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 ?
Best Answer
If $p$ is a point on a compact Riemann surface of genus one, we have the exact sequence of sheaves $0 \to \mathcal O \to \mathcal O(p)\to \mathbb C_p \to 0$ (the last non zero sheaf being a sky-scraper sheaf) .
The sequence of global sections is $$ 0 \to \mathbb C = \mathbb C \stackrel {0} {\to}\mathbb C \to 0 $$ and is thus not exact.