The Stacks project

Lemma 37.30.2. 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

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

  2. $S$ 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 37.30.1 is bijective and hence a closed immersion. But since $S$ is reduced, $j$ is an isomorphism. $\square$


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