A Note on Singularity Categories and Triangular Matrix Algebras

Let \(\Lambda = \left[ \begin{array}{cc} A &{} 0 \\ M &{} B \end{array}\right] \) $\mathrm{\Lambda }=\left[\begin{array}{cc}A& 0\\ M& B\end{array}\right]$ be an Artin algebra and \(_BM_A\) ${}_B{M}_A$ a B-A-bimodule. We prove that there is a triangle equivalence \(D_{sg}(\Lambda ) \cong D_{sg}(A)\coprod D_{sg}(B)\) $D_{\mathit{sg}}(\mathrm{\Lambda })\cong D_{\mathit{sg}}(A)\coprod D_{\mathit{sg}}(B)$ between the corresponding singularity categories if \(_BM\) ${}_{B}M$ is semi-simple and \(M_A\) $M_A$ is projective. As a result, we obtain a new method for describing the singularity categories of certain bounded quiver algebras.