Lemma 31.32.5 (Universal property blowing up). Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{C}$ be the full subcategory of $(\mathit{Sch}/X)$ consisting of $Y \to X$ such that the inverse image of $Z$ is an effective Cartier divisor on $Y$. Then the blowing up $b : X' \to X$ of $Z$ in $X$ is a final object of $\mathcal{C}$.

Proof. We see that $b : X' \to X$ is an object of $\mathcal{C}$ according to Lemma 31.32.4. Let $f : Y \to X$ be an object of $\mathcal{C}$. We have to show there exists a unique morphism $Y \to X'$ over $X$. Let $D = f^{-1}(Z)$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal sheaf of $Z$ and let $\mathcal{I}_ D$ be the ideal sheaf of $D$. Then $f^*\mathcal{I} \to \mathcal{I}_ D$ is a surjection to an invertible $\mathcal{O}_ Y$-module. This extends to a map $\psi : \bigoplus f^*\mathcal{I}^ d \to \bigoplus \mathcal{I}_ D^ d$ of graded $\mathcal{O}_ Y$-algebras. (We observe that $\mathcal{I}_ D^ d = \mathcal{I}_ D^{\otimes d}$ as $D$ is an effective Cartier divisor.) By the material in Constructions, Section 27.16 the triple $(1, f : Y \to X, \psi )$ defines a morphism $Y \to X'$ over $X$. The restriction

$Y \setminus D \longrightarrow X' \setminus b^{-1}(Z) = X \setminus Z$

is unique. The open $Y \setminus D$ is scheme theoretically dense in $Y$ according to Lemma 31.13.4. Thus the morphism $Y \to X'$ is unique by Morphisms, Lemma 29.7.10 (also $b$ is separated by Constructions, Lemma 27.16.9). $\square$

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