Homological epimorphisms in functor categories and singularity categories

Abstract

Given a homological epimorphism $\ddot{I}€:\mathcal{C}⟶\mathcal{C}/\mathcal{I}$ 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 $\hat{I}›:=\left[\begin{array}{cc}{\mathcal{C}}_K& 0\\ \underline{M}& \mathcal{U}\end{array}\right].$

19 pages