  We present a proof that quantum Yang-Mills theory can be consistently defined
as a renormalized, perturbative quantum field theory on an arbitrary globally
hyperbolic curved, Lorentzian spacetime. To this end, we construct the
non-commutative algebra of observables, in the sense of formal power series, as
well as a space of corresponding quantum states. The algebra contains all gauge
invariant, renormalized, interacting quantum field operators (polynomials in
the field strength and its derivatives), and all their relations such as
commutation relations or operator product expansion. It can be viewed as a
deformation quantization of the Poisson algebra of classical Yang-Mills theory
equipped with the Peierls bracket. The algebra is constructed as the cohomology
of an auxiliary algebra describing a gauge fixed theory with ghosts and
anti-fields. A key technical difficulty is to establish a suitable hierarchy of
Ward identities at the renormalized level that ensure conservation of the
interacting BRST-current, and that the interacting BRST-charge is nilpotent.
The algebra of physical interacting field observables is obtained as the
cohomology of this charge. As a consequence of our constructions, we can prove
that the operator product expansion closes on the space of gauge invariant
operators. Similarly, the renormalization group flow is proved not to leave the
space of gauge invariant operators.
