Lemma 71.18.5 (0866)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 71: Divisors on Algebraic Spaces
Section 71.18: Strict transform
Lemma 71.18.5 (cite)

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Lemma 71.18.5. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$ . Let $Z \subset B$ be a closed subspace. Let $b : B' \to B$ be the blowing up of $Z$ in $B$ . Let $g : X \to Y$ be an affine morphism of spaces over $B$ . Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$ . Let $g' : X \times _ B B' \to Y \times _ B B'$ be the base change of $g$ . Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$ relative to $b$ . Then $g'_*\mathcal{F}'$ is the strict transform of $g_*\mathcal{F}$ .

Proof. Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 31.33.4) by étale localization (Lemma 71.18.2). $\square$

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