Relative Buchweitz-Happel theorem respect to a self-orthogonal class

Authors : Haiyu LİU, Yuxian GENG, Rongmin ZHU

Pages : 374-390

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Publication Date : 2023-03-31

Article Type : Research Paper

Abstract :Let $R$ be a ring, $F$ a subbifunctor of the functor Ext$^{1}_{R}insert ignore into journalissuearticles values(-,-);$, $\\mathcal{W}_{F}$ a self-orthogonal class of left $R$-modules respect to $F$. We introduce $\\mathcal{W}_{F}$-Gorenstein modules $\\mathcal{G}insert ignore into journalissuearticles values(\\mathcal{W}_{F});$ as a generalization of $\\mathcal{W}$-Gorenstein modules insert ignore into journalissuearticles values(Geng and Ding, 2011);, $F$-Gorenstein projective and $F$-Gorenstein injective modules insert ignore into journalissuearticles values(Tang, 2014);. We introduce the notion of relative singularity category $D_{\\mathcal{W}_{F}} insert ignore into journalissuearticles values(R);$ with respect to $\\mathcal{W}_{F}$. Moreover, we give a necessary and sufficient condition such that the stable category $\\underline{\\mathcal{G}insert ignore into journalissuearticles values(\\mathcal{W}_{F});}$ and the relative singularity category $D_{\\mathcal{W}_{F}} insert ignore into journalissuearticles values(R);$ are triangle-equivalence.
Keywords : Buchweitz Happel Theorem, self orthogonal class, triangle equivalence