A note on singular equivalences and idempotents

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

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