Let $X$ be a smooth algebraic curve, and $F$ be a sheaf of $\mathcal O$ module such that for every point $x \in X$, the stalk $F_x$ is isomorphic to $\mathcal O_x$. Does it follow that $F$ is an invertible sheaf ?
Clearly, the converse holds. But I have no ideas for the first statement : I know that for any $x$, there is a germ $s_x$ which generate $F_x$ with trivial annihilator. So if I pick any $f \in F(U)$, with $U$ a neighborhood of $x$ I can take a neighborhood $V$ such that $f = \lambda.s$ where $\lambda \in \mathcal O(V)$. But the problem is, possibly $V$ depends on $f$, and I don't see how to pick a good $V$. Maybe this is wrong but I don't see how to build a counter-example.
(This is an exercise in the book of Rick Miranda, Algebraic curves and Riemann surfaces.)