Lemma 76.40.3 (088S)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.40: Chow's lemma
Lemma 76.40.3 (cite)

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Lemma 76.40.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ . Assume $f$ separated, of finite type, and $Y$ Noetherian. Then there exists a dense open subspace $U \subset X$ and a commutative diagram

$\xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & Y & \mathbf{P}^ n_ Y \ar[l] }$

where the arrows with source $U$ are open immersions, $X' \to X$ is a $U$ -admissible blowup, $X' \to Z'$ is an open immersion, $Z' \to Y$ is a proper and representable morphism of algebraic spaces. More precisely, $Z' \to Z$ is a $U$ -admissible blowup and $Z \to \mathbf{P}^ n_ Y$ is a closed immersion.

Proof. By Limits of Spaces, Lemma 70.13.3 there exists a dense open subspace $U \subset X$ and an immersion $U \to \mathbf{A}^ n_ Y$ over $Y$ . Composing with the open immersion $\mathbf{A}^ n_ Y \to \mathbf{P}^ n_ Y$ we obtain a situation as in Lemma 76.40.2 and the result follows. $\square$

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