Lemma 71.17.5 (Universal property blowing up). Let S be a scheme. Let X be an algebraic space over S. Let Z \subset X be a closed subspace. Let \mathcal{C} be the full subcategory of (\textit{Spaces}/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}.
Blow up a closed subset to make it Cartier.
Proof. We see that b : X' \to X is an object of \mathcal{C} according to Lemma 71.17.4. 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 Lemma 71.11.11. the triple (f : Y \to X, \mathcal{I}_ D, \psi ) defines a morphism Y \to X' over X. The restriction
is unique. The open Y \setminus D is scheme theoretically dense in Y according to Lemma 71.6.4. Thus the morphism Y \to X' is unique by Morphisms of Spaces, Lemma 67.17.8 (also b is separated by Lemma 71.11.6). \square
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