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The Stacks project

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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