Let $\alpha$ be a non vanishing one form on a manifold which which defines a codimension one foliation. With this $\alpha$ we define the following complex:

$$\phi:\Omega^{i}(M)\to \Omega^{i+2}(M)\;\;\;\phi(\beta)=d(\alpha\wedge \beta)$$ Obviously $\phi$ satisfies $\phi \circ \phi=0$, so we have cohomologies associated with this complex. The total cohomology is denoted by $H^{*}(\alpha)$

Are these cohomologies finite dimensional vector space?

Are there some dynamical information in this cohomology?

Is this cohomology independent of choosing the one form $\alpha$ which kernel is tangent to the foliation? This means that: Is it true to say $H^{*}(\alpha) \simeq H^{*}(f\alpha)$ for a non vanishing smooth function $f$?