  We show that Higher Hochschild complex associated to a connected pointed
simplicial set commutes with localization of commutative algebras over a field
of characteristic zero. Then, we define in two ways higher order Hochschild
cohomology of schemes over a field of characteristic zero. Originally, we can
take the hyperext functor of the sheaf associated to Higher Hochschild
presheaf. We obtain a Hodge decomposition for higher order Hochschild
cohomology of smooth algebraic varieties over a field of characteristic zero
which generalizes Pirashvili's Hodge decomposition. We can also define the
Higher Hochschild cohomology of order d of a separated scheme by taking the ext
functor of its structure presheaf over the Higher Hochschild presheaf of order
d -- 1. These two definitions are really close to those of Swanfor classical
Hochschild cohomology, but our tools are model categories and derived functors.
We also generalize the equivalence of Swan's definitions to any separated
schemes over a field.
