The Stacks project

Proof. Part (1) follows from Decent Spaces, Lemma 68.22.5 (and the fact that the Noetherian algebraic spaces $X$ and $Y$ are quasi-separated and hence decent). Part (2) follows from part (3) and Lemma 72.3.2 (and the fact that finite morphisms have finite fibres). Let $x \in |X|$ be as in (3). By Cohomology of Spaces, Lemma 69.23.2 (which applies by Decent Spaces, Lemma 68.18.10) we may assume $f$ is finite. Choose an affine scheme $X'$ and an étale morphism $X' \to X$ and a point $x' \in X$ mapping to $x$. It suffices to show there exists an open neighbourhood $U'$ of $x' \in X'$ such that $Y \times _ X X' \to X'$ is an isomorphism over $U'$ (namely, then $U$ contains the image of $U'$ in $X$, see Spaces, Lemma 65.5.6). Then $Y \times _ X X' \to X$ is a finite birational (Decent Spaces, Lemma 68.22.6) morphism. Since a finite morphism is affine we reduce to the case of a finite birational morphism of Noetherian affine schemes $Y \to X$ and $x \in X$ such that $\mathcal{O}_{X, x}$ is a normal domain. This is treated in Varieties, Lemma 33.17.3. $\square$