The Stacks project

Lemma 70.17.11. 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. Let $b : X' \to X$ be the blowing up of $X$ in the ideal sheaf $\mathcal{I}$. If $\mathcal{I}$ is of finite type, then $b : X' \to X$ is a proper morphism.

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective ├ętale morphism. As blowing up commutes with flat base change (Lemma 70.17.3) we can prove each of these statements after base change to $U$ (see Morphisms of Spaces, Lemma 66.40.2). This reduces us to the case of schemes. In this case the morphism $b$ is projective by Divisors, Lemma 31.32.13 hence proper by Morphisms, Lemma 29.43.5. $\square$


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