The bounded derived category of coherent sheaves on a smooth projective variety X is an invariant that encodes several geometric properties of the variety. Remarkable invariants are the topological, Kodaira, and numerical dimensions, as well as some Hodge numbers. In this talk I will consider bounded derived categories of coherent sheaves supported on a closed subset Z in X and will study equivalences between them. While the existence of a Serre functor immediately yields the invariance of the dimension of the ambient space, it is an open question whether the dimension of the support Z is invariant as well. During the talk I will show a proof of this invariance when the canonical bundle restricted to the support is ample.