Lemma 31.33.5. Let $S$ be a scheme. Let $Z \subset S$ be a closed subscheme. Let $D \subset S$ be an effective Cartier divisor. Let $Z' \subset S$ be the closed subscheme cut out by the product of the ideal sheaves of $Z$ and $D$. Let $S' \to S$ be the blowup of $S$ in $Z$.

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

Let $f : X \to S$ be a morphism of schemes and let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $\mathcal{F}$ has no nonzero local sections supported in $f^{-1}D$, 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 $S$ in $Z'$.

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