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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 étale 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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