The Stacks project

Lemma 71.16.4. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_ X$-module. There is a bijection

\[ \left\{ \begin{matrix} \text{sections }\sigma \text{ of the } \\ \text{morphism } \mathbf{P}(\mathcal{E}) \to X \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{surjections }\mathcal{E} \to \mathcal{L}\text{ where} \\ \mathcal{L}\text{ is an invertible }\mathcal{O}_ X\text{-module} \end{matrix} \right\} \]

In this case $\sigma $ is a closed immersion and there is a canonical isomorphism

\[ \mathop{\mathrm{Ker}}(\mathcal{E} \to \mathcal{L}) \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes -1} \longrightarrow \mathcal{C}_{\sigma (X)/\mathbf{P}(\mathcal{E})} \]

Both the bijection and isomorphism are compatible with base change.

Proof. Because the constructions are compatible with base change, it suffices to check the statement ├ętale locally on $X$. Thus we may assume $X$ is a scheme and the result is Divisors, Lemma 31.31.6. $\square$


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