Lemma 37.33.2 (0EX7)—The Stacks project

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Lemma 37.33.2. Let $f : X \to S$ be a flat, proper morphism of finite presentation such that $f_*\mathcal{O}_ X = \mathcal{O}_ S$ and this remains true after arbitrary base change. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$ -module. Assume

$\mathcal{E}|_{X_ s}$ is isomorphic to $\mathcal{O}_{X_ s}^{\oplus r_ s}$ for all $s \in S$ , and
$S$ is reduced.

Then $\mathcal{E} = f^*\mathcal{N}$ for some finite locally free $\mathcal{O}_ S$ -module $\mathcal{N}$ .

Proof. Namely, in this case the locally closed immersion $j : Z \to S$ of Lemma 37.33.1 is bijective and hence a closed immersion. But since $S$ is reduced, $j$ is an isomorphism. $\square$

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