Is this sequence of presheaves exact?

Question

On p.298 of his Homological Algebra text, Rotman considers the sequence of presheaves on $X=\mathbb{C}-\{0\}$ : $0 \to \mathbb{Z}\to \mathcal{O}\to \mathcal{O}^\times \to 0$ where $\mathbb{Z}$ is the constant sheaf, $\mathcal{O}$ is the presheaf of holomorphic functions and $\mathcal{O}^\times$ is the presheaf of non zero holomorphic functions. The last map is given by $f\mapsto e^{2 \pi i f}$. He mentions that this sequence of presheaves is exact. However it seems to me that this is incorrect since if the presheaf sequence were exact so would be the sequence of abelian groups obtained by taking the global sections. However the function $f(z)=z$ does not come from any global section of $\mathcal{O}$. So the sequence of presheaves is not exact. Please advise if I am making some mistake. I think the correct statement should have been that the sequence of sheaves is exact.

Answer 1

You are right in all points. You can find here a list of other errata. Perhaps you should mail Rotman. By the way, the book can be downloaded at the archive.