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
\Phi : f^*f_*\mathcal{E} \longrightarrow \mathcal{E}
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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