  In this article we consider quantum phase space reduction when zero is a
regular value of the momentum map. By analogy with the classical case we define
the BRST cohomology in the framework of deformation quantization. We compute
the quantum BRST cohomology in terms of a `quantum' Chevalley-Eilenberg
cohomology of the Lie algebra on the constraint surface. To prove this result,
we construct an explicit chain homotopy, both in the classical and quantum
case, which is constructed out of a prolongation of functions on the constraint
surface. We have observed the phenomenon that the quantum BRST cohomology
cannot always be used for quantum reduction, because generally its zero part is
no longer a deformation of the space of all smooth functions on the reduced
phase space. But in case the group action is `sufficiently nice', e.g. proper
(which is the case for all compact Lie group actions), it is shown for a
strongly invariant star product that the BRST procedure always induces a star
product on the reduced phase space in a rather explicit and natural way. Simple
examples and counter examples are discussed.
