The Stacks project

Proof. The question is étale local on $Y$, see Lemma 67.48.4. Thus we may assume $Y = \mathop{\mathrm{Spec}}(R)$ is affine. Then $R$ is a Noetherian Nagata ring and we have to show that the integral closure of $R$ in $\Gamma (X, \mathcal{O}_ X)$ is finite over $R$. Since $f$ is quasi-compact we see that $X$ is quasi-compact. Choose an affine scheme $U$ and a surjective étale morphism $U \to X$ (Properties of Spaces, Lemma 66.6.3). Then $\Gamma (X, \mathcal{O}_ X) \subset \Gamma (U, \mathcal{O}_ X)$. Since $R$ is Noetherian it suffices to show that the integral closure of $R$ in $\Gamma (U, \mathcal{O}_ U)$ is finite over $R$. As $U \to Y$ is of finite type this follows from Morphisms, Lemma 29.53.15. $\square$