Lemma 38.39.6. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. Assume
$f$ is flat and proper and $\mathcal{O}_ S = f_*\mathcal{O}_ X$,
$S$ is a normal Noetherian scheme,
the pullback of $\mathcal{E}$ to $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ is free for every codimension $1$ point $s \in S$.
Then $\mathcal{E}$ is isomorphic to the pullback of a finite locally free $\mathcal{O}_ S$-module.
Proof. We will prove the canonical map
is an isomorphism. By flat base change (Cohomology of Schemes, Lemma 30.5.2) and assumptions (1) and (3) we see that the pullback of this to $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ is an isomorphism for every codimension $1$ point $s \in S$. By Divisors, Lemma 31.2.11 it suffices to prove that $\text{depth}((f^*f_*\mathcal{E})_ x) \geq 2$ for any point $x \in X$ mapping to a point $s \in S$ of codimension $\geq 2$. Since $f$ is flat and $(f^*f_*\mathcal{E})_ x = (f_*\mathcal{E})_ s \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{X, x}$, it suffices to prove that $\text{depth}((f_*\mathcal{E})_ s) \geq 2$, see Algebra, Lemma 10.163.2. Since $S$ is a normal Noetherian scheme and $\dim (\mathcal{O}_{S, s}) \geq 2$ we have $\text{depth}(\mathcal{O}_{S, s}) \geq 2$, see Properties, Lemma 28.12.5. Thus we get what we want from Divisors, Lemma 31.6.6. $\square$
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