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Let $W$ be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra $SP$ with homogeneous generators $p_1,\cdots ,p_r$. We show that for $W$ acyclic, the cohomology of the quotient $H(W/$ $\<p_1,\cdots ,p_r\>)$ 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 $𝒲\left(𝔤\right)=U𝔤\otimes \text{Cl}𝔤$ for $𝔤$...