The Stacks project

Lemma 70.18.3. In the situation of Definition 70.18.1.

  1. The strict transform $X'$ of $X$ is the blowup of $X$ in the closed subspace $f^{-1}Z$ of $X$.

  2. For a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the strict transform $\mathcal{F}'$ is canonically isomorphic to the pushforward along $X' \to X \times _ B B'$ of the strict transform of $\mathcal{F}$ relative to the blowing up $X' \to X$.

Proof. Let $X'' \to X$ be the blowup of $X$ in $f^{-1}Z$. By the universal property of blowing up (Lemma 70.17.5) there exists a commutative diagram

\[ \xymatrix{ X'' \ar[r] \ar[d] & X \ar[d] \\ B' \ar[r] & B } \]

whence a morphism $i : X'' \to X \times _ B B'$. The first assertion of the lemma is that $i$ is a closed immersion with image $X'$. The second assertion of the lemma is that $\mathcal{F}' = i_*\mathcal{F}''$ where $\mathcal{F}''$ is the strict transform of $\mathcal{F}$ with respect to the blowing up $X'' \to X$. We can check these assertions ├ętale locally on $X$, hence we reduce to the case of schemes (Divisors, Lemma 31.33.2). Some details omitted. $\square$


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