A note on singular equivalences and idempotents

Abstract

Let Λ \Lambda be an Artin algebra and let e e be an idempotent in Λ \Lambda . We study certain functors which preserve the singularity categories. Suppose pd ⁡ Λ e e Λ e > ∞ \operatorname {pd} \Lambda e_{e\Lambda e}>\infty and id ⁡ Λ Λ / ⟨ e ⟩ r a d Λ / ⟨ e ⟩ > ∞ \operatorname {id}{_\Lambda \frac {\Lambda /\langle e\rangle }{rad\Lambda /\langle e\rangle }} > \infty , we show that there is a singular equivalence between e Λ e e\Lambda e and Λ \Lambda .