Lemma 70.17.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a closed subspace. The blowing up $b : X' \to X$ of $Z$ in $X$ has the following properties:

1. $b|_{b^{-1}(X \setminus Z)} : b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism,

2. the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor on $X'$,

3. there is a canonical isomorphism $\mathcal{O}_{X'}(-1) = \mathcal{O}_{X'}(E)$

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 70.17.3) 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 31.32.4. $\square$

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