  Derived equivalences between finite dimensional algebras do, in general, not
pass to centraliser (or other) subalgebras, nor do they preserve homological
invariants of the algebras, such as global or dominant dimension. We show that,
however, they do so for large classes of algebras described in this article.
  Algebras $A$ of $\nu$-dominant dimension at least one have unique largest
non-trivial self-injective centraliser subalgebras $H_A$. A derived restriction
theorem is proved: A derived equivalence between $A$ and $B$ implies a derived
equivalence between $H_A$ and $H_B$.
  Two methods are developed to show that global and dominant dimension are
preserved by derived equivalences between algebras of $\nu$-dominant dimension
at least one with anti-automorphisms preserving simples, and also between
almost self-injective algebras. One method is based on identifying particular
derived equivalences preserving homological dimensions, while the other method
identifies homological dimensions inside certain derived categories.
  In particular, derived equivalent cellular algebras have the same global
dimension. As an application, the global and dominant dimensions of blocks of
quantised Schur algebras with $n \geq r$ are completely determined.
