Explicitly showing cokernel of exponential sequence is not a sheaf In the classical example of short exact sequence of presheaves $$0\rightarrow \mathbb Z \rightarrow \mathcal O_X \rightarrow \mathcal F \rightarrow 1,$$ it is well known that $\mathcal F$ is not a sheaf because there are functions that admit a logarithm locally but not globally. I'm trying to make sense this statement more vividly by giving explicit open sets $U_i$ such that I can find $f_i \in \mathcal F(U_i)$ which agree on the intersections but cannot be lifted to a global $f\in\mathcal F(\cup U_i)$. After all, that's the way of showing that $\mathcal F$ is not a sheaf, right?
My idea was to show that I cannot patch up $log(x)$ over some non-simply connected open set that goes around the origin. Of course, $log(x)$ cannot exist in such a set, but my problem is in finding a cover of such a set such that a $log(x)$ does exist in each component and agree on the intersections. Making them agree is the hard part, and the best I could do was the following:
Pick $U_1$ to be the pointed disk around the origin with a cut along the negative real axis and $U_2$ some open, simply connected set meeting the cut but not the origin. In this way $U_2$ would "bridge over the cut", and the union of the two sets would not be able to contain some $f$ which restricts to $log(x)$ in $U_1, U_2$. The problem is that my $U_1$ and $U_2$ do not satisfy the glueability condition of having some $f_i$ which agree on the intersection. 
 A: If $U$ is a simply connected open of $\Bbb C$, you have an exact sequence $0(U) \to \Bbb Z(U) \xrightarrow{2i\pi} \Gamma(U,\Bbb C) \xrightarrow{\exp} \Gamma(U,\Bbb C^*) \to 0(U)$.
The sheaf who is locally the holomorphic functions to $\Bbb C^*$, is the sheaf of holomorphic functions to $\Bbb C^*$.  
But if you pick $U = \Bbb C^*$, the sequence of abelian groups $0(\Bbb C^*) \to \Bbb Z(\Bbb C^*) \xrightarrow{2i\pi} \Gamma(\Bbb C^*,\Bbb C) \xrightarrow{\exp} \Gamma(\Bbb C^*,\Bbb C^*) \to 0(\Bbb C^*)$ is not exact. For example, the identity map $id_{\Bbb C^*} \in \Gamma(\Bbb C^*,\Bbb C^*)$ is not in the image of $\Gamma(\Bbb C^*,\Bbb C)$ by $\exp$ (because there is no logarithm function defined on all $\Bbb C^*$).
If you define $\mathcal F(U) = coker (\Bbb Z(U) \xrightarrow{2i\pi} \Gamma(U,\Bbb C))$, then this is not a sheaf : let $U_\pm = \Bbb C \setminus \Bbb R^\pm$, you have elements $\log_\pm \in \mathcal F(U_\pm)$ given by the classes of the logarithm functions $\log_\pm : U_\pm \to \Bbb C$ modulo $\Bbb Z(U_\pm)$. They agree on $U_+ \cap U_-$. Not because the functions themselves agree, but because $(\log_+ - \log_-)/2i\pi$ is an element of $\Bbb Z(U_+ \cap U_-)$ : it is an integer-valued locally constant function.
But their glueing doesn't correspond to the class of any function $\log : \Bbb C^* \to \Bbb C$. So there is no element in $\mathcal F(\Bbb C^*)$ whose restrictions are the $\log_\pm$. 
