My question refers to a step in the proof of THM 2.1.1 (Grotherdieck) from excerpt from "Vector Bundles on Complex Projective Spaces" by Christian Okonek, Michael Schneider, Heinz Spindler (page 12):
Consider a non trivial section $s \in H^0(\mathbb{P}^1, E(k_0)$ such that $s(x)=0$ for a point $x \in \mathbb{P}^1$.
Equivalently: $s(x) =0 \in E(k_0)_{\mathbb{P}^1,x}/m_x$ or $s(x) \in m_x$ in the unique max ideal of the stalk $E(k_0)_{\mathbb{P}^1,x}$.
I have two questions:
1.: Why does it imply that $s$ in a section of the tensor sheaf $E(k_0) \otimes_{\mathbb{P}^1} J_x$ where $J_x$ is the corresponding ideal sheaf of the point $x$ interpreted as closed subscheme of $\mathbb{P}^1$.
- Why $J_x = \mathcal{O}_{\mathcal{\mathbb{P}^1}}(-1)$ as sheaf?
I know that $J_x$ naturally belongs to exact sequence
$$0 \to J_x \to \mathcal{O}_{\mathcal{\mathbb{P}^1}} \to \mathcal{O}_x \to 0$$
where $\mathcal{O}_x$ is skyscraper with only non trivial stalk $k(x)$ in $x$.
But why $J_x = \mathcal{O}_{\mathcal{\mathbb{P}^1}}(-1)$?