  We introduce Hochschild (co-)homology of morphisms of schemes or analytic
spaces and study its fundamental properties. In analogy with the cotangent
complex we introduce the so called (derived) Hochschild complex of a morphism;
the Hochschild cohomology and homology groups are then the Ext and Tor groups
of that complex. We prove that these objects are well defined, extend the known
cases, and have the expected functorial and homological properties such as
graded commutativity of Hochschild cohomology and existence of the
characteristic homomorphism from Hochschild cohomology to the (graded) centre
of the derived category.
