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The Stacks project

Lemma 71.18.4. In the situation of Definition 71.18.1.

  1. If X is flat over B at all points lying over Z, then the strict transform of X is equal to the base change X \times _ B B'.

  2. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. If \mathcal{F} is flat over B at all points lying over Z, then the strict transform \mathcal{F}' of \mathcal{F} is equal to the pullback \text{pr}_ X^*\mathcal{F}.

Proof. Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 31.33.3) by étale localization (Lemma 71.18.2). \square


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