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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