TITLE: Exact sequence of sheaves with non exact sequence of global sections
QUESTION [13 upvotes]: Let $X$ be some topological space. By $\mathcal{F}_i$ we denote some sheaves of abelian groups on $X$. The sequence of sheaves and morphisms  $$\mathcal{F}_1\longrightarrow \mathcal{F}_2\longrightarrow \mathcal{F}_3\longrightarrow... $$ is said to be exact if for each $x\in X$ the corresponding sequence of stalks $$(\mathcal{F}_1)_x\longrightarrow (\mathcal{F}_2)_x\longrightarrow (\mathcal{F}_3)_x\longrightarrow... $$ is exact. However if the sequence of sheaves is exact than the sequence of global sections is not necessarily exact! (The most famous example is the sequence of sheaves $$0\longrightarrow\mathbb{Z}\hookrightarrow \mathcal{O}\stackrel{\exp}{\longrightarrow} \mathcal{O}^*\longrightarrow0$$ considered as sheaves on $\mathbb{C}-\{0\}$, where $\mathcal{O}$ is a sheaf of holomorphic functions, $\mathcal{O}^*$ is a sheaf of holomorphic functions with no zeros).

So, could you give me some easy examples of such phenomenon ?

REPLY [4 votes]: Let $M$ be a smooth manifold and consider the de Rham sheaf sequence on $M$:
$$0\to\mathbf{R} \to \mathcal{O}_M \to \Omega^1_M\to \Omega^2_M\to \dots$$
where the first map is inclusion and the other maps are exterior differentiation. It is exact as a sequence of sheaves by the Poincaré lemma, but on global sections it is the usual de Rham complex, whose cohomology is the de Rham cohomology. Of course unless $M$ is 1-dimensional, is not a short exact sequence, but nevertheless, the phenomenon is the same.