  We characterize the dependence on doublets of the cohomology of an arbitrary
nilpotent differential s (including BRST differentials and classical linearized
Slavnov-Taylor (ST) operators) in terms of the cohomology of the
doublets-independent component of s. All cohomologies are computed in the space
of local integrated formal power series. We drop the usual assumption that the
counting operator for the doublets commutes with s (decoupled doublets) and
discuss the general case where the counting operator does not commute with s
(coupled doublets). The results are purely algebraic and do not rely on
power-counting arguments.
