  Twisted separable functors generalize the separable functors of Nastasescu,
Van den Bergh and Van Oystaeyen, and provide a convenient tool to compare
various projective dimensions. We discuss when an adjoint functor is twisted
separable, obtaining a version of Rafael's Theorem in the twisted case. As an
application, we show that if $R$ is Hopf-Galois object over a Hopf algebra $A$,
then their Hochschild cohomological dimension coincide, provided that the
cohomological dimension of $A$ is finite and that $R$ has a unital twisted
trace with respect to a semi-colinear automorphism.
