The Stacks project

Exercise 111.52.3. Let $k$ be a field. Let $\mathcal{E}$ be a vector bundle on $\mathbf{P}^2_ k$, i.e., a finite locally free $\mathcal{O}_{\mathbf{P}^2_ k}$-module. We say $\mathcal{E}$ is split if $\mathcal{E}$ is isomorphic to a direct sum invertible $\mathcal{O}_{\mathbf{P}^2_ k}$-modules.

1. Show that $\mathcal{E}$ is split if and only if $\mathcal{E}(n)$ is split.

2. Show that if $\mathcal{E}$ is split then $H^1({\mathbf{P}^2_ k}, \mathcal{E}(n)) = 0$ for all $n \in \mathbf{Z}$.

3. Let

   \[ \varphi : \mathcal{O}_{\mathbf{P}^2_ k} \longrightarrow \mathcal{O}_{\mathbf{P}^2_ k}(1) \oplus \mathcal{O}_{\mathbf{P}^2_ k}(1) \oplus \mathcal{O}_{\mathbf{P}^2_ k}(1) \]

   be given by linear forms $L_0, L_1, L_2 \in \Gamma (\mathbf{P}^2_ k, \mathcal{O}_{\mathbf{P}^2_ k}(1))$. Assume $L_ i \not= 0$ for some $i$. What is the condition on $L_0, L_1, L_2$ such that the cokernel of $\varphi $ is a vector bundle? Why?

4. Given an example of such a $\varphi $.

5. Show that $\mathop{\mathrm{Coker}}(\varphi )$ is not split (if it is a vector bundle).