The question concerns a sheaf $S$ of abelian groups over a compact space $X$. Suppose each stalk $S_x$ is finite rank free. Is the group of global sections free?

Well, each stalk $S_x$ is generated by a finite number of sections over an open containing $x$. We can cover $X$ by such opens, and then find a finite subcover. This should, I hope, give some information about the group of global sections.
– David RobertsSep 13 '11 at 4:53

Is your space at least Hausdorff or you are working in an algebraic category?
– Anton FonarevSep 13 '11 at 7:15

1

For the constant sheaf $\mathbb Z$, it is true: the global sections may be identified with a subgroup of the group of bounded functions $X\to\mathbb Z$, and the latter group is free (G. Nöbeling, Verallgemeinerung eines Satzes von Herrn E. Specker, Invent. Math. 6 (1968) 41-55).
– user2035Sep 13 '11 at 13:43

The space is not necessarily Hausdorff, but an affirmative answer in the Hausdorff case would be interesting. The case of the constant sheaf mentioned is the motivation for this question. Specker's proof of freeness in that case doesn't seem to go over very well to this more general case.
– GMarkSep 13 '11 at 15:24