The Stacks project

69.8 Modifications and alterations

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

Definition 69.8.1. 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 65.22.1.

Lemma 69.8.2. Let $f : X' \to X$ be a modification as in Definition 69.8.1. There exists a nonempty open $U \subset X$ such that $f^{-1}(U) \to U$ is an isomorphism.

Proof. By Lemma 69.5.1 there exists a nonempty $U \subset X$ such that $f^{-1}(U) \to U$ is finite. By generic flatness (Morphisms of Spaces, Proposition 64.32.1) 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 64.46.6). 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 69.8.3. 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 69.8.4. 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 69.5.1 hold.

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

Lemma 69.8.5. 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 65.10.2). 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 65.12.4 and Topology, Lemma 5.12.8.) Let $X'$ be the reduced induced closed subspace structure on $\overline{\{ x\} } \subset |X|$ (Properties of Spaces, Definition 63.12.6. 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 65.18.6 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 65.17.2 and 65.17.5). Thus $f'$ is an alteration by Lemma 69.8.4. $\square$

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