The Stacks project

Lemma 31.33.4. Let $S$ be a scheme. Let $Z \subset S$ be a closed subscheme. Let $b : S' \to S$ be the blowing up of $Z$ in $S$. Let $g : X \to Y$ be an affine morphism of schemes over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $g' : X \times _ S S' \to Y \times _ S S'$ 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. Observe that $g'_*\text{pr}_ X^*\mathcal{F} = \text{pr}_ Y^*g_*\mathcal{F}$ by Cohomology of Schemes, Lemma 30.5.1. Let $\mathcal{K} \subset \text{pr}_ X^*\mathcal{F}$ be the subsheaf of sections supported in the inverse image of $Z$ in $X \times _ S S'$. By Properties, Lemma 28.24.7 the pushforward $g'_*\mathcal{K}$ is the subsheaf of sections of $\text{pr}_ Y^*g_*\mathcal{F}$ supported in the inverse image of $Z$ in $Y \times _ S S'$. As $g'$ is affine (Morphisms, Lemma 29.11.8) we see that $g'_*$ is exact, hence we conclude. $\square$

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