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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 étale 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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