67.21 Generically finite morphisms

This section discusses for morphisms of algebraic spaces the material discussed in Morphisms, Section 29.51 and Varieties, Section 33.17 for morphisms of schemes.

Lemma 67.21.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is quasi-separated of finite type. Let $y \in |Y|$ be a point of codimension $0$ on $Y$. The following are equivalent:

1. the space $|X_ k|$ is finite where $\mathop{\mathrm{Spec}}(k) \to Y$ represents $y$,

2. $X \to Y$ is quasi-finite at all points of $|X|$ over $y$,

3. there exists an open subspace $Y' \subset Y$ with $y \in |Y'|$ such that $Y' \times _ Y X \to Y'$ is finite.

If $Y$ is decent these are also equivalent to

1. the set $f^{-1}(\{ y\} )$ is finite.

Proof. The equivalence of (1) and (2) follows from Lemma 67.18.10 (and the fact that a quasi-separated morphism is decent by Lemma 67.17.2).

Assume the equivalent conditions of (1) and (2). Choose an affine scheme $V$ and an étale morphism $V \to Y$ mapping a point $v \in V$ to $y$. Then $v$ is a generic point of an irreducible component of $V$ by Properties of Spaces, Lemma 65.11.1. Choose an affine scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Then $U \to V$ is of finite type. The morphism $U \to V$ is quasi-finite at every point lying over $v$ by (2). It follows that the fibre of $U \to V$ over $v$ is finite (Morphisms, Lemma 29.20.14). By Morphisms, Lemma 29.51.1 after shrinking $V$ we may assume that $U \to V$ is finite. Let

$R = U \times _{V \times _ Y X} U$

Since $f$ is quasi-separated, we see that $V \times _ Y X$ is quasi-separated and hence $R$ is a quasi-compact scheme. Moreover the morphisms $R \to V$ is quasi-finite as the composition of an étale morphism $R \to U$ and a finite morphism $U \to V$. Hence we may apply Morphisms, Lemma 29.51.1 once more and after shrinking $V$ we may assume that $R \to V$ is finite as well. This of course implies that the two projections $R \to V$ are finite étale. It follows that $V/R = V \times _ Y X$ is an affine scheme, see Groupoids, Proposition 39.23.9. By Morphisms, Lemma 29.41.9 we conclude that $V \times _ Y X \to V$ is proper and by Morphisms, Lemma 29.44.11 we conclude that $V \times _ Y X \to V$ is finite. Finally, we let $Y' \subset Y$ be the open subspace of $Y$ corresponding to the image of $|V| \to |Y|$. By Morphisms of Spaces, Lemma 66.45.3 we conclude that $Y' \times _ Y X \to Y'$ is finite as the base change to $V$ is finite and as $V \to Y'$ is a surjective étale morphism.

If $Y$ is decent and $f$ is quasi-separated, then we see that $X$ is decent too; use Lemmas 67.17.2 and 67.17.5. Hence Lemma 67.18.10 applies to show that (4) implies (1) and (2). On the other hand, we see that (2) implies (4) by Morphisms of Spaces, Lemma 66.27.9. $\square$

Lemma 67.21.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is quasi-separated and locally of finite type and $Y$ quasi-separated. Let $y \in |Y|$ be a point of codimension $0$ on $Y$. The following are equivalent:

1. the set $f^{-1}(\{ y\} )$ is finite,

2. the space $|X_ k|$ is finite where $\mathop{\mathrm{Spec}}(k) \to Y$ represents $y$,

3. there exist open subspaces $X' \subset X$ and $Y' \subset Y$ with $f(X') \subset Y'$, $y \in |Y'|$, and $f^{-1}(\{ y\} ) \subset |X'|$ such that $f|_{X'} : X' \to Y'$ is finite.

Proof. Since quasi-separated algebraic spaces are decent, the equivalence of (1) and (2) follows from Lemma 67.18.10. To prove that (1) and (2) imply (3) we may and do replace $Y$ by a quasi-compact open containing $y$. Since $f^{-1}(\{ y\} )$ is finite, we can find a quasi-compact open subspace of $X' \subset X$ containing the fibre. The restriction $f|_{X'} : X' \to Y$ is quasi-compact and quasi-separated by Morphisms of Spaces, Lemma 66.8.10 (this is where we use that $Y$ is quasi-separated). Applying Lemma 67.21.1 to $f|_{X'} : X' \to Y$ we see that (3) holds. We omit the proof that (3) implies (2). $\square$

Lemma 67.21.3. 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. Let $X^0 \subset |X|$, resp. $Y^0 \subset |Y|$ denote the set of codimension $0$ points of $X$, resp. $Y$. Let $y \in Y^0$. The following are equivalent

1. $f^{-1}(\{ y\} ) \subset X^0$,

2. $f$ is quasi-finite at all points lying over $y$,

3. $f$ is quasi-finite at all $x \in X^0$ lying over $y$.

Proof. Let $V$ be a scheme and let $V \to Y$ be a surjective étale morphism. Let $U$ be a scheme and let $U \to V \times _ Y X$ be a surjective étale morphism. Then $f$ is quasi-finite at the image $x$ of a point $u \in U$ if and only if $U \to V$ is quasi-finite at $u$. Moreover, $x \in X^0$ if and only if $u$ is the generic point of an irreducible component of $U$ (Properties of Spaces, Lemma 65.11.1). Thus the lemma reduces to the case of the morphism $U \to V$, i.e., to Morphisms, Lemma 29.51.4. $\square$

Lemma 67.21.4. 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. Let $X^0 \subset |X|$, resp. $Y^0 \subset |Y|$ denote the set of codimension $0$ points of $X$, resp. $Y$. Assume

1. $Y$ is decent,

2. $X^0$ and $Y^0$ are finite and $f^{-1}(Y^0) = X^0$,

3. either $f$ is quasi-compact or $f$ is separated.

Then there exists a dense open $V \subset Y$ such that $f^{-1}(V) \to V$ is finite.

Proof. By Lemmas 67.20.4 and 67.20.1 we may assume $Y$ is a scheme with finitely many irreducible components. Shrinking further we may assume $Y$ is an irreducible affine scheme with generic point $y$. Then the fibre of $f$ over $y$ is finite.

Assume $f$ is quasi-compact and $Y$ affine irreducible. Then $X$ is quasi-compact and we may choose an affine scheme $U$ and a surjective étale morphism $U \to X$. Then $U \to Y$ is of finite type and the fibre of $U \to Y$ over $y$ is the set $U^0$ of generic points of irreducible components of $U$ (Properties of Spaces, Lemma 65.11.1). Hence $U^0$ is finite (Morphisms, Lemma 29.20.14) and after shrinking $Y$ we may assume that $U \to Y$ is finite (Morphisms, Lemma 29.51.1). Next, consider $R = U \times _ X U$. Since the projection $s : R \to U$ is étale we see that $R^0 = s^{-1}(U^0)$ lies over $y$. Since $R \to U \times _ Y U$ is a monomorphism, we conclude that $R^0$ is finite as $U \times _ Y U \to Y$ is finite. And $R$ is separated (Properties of Spaces, Lemma 65.6.4). Thus we may shrink $Y$ once more to reach the situation where $R$ is finite over $Y$ (Morphisms, Lemma 29.51.5). In this case it follows that $X = U/R$ is finite over $Y$ by exactly the same arguments as given in the proof of Lemma 67.21.1 (or we can simply apply that lemma because it follows immediately that $X$ is quasi-separated as well).

Assume $f$ is separated and $Y$ affine irreducible. Choose $V \subset Y$ and $U \subset X$ as in Lemma 67.21.2. Since $f|_ U : U \to V$ is finite, we see that $U \subset f^{-1}(V)$ is closed as well as open (Morphisms of Spaces, Lemmas 66.40.6 and 66.45.9). Thus $f^{-1}(V) = U \amalg W$ for some open subspace $W$ of $X$. However, since $U$ contains all the codimension $0$ points of $X$ we conclude that $W = \emptyset$ (Properties of Spaces, Lemma 65.11.2) as desired. $\square$

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