The Stacks project

Lemma 70.8.3. Let $S$ be a scheme. Let $X$ be a regular Noetherian separated algebraic space over $S$. Let $U \subset X$ be a dense affine open. Then there exists an effective Cartier divisor $D \subset X$ with $U = X \setminus D$.

Proof. We claim that the reduced induced algebraic space structure $D$ on $X \setminus U$ (Properties of Spaces, Definition 65.12.5) is the desired effective Cartier divisor. The construction of $D$ commutes with ├ętale localization, see proof of Properties of Spaces, Lemma 65.12.3. Let $X' \to X$ be a surjective ├ętale morphism with $X'$ affine. Since $X$ is separated, we see that $U' = X' \times _ X U$ is affine. Since $|X'| \to |X|$ is open, we see that $U'$ is dense in $X'$. Since $D' = X' \times _ X D$ is the reduced induced scheme structure on $X' \setminus U'$, we conclude that $D'$ is an effective Cartier divisor by Divisors, Lemma 31.16.6 and its proof. This is what we had to show. $\square$

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