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

Abstract

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