Definition 71.17.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals, and let $Z \subset X$ be the closed subspace corresponding to $\mathcal{I}$ (Morphisms of Spaces, Lemma 67.13.1). The *blowing up of $X$ along $Z$*, or the *blowing up of $X$ in the ideal sheaf $\mathcal{I}$* is the morphism

The *exceptional divisor* of the blowup is the inverse image $b^{-1}(Z)$. Sometimes $Z$ is called the *center* of the blowup.

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