Lemma 70.18.7. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $Z \subset B$ be a closed subspace. Let $b : B' \to B$ be the blowing up with center $Z$. Let $Z' \subset B'$ be a closed subspace. Let $B'' \to B'$ be the blowing up with center $Z'$. Let $Y \subset B$ be a closed subscheme such that $|Y| = |Z| \cup |b|(|Z'|)$ and the composition $B'' \to B$ is isomorphic to the blowing up of $B$ in $Y$. In this situation, given any scheme $X$ over $B$ and $\mathcal{F} \in \mathit{QCoh}(\mathcal{O}_ X)$ we have

1. the strict transform of $\mathcal{F}$ with respect to the blowing up of $B$ in $Y$ is equal to the strict transform with respect to the blowup $B'' \to B'$ in $Z'$ of the strict transform of $\mathcal{F}$ with respect to the blowup $B' \to B$ of $B$ in $Z$, and

2. the strict transform of $X$ with respect to the blowing up of $B$ in $Y$ is equal to the strict transform with respect to the blowup $B'' \to B'$ in $Z'$ of the strict transform of $X$ with respect to the blowup $B' \to B$ of $B$ in $Z$.

Proof. Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 31.33.6) by étale localization (Lemma 70.18.2). $\square$

There are also:

• 2 comment(s) on Section 70.18: Strict transform

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).