The Stacks project

Lemma 72.3.3. 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 66.7.6), 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 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 ├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 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. $\square$

Comments (0)

There are also:

  • 1 comment(s) on Section 72.3: Generically finite morphisms

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BBQ. Beware of the difference between the letter 'O' and the digit '0'.