  We study the quantization of chiral QED by means of an extended
Slavnov-Taylor (ST) identity under which an external source and the ghost field
form a doublet. This ST identity incorporates the Adler-Bardeen anomaly. We
prove that the cohomology classes of the extended classical linearized ST
operator ${\cal S}'_0$ are modified with respect to the ones of the classical
BRST differential in the FP neutral sector (physical observables). This
provides a counterexample showing that the introduction of a doublet can modify
the cohomology of the model, as a consequence of the fact that the counting
operator for the doublet does not commute with ${\cal S}'_0$.
