  A crucial property of the standard antifield-BRST cohomology at non negative
ghost number is that any cohomological class is completely determined by its
antifield independent part. In particular, a BRST cocycle that vanishes when
the antifields are set equal to zero is necessarily exact.\ \ This property,
which follows from the standard theorems of homological perturbation theory,
holds not only in the algebra of local functions, but also in the space of
local functionals. The present paper stresses how important it is that the
antifields in question be the usual antifields associated with the gauge
invariant description. By means of explicit counterexamples drawn from the free
Maxwell-Klein-Gordon system, we show that the property does not hold, in the
case of local functionals, if one replaces the antifields of the gauge
invariant description by new antifields adapted to the gauge fixation. In terms
of these new antifields, it is not true that a local functional weakly
annihilated by the gauge-fixed BRST generator determines a BRST cocycle; nor
that a BRST cocycle which vanishes when the antifields are set equal to zero is
necessarily exact.
