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 ├ętale localization (Lemma 71.18.2). $\square$


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