Homological epimorphisms in functor categories and singularity categories

Abstract

Given a homological epimorphism $\pi:\mathcal{C}\longrightarrow \mathcal{C}/\mathcal{I}$ between $K$-categories, we show that if the ideal $\mathcal{I}$ satisfies certain conditions, then there exists an equivalence between the singularity categories $\mathbf{D}_{sg}(\mathrm{Mod}(\mathcal{C}))$ and $ \mathbf{D}_{sg}(\mathrm{Mod}(\mathcal{C}/\mathcal{I}))$. This result generalizes the one obtained by Xiao-Wu Chen in [6]. We apply our result to the one point extension category and show that there is a singular equivalence between a $K$-category $\mathcal{U}$ and its one point extension category $\Lambda:=\left[ \begin{smallmatrix} \mathcal{C}_{K} & 0 \\ \underline{M} & \mathcal{U} \end{smallmatrix}\right].$