Blowing up commutes with flat base change.

Lemma 31.32.3. Let $X_1 \to X_2$ be a flat morphism of schemes. Let $Z_2 \subset X_2$ be a closed subscheme. Let $Z_1$ be the inverse image of $Z_2$ in $X_1$. Let $X'_ i$ be the blowup of $Z_ i$ in $X_ i$. Then there exists a cartesian diagram

$\xymatrix{ X_1' \ar[r] \ar[d] & X_2' \ar[d] \\ X_1 \ar[r] & X_2 }$

of schemes.

Proof. Let $\mathcal{I}_2$ be the ideal sheaf of $Z_2$ in $X_2$. Denote $g : X_1 \to X_2$ the given morphism. Then the ideal sheaf $\mathcal{I}_1$ of $Z_1$ is the image of $g^*\mathcal{I}_2 \to \mathcal{O}_{X_1}$ (by definition of the inverse image, see Schemes, Definition 26.17.7). By Constructions, Lemma 27.16.10 we see that $X_1 \times _{X_2} X_2'$ is the relative Proj of $\bigoplus _{n \geq 0} g^*\mathcal{I}_2^ n$. Because $g$ is flat the map $g^*\mathcal{I}_2^ n \to \mathcal{O}_{X_1}$ is injective with image $\mathcal{I}_1^ n$. Thus we see that $X_1 \times _{X_2} X_2' = X_1'$. $\square$

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