The Stacks project

Lemma 70.3.3. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $f : Y \to X$ be a birational proper morphism of algebraic spaces with $Y$ reduced. Let $U \subset X$ be the maximal open over which $f$ is an isomorphism. Then $U$ contains

  1. every point of codimension $0$ in $X$,

  2. every $x \in |X|$ of codimension $1$ on $X$ such that the local ring of $X$ at $x$ is normal (Properties of Spaces, Remark 64.7.6), and

  3. every $x \in |X|$ such that the fibre of $|Y| \to |X|$ over $x$ is finite and such that the local ring of $X$ at $x$ is normal.

Proof. Part (1) follows from Decent Spaces, Lemma 66.22.5 (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 70.3.2 (and the fact that finite morphisms have finite fibres). Let $x \in |X|$ be as in (3). By Cohomology of Spaces, Lemma 67.22.2 (which applies by Decent Spaces, Lemma 66.18.10) we may assume $f$ is finite. Choose an affine scheme $X'$ and an ├ętale 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 63.5.6). Then $Y \times _ X X' \to X$ is a finite birational (Decent Spaces, Lemma 66.22.6) 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. $\square$


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