A note on singular equivalences and idempotents

Let $\Lambda$ be an Artin algebra and let $e$ be an idempotent in $\Lambda$. We study certain functors which preserve the singularity categories. Suppose $\mathrm{pd}\Lambda e_{e\Lambda e}<\infty$ and $\mathrm{id}_\Lambda\tfrac{\Lambda/\langle e\rangle}{\mathrm{rad}\Lambda/\langle e\rangle} < \infty$, we show that there is a singular equivalence between $e\Lambda e$ and $\Lambda$.