A cohomology associated with a codimension one foliation

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 $f$?

First, it's not finite dimensional, even in the case of a torus. Just let $x,y$ be the $2\pi$-periodic functions on the torus and take $\alpha = \mathrm{d} x$, and you'll see that $H^0$ is all the functions of the form $f(x)$. On the other hand, if you let $\alpha = \mathrm{d} x + \sqrt{2}\,\mathrm{d} y$, then $H^0$ just consists of the constants, so it definitely depends on the foliation.
• Thank you very much for your answer. According to your example, is it natural to ask "what is a dynamical interpretation for finite dimensionality of $H^{*}(\alpha)$ for all dimension $*$, where the manifold is not necessarilly a low dim manifold. Moreover is wedge product well defind in this cohomology? And finally is it easy to compute these cohomology for some higher dimension:ex $S^{3}$ with the Reab foliation? Thanks again for your answer. – Ali Taghavi Jan 30 '15 at 12:12