  Let $W$ be a differential (not necessarily commutative) algebra which carries
a free action of a polynomial algebra $SP$ with homogeneous generators $p_1,
>..., p_r$. We show that for $W$ acyclic, the cohomology of the quotient
$H(W/<p_1, ..., p_r>)$ is isomorphic to a Clifford algebra $\text{Cl}(P,B)$,
where the (possibly degenerate) bilinear form $B$ depends on $W$. This
observation is an analogue of an old result of Borel in a non-commutative
context. As an application, we study the case of $W$ given by the quantized
Weil algebra $\qWg = \Ug \otimes \Clg$ for $\Lieg$ a reductive Lie algebra. The
resulting cohomology of the canonical Weil differential gives a Clifford
algebra, but the bilinear form vanishes on the space of primitive invariants of
the semi-simple part. As an application, we consider the deformed Weil
differential (following Freed, Hopkins and Teleman).
