A localisation theorem for singularity categories of proper dg algebras

Given a recollement of three proper dg algebras over a noetherian commutative ring, e.g. three algebras which are finitely generated over the base ring, which extends one step downwards, it is shown that there is a short exact sequence of their singularity categories. This allows us to recover and generalise some known results on singularity categories of finite-dimensional algebras.