Lemma 31.32.12. Let X be a scheme. Let \mathcal{I}, \mathcal{J} \subset \mathcal{O}_ X be quasi-coherent sheaves of ideals. Let b : X' \to X be the blowing up of X in \mathcal{I}. Let b' : X'' \to X' be the blowing up of X' in b^{-1}\mathcal{J} \mathcal{O}_{X'}. Then X'' \to X is canonically isomorphic to the blowing up of X in \mathcal{I}\mathcal{J}.
Proof. Let E \subset X' be the exceptional divisor of b which is an effective Cartier divisor by Lemma 31.32.4. Then (b')^{-1}E is an effective Cartier divisor on X'' by Lemma 31.32.11. Let E' \subset X'' be the exceptional divisor of b' (also an effective Cartier divisor). Consider the effective Cartier divisor E'' = E' + (b')^{-1}E. By construction the ideal of E'' is (b \circ b')^{-1}\mathcal{I} (b \circ b')^{-1}\mathcal{J} \mathcal{O}_{X''}. Hence according to Lemma 31.32.5 there is a canonical morphism from X'' to the blowup c : Y \to X of X in \mathcal{I}\mathcal{J}. Conversely, as \mathcal{I}\mathcal{J} pulls back to an invertible ideal we see that c^{-1}\mathcal{I}\mathcal{O}_ Y defines an effective Cartier divisor, see Lemma 31.13.9. Thus a morphism c' : Y \to X' over X by Lemma 31.32.5. Then (c')^{-1}b^{-1}\mathcal{J}\mathcal{O}_ Y = c^{-1}\mathcal{J}\mathcal{O}_ Y which also defines an effective Cartier divisor. Thus a morphism c'' : Y \to X'' over X'. We omit the verification that this morphism is inverse to the morphism X'' \to Y constructed earlier. \square
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