Assume that we have a codimension one foliation of a manifold $M$ which is generated by a one form $\alpha$. So the following $\phi$ satisfies $\phi \circ \phi =0$:$$\phi:\Omega^{i}(M)\to \Omega^{i+2}(M): \phi(\beta)=\alpha\wedge d\beta$$ Then we obtain a cohomology. The total cohomology is denoted by $H^{*}(\alpha)$ which is a $H^{0}(\alpha)$-module. $H^{0}(\alpha)$ consist all smooth functions which are constant along leaves of the foliation.

On the other hand, differential forms correspond to singular cochains, so we have a similar complex as above in the following way: We consider $\alpha$ as a 1-cochain then we define $$\phi:C^{i}(M,\mathbb{C})\to C^{i+2}(M, \mathbb{C}): \phi(\beta)=\alpha \smile\sigma \beta$$ The total cohomology is denoted by $H^{*}(\alpha)$. We reduce $H^{0}(\alpha)$ to all continuous functions constant along leaves.(We ignor non continuos functions). Similar to the above smooth case, we have that $H^{*}(\alpha)$ is a $H^{0}(\alpha)$-module.

In the following two questions we concern with the singular but not smooth version

Question 1: Is $H^{*}(\alpha)$ a finitely generated projective module over the commutative algebra $H^{0}(\alpha)$?

We know that for some particular foliations, with non Hausdorff holonomy groupoid, the commutative algebra $H^{0}(\alpha)$ does not contain enough information about the foliation $F$. Indeed there is a non commutative remedy $C^{*}(F)$. So our next question is that

Question 2: Assume that a codimension one foliation $F$ is generated by a one form. Can we assign a finitely generated projective $C^{*}(F)$- module(A NC vector bundle) which constructions is based on the above processess and contains useful information about the dynamics of the foliation?