The Stacks project

70.3 Generically finite morphisms

This section continues the discussion in Decent Spaces, Section 66.21 and the analogue for morphisms of algebraic spaces of Varieties, Section 33.17.

Lemma 70.3.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is locally Noetherian. Let $y \in |Y|$ be a point of codimension $\leq 1$ on $Y$. Let $X^0 \subset |X|$ be the set of points of codimension $0$ on $X$. Assume in addition one of the following conditions is satisfied

  1. for every $x \in X^0$ the transcendence degree of $x/f(x)$ is $0$,

  2. for every $x \in X^0$ with $f(x) \leadsto y$ the transcendence degree of $x/f(x)$ is $0$,

  3. $f$ is quasi-finite at every $x \in X^0$,

  4. $f$ is quasi-finite at a dense set of points of $|X|$,

  5. add more here.

Then $f$ is quasi-finite at every point of $X$ lying over $y$.

Proof. We want to reduce the proof to the case of schemes. To do this we choose a commutative diagram

\[ \xymatrix{ U \ar[r] \ar[d]_ g & X \ar[d]^ f \\ V \ar[r] & Y } \]

where $U$, $V$ are schemes and where the horizontal arrows are ├ętale and surjective. Pick $v \in V$ mapping to $y$. Observe that $V$ is locally Noetherian and that $\dim (\mathcal{O}_{V, v}) \leq 1$ (see Properties of Spaces, Definitions 64.10.2 and Remark 64.7.3). The fibre $U_ v$ of $U \to V$ over $v$ surjects onto $f^{-1}(\{ y\} ) \subset |X|$. The inverse image of $X^0$ in $U$ is exactly the set of generic points of irreducible components of $U$ (Properties of Spaces, Lemma 64.11.1). If $\eta \in U$ is such a point with image $x \in X^0$, then the transcendence degree of $x / f(x)$ is the transcendence degree of $\kappa (\eta )$ over $\kappa (g(\eta ))$ (Morphisms of Spaces, Definition 65.33.1). Observe that $U \to V$ is quasi-finite at $u \in U$ if and only if $f$ is quasi-finite at the image of $u$ in $X$.

Case (1). Here case (1) of Varieties, Lemma 33.17.1 applies and we conclude that $U \to V$ is quasi-finite at all points of $U_ v$. Hence $f$ is quasi-finite at every point lying over $y$.

Case (2). Let $u \in U$ be a generic point of an irreducible component whose image in $V$ specializes to $v$. Then the image $x \in X^0$ of $u$ has the property that $f(x) \leadsto y$. Hence we see that case (2) of Varieties, Lemma 33.17.1 applies and we conclude as before.

Case (3) follows from case (3) of Varieties, Lemma 33.17.1.

In case (4), since $|U| \to |X|$ is open, we see that the set of points where $U \to V$ is quasi-finite is dense as well. Hence case (4) of Varieties, Lemma 33.17.1 applies. $\square$

Lemma 70.3.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper and $Y$ is locally Noetherian. Let $y \in Y$ be a point of codimension $\leq 1$ in $Y$. Let $X^0 \subset |X|$ be the set of points of codimension $0$ on $X$. Assume in addition one of the following conditions is satisfied

  1. for every $x \in X^0$ the transcendence degree of $x/f(x)$ is $0$,

  2. for every $x \in X^0$ with $f(x) \leadsto y$ the transcendence degree of $x/f(x)$ is $0$,

  3. $f$ is quasi-finite at every $x \in X^0$,

  4. $f$ is quasi-finite at a dense set of points of $|X|$,

  5. add more here.

Then there exists an open subspace $Y' \subset Y$ containing $y$ such that $Y' \times _ Y X \to Y'$ is finite.

Proof. By Lemma 70.3.1 the morphism $f$ is quasi-finite at every point lying over $y$. Let $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ be a geometric point lying over $y$. Then $|X_{\overline{y}}|$ is a discrete space (Decent Spaces, Lemma 66.18.10). Since $X_{\overline{y}}$ is quasi-compact as $f$ is proper we conclude that $|X_{\overline{y}}|$ is finite. Thus we can apply Cohomology of Spaces, Lemma 67.22.2 to conclude. $\square$

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. $\square$


Comments (1)

Comment #1776 by Weizhe Zheng on

I think in the first line it should be "Decent Spaces, Section 55.19". The two sections have the same \label{section-generically-finite}.


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 0ACY. Beware of the difference between the letter 'O' and the digit '0'.