TITLE: Cohomology of a sheaf of functions locally constant along a foliation
QUESTION [17 upvotes]: Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known about Chech cohomology of such a sheaf?
I am pretty sure that such  a question was studied (and maybe even has a complete answer), but I don't know a reference. 
A more specific question is: what happen when $F$ is 1-dimensional, given by integral trajectories of a  non-vanishing vector field? Or even more specifically, suppose $H^1(M^n)=0$ and we consider a Killing vector field $v$ on $M^n$ (i.e. $v$ is preserving a metric). Is it true the the sheaf of functions  $\cal F$ locally constant along trajectories of $v$ is acyclic? (we need $H^1(M^n)=0$, otherwise $S^1$ will be an obvious counterexample).
An example of a foliation. Consider the unit sphere $S^3$ in $\mathbb C^2$ and conisder the action of $\mathbb R$ via diagonal matrixes : $(z,w)\to (e^{ita}z, e^{itb}w)$ with $\frac{a}{b}$ irrational.

REPLY [2 votes]: I am joining this discussion a bit late, but let me add an example. If you consider a smooth minimal action of Z on the circle S^1 the suspension gives a flow on the torus. If the action is C^2 conjugate to an irrational rotation, then the transverse basic cohomology is finite dimensional. But if the action is only topologically conjugate to a rotation, then the basic cohomology may be infinite. The literature on this is quite a long time ago, in the 1970's perhaps. here is one reference
Haefliger, A.and Banghe, Li
Currents on a circle invariant by a Fuchsian group. Geometric dynamics (Rio de Janeiro, 1981), 369–378, Lecture Notes in Math., 1007, Springer, Berlin, 1983. 
Here is a more recent article
Avila, Artur and Kocsard, Alejandro
Cohomological equations and invariant distributions for minimal circle diffeomorphisms. Duke Math. J. 158 (2011), no. 3, 501–536. 
and there is one more artcile that is likely relevant to the question
Lott, John
Invariant currents on limit sets. Comment. Math. Helv. 75 (2000), no. 2, 319–350.