The Stacks project

70.8 Modifications and alterations

Using our notion of an integral algebraic space we can define a modification as follows.

Definition 70.8.1. Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. A modification of $X$ is a birational proper morphism $f : X' \to X$ of algebraic spaces over $S$ with $X'$ integral.

For birational morphisms of algebraic spaces, see Decent Spaces, Definition 66.22.1.

Lemma 70.8.2. Let $f : X' \to X$ be a modification as in Definition 70.8.1. There exists a nonempty open $U \subset X$ such that $f^{-1}(U) \to U$ is an isomorphism.

Proof. By Lemma 70.5.1 there exists a nonempty $U \subset X$ such that $f^{-1}(U) \to U$ is finite. By generic flatness (Morphisms of Spaces, Proposition 65.32.1) we may assume $f^{-1}(U) \to U$ is flat and of finite presentation. So $f^{-1}(U) \to U$ is finite locally free (Morphisms of Spaces, Lemma 65.46.6). Since $f$ is birational, the degree of $X'$ over $X$ is $1$. Hence $f^{-1}(U) \to U$ is finite locally free of degree $1$, in other words it is an isomorphism. $\square$

Definition 70.8.3. Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. An alteration of $X$ is a proper dominant morphism $f : Y \to X$ of algebraic spaces over $S$ with $Y$ integral such that $f^{-1}(U) \to U$ is finite for some nonempty open $U \subset X$.

If $f : Y \to X$ is a dominant and proper morphism between integral algebraic spaces, then it is an alteration as soon as the induced extension of residue fields in generic points is finite. Here is the precise statement.

Lemma 70.8.4. Let $S$ be a scheme. Let $f : X \to Y$ be a proper dominant morphism of integral algebraic spaces over $S$. Then $f$ is an alteration if and only if any of the equivalent conditions (1) – (6) of Lemma 70.5.1 hold.

Proof. Immediate consequence of the lemma referenced in the statement. $\square$

Lemma 70.8.5. Let $S$ be a scheme. Let $f : X \to Y$ be a proper surjective morphism of algebraic spaces over $S$. Assume $Y$ is integral. Then there exists an integral closed subspace $X' \subset X$ such that $f' = f|_{X'} : X' \to Y$ is an alteration.

Proof. Let $V \subset Y$ be a nonempty open affine (Decent Spaces, Theorem 66.10.2). Let $\eta \in V$ be the generic point. Then $X_\eta $ is a nonempty proper algebraic space over $\eta $. Choose a closed point $x \in |X_\eta |$ (exists because $|X_\eta |$ is a quasi-compact, sober topological space, see Decent Spaces, Proposition 66.12.4 and Topology, Lemma 5.12.8.) Let $X'$ be the reduced induced closed subspace structure on $\overline{\{ x\} } \subset |X|$ (Properties of Spaces, Definition 64.12.5. Then $f' : X' \to Y$ is surjective as the image contains $\eta $. Also $f'$ is proper as a composition of a closed immersion and a proper morphism. Finally, the fibre $X'_\eta $ has a single point; to see this use Decent Spaces, Lemma 66.18.6 for both $X \to Y$ and $X' \to Y$ and the point $\eta $. Since $Y$ is decent and $X' \to Y$ is separated we see that $X'$ is decent (Decent Spaces, Lemmas 66.17.2 and 66.17.5). Thus $f'$ is an alteration by Lemma 70.8.4. $\square$


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