}$. We show that the following equality holds under some assumptions: $$sup{xitextrm{-}mathcal{G}{rm pd}M | textrm{for} textrm{any} Min{mathcal{C}}}=sup{xitextrm{-}mathcal{G}{rm id}M | textrm{for} textrm{any} Min{mathcal{C}}},$$ where $xitextrm{-}mathcal{G}{rm pd}M$ (resp. $xitextrm{-}mathcal{G}{rm id}M$) denotes $xi$-$mathcal{G}$projective (resp. $xi$-$mathcal{G}$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." @default.
- W4288272488 created "2022-07-28" @default.
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- W4288272488 date "2019-08-01" @default.
- W4288272488 modified "2023-10-15" @default.
- W4288272488 title "Gorenstein homological dimensions for extriangulated categories" @default.
- W4288272488 doi "https://doi.org/10.48550/arxiv.1908.00931" @default.
- W4288272488 hasPublicationYear "2019" @default.
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