In this question all objects are real analytic.(manifolds, differential forms..)

Assume that $M$ is a compact manifold and $\alpha \in \Omega^{1}(M)$ is a one form.

We define a map $\phi:\Omega^{*}(M)\to \Omega^{*+1}(M)$ with wedge product; $\phi(\beta)=\alpha \wedge \beta$. Then $\phi \circ \phi=0$. Then we have a complex of vector spaces. So we naturally obtain a cohomology.

Is each cohomology, a finite dimensional vector space?