  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 $\Lambda$.
