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.



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> Is each cohomology, a finite dimensional vector space?
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A cohomology associated to a 1- form
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.


