  To study coisotropic reduction in the context of deformation quantization we
introduce constraint manifolds and constraint algebras as the basic objects
encoding the additional information needed to define a reduction. General
properties of various categories of constraint objects and their compatiblity
with reduction are examined. A constraint Serre-Swan theorem, identifying
constraint vector bundles with certain finitely generated projective constraint
modules, as well as a constraint symbol calculus are proved. After developing
the general deformation theory of constraint algebras, including constraint
Hochschild cohomology and constraint differential graded Lie algebras, the
second constraint Hochschild cohomology for the constraint algebra of functions
on a constraint flat space is computed.
