Lemma 31.33.6. Let $S$ be a scheme. Let $Z \subset S$ be a closed subscheme. Let $b : S' \to S$ be the blowing up with center $Z$. Let $Z' \subset S'$ be a closed subscheme. Let $S'' \to S'$ be the blowing up with center $Z'$. Let $Y \subset S$ be a closed subscheme such that $Y = Z \cup b(Z')$ set theoretically and the composition $S'' \to S$ is isomorphic to the blowing up of $S$ in $Y$. In this situation, given any scheme $X$ over $S$ 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 $S$ in $Y$ is equal to the strict transform with respect to the blowup $S'' \to S'$ in $Z'$ of the strict transform of $\mathcal{F}$ with respect to the blowup $S' \to S$ of $S$ in $Z$, and

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

Proof. Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$ with respect to the blowup $S' \to S$ of $S$ in $Z$. Let $\mathcal{F}''$ be the strict transform of $\mathcal{F}'$ with respect to the blowup $S'' \to S'$ of $S'$ in $Z'$. Let $\mathcal{G}$ be the strict transform of $\mathcal{F}$ with respect to the blowup $S'' \to S$ of $S$ in $Y$. We also label the morphisms

$\xymatrix{ X \times _ S S'' \ar[r]_ q \ar[d]^{f''} & X \times _ S S' \ar[r]_ p \ar[d]^{f'} & X \ar[d]^ f \\ S'' \ar[r] & S' \ar[r] & S }$

By definition there is a surjection $p^*\mathcal{F} \to \mathcal{F}'$ and a surjection $q^*\mathcal{F}' \to \mathcal{F}''$ which combine by right exactness of $q^*$ to a surjection $(p \circ q)^*\mathcal{F} \to \mathcal{F}''$. Also we have the surjection $(p \circ q)^*\mathcal{F} \to \mathcal{G}$. Thus it suffices to prove that these two surjections have the same kernel.

The kernel of the surjection $p^*\mathcal{F} \to \mathcal{F}'$ is supported on $(f \circ p)^{-1}Z$, so this map is an isomorphism at points in the complement. Hence the kernel of $q^*p^*\mathcal{F} \to q^*\mathcal{F}'$ is supported on $(f \circ p \circ q)^{-1}Z$. The kernel of $q^*\mathcal{F}' \to \mathcal{F}''$ is supported on $(f' \circ q)^{-1}Z'$. Combined we see that the kernel of $(p \circ q)^*\mathcal{F} \to \mathcal{F}''$ is supported on $(f \circ p \circ q)^{-1}Z \cup (f' \circ q)^{-1}Z' = (f \circ p \circ q)^{-1}Y$. By construction of $\mathcal{G}$ we see that we obtain a factorization $(p \circ q)^*\mathcal{F} \to \mathcal{F}'' \to \mathcal{G}$. To finish the proof it suffices to show that $\mathcal{F}''$ has no nonzero (local) sections supported on $(f \circ p \circ q)^{-1}(Y) = (f \circ p \circ q)^{-1}Z \cup (f' \circ q)^{-1}Z'$. This follows from Lemma 31.33.5 applied to $\mathcal{F}'$ on $X \times _ S S'$ over $S'$, the closed subscheme $Z'$ and the effective Cartier divisor $b^{-1}Z$. $\square$

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