The Stacks project

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. $\square$