The Stacks project

Proof. We have to show that $\nu$ is integral, universally injective, and surjective, see Lemma 29.45.5. By Lemma 29.54.5 the morphism $\nu$ is integral. Let $x \in X$ and set $A = \mathcal{O}_{X, x}$. Since $X$ is irreducible we see that $A$ has a single minimal prime $\mathfrak p$ and $A_{red} = A/\mathfrak p$. By Lemma 29.54.4 the stalk $A' = (\nu _*\mathcal{O}_{X^\nu })_ x$ is the integral closure of $A$ in the fraction field of $A_{red}$. By More on Algebra, Definition 15.106.1 we see that $A'$ has a single prime $\mathfrak m'$ lying over $\mathfrak m_ x \subset A$ and $\kappa (\mathfrak m')/\kappa (x)$ is purely inseparable. Hence $\nu$ is bijective (hence surjective) and universally injective by Lemma 29.10.2. $\square$