Quotients of triangulated categories and Equivalences of Buchweitz, Orlov, and Amiot-Guo-Keller

We give a simple sufficient condition for a Verdier quotient ${\cal T}/\ {\cal S}$ of a triangulated category ${\cal T}$ by a thick subcategory ${\cal S}$ to be realized inside of ${\cal T}$ as an ideal quotient. As applications, we deduce three significant results by Buchweitz, Orlov and Amiot-Guo-Keller.