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 $H^{*}(\alpha)$. 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 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?