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 are corresponded to singular cochains, so we have a similar complex as above in the following way: We consider the one form $\alpha$ as a 1-cochain then we define $$\phi:C^{i}(M,\mathbb{C})\to C^{i+2}(M, \mathbb{C}): \phi(\beta)=\alpha \smile\sigma \beta$$ where $C^{i}(M,\mathbb{C})$ is the complex vector space generated by all $i$-cochains in $M$.
The total cohomology is denoted by $H^{*}(\alpha)$. $H^{0}(\alpha)$ is the space of all (not necessarily continuous) complex functions on $M$ which are constant along leaves. We reduce $H^{0}(\alpha)$ to only continuous functions constant along leaves and with some abuse of notation we denoted it again by $H^{0}(\alpha)$.(We ignor non continuous 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 or non hausdorff leaf space,, the commutative algebra $H^{0}(\alpha)$ does not contain enough information about the foliation $F$. Indeed there is a non commutative remedy, the noncommutative $C^{*}$ algebra of a foliation denoted by $C^{*}(F)$. 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 some useful information about the dynamics of the foliation?In particular some invariant or character of this noncommutative vector bundle would be related to some dynamical invariants of the foliation?
