  To all orders of perturbation theory, the renormalization of the topological
charge density in dimensionally regularized QCD is shown to require no more
than an additive renormalization proportional to the divergence of the
flavour-singlet axial current. The proof is based on the standard BRS analysis
of the QCD vertex functional in the background gauge and exploits the special
algebraic properties of the charge density through the Stora--Zumino chain of
descent equations.
