The Stacks project

Lemma 71.17.7. 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. If $X$ is reduced, then the blowup $X'$ of $X$ in $\mathcal{I}$ is reduced.

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 71.17.3) we can prove each of these statements after base change to $U$. This reduces us to the case of schemes. In this case the result is Divisors, Lemma 31.32.8. $\square$

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