The Stacks project

Lemma 70.18.6. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $Z \subset B$ be a closed subspace. Let $D \subset B$ be an effective Cartier divisor. Let $Z' \subset B$ be the closed subspace cut out by the product of the ideal sheaves of $Z$ and $D$. Let $B' \to B$ be the blowup of $B$ in $Z$.

  1. The blowup of $B$ in $Z'$ is isomorphic to $B' \to B$.

  2. Let $f : X \to B$ be a morphism of algebraic spaces and let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If the subsheaf of $\mathcal{F}$ of sections supported on $|f^{-1}D|$ is zero, then the strict transform of $\mathcal{F}$ relative to the blowing up in $Z$ agrees with the strict transform of $\mathcal{F}$ relative to the blowing up of $B$ in $Z'$.

Proof. Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 31.33.5) by ├ętale localization (Lemma 70.18.2). $\square$


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