Lemma 70.18.2 (Étale localization and strict transform). In the situation of Definition 70.18.1. Let

be a commutative diagram of morphisms with $U$ and $V$ schemes and étale horizontal arrows. Let $V' \to V$ be the blowup of $V$ in $Z \times _ B V$. Then

$V' = V \times _ B B'$ and the maps $V' \to B'$ and $U \times _ V V' \to X \times _ B B'$ are étale,

the strict transform $U'$ of $U$ relative to $V' \to V$ is equal to $X' \times _ X U$ where $X'$ is the strict transform of $X$ relative to $B' \to B$, and

for a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the restriction of the strict transform $\mathcal{F}'$ to $U \times _ V V'$ is the strict transform of $\mathcal{F}|_ U$ relative to $V' \to V$.

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