On singular equivalences of Morita type

Stable equivalences of Morita type preserve many interesting properties and are proved to be the appropriate concept for studying equivalences between stable categories. Recently the singularity category attained much attraction and Xiao-Wu Chen and Long-Gang Sun gave an appropriate definition of singular equivalence of Morita type. We shall show that under some conditions singular equivalences of Morita type have some biadjoint functor properties and preserve positive degree Hochschild homology.