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Suppose that $\alpha$ is nowhere zero. A differential form $\beta$ satisfies $\alpha\wedge \beta=0$ just when $\beta=\alpha \wedge \gamma$ for some $\gamma$ by Cartan's lemma. So the cohomology vanishes, finite dimensional. On the other hand, take $\alpha=0$. Then the kernel is everything, the image nothing, so the quotient is everything, infinite dimensional.