The Stacks project

Lemma 71.6.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y$ be two closed subspaces of $X$ with ideal sheaves $\mathcal{I}$ and $\mathcal{J}$. If $\mathcal{I}\mathcal{J}$ defines an effective Cartier divisor $D \subset X$, then $Z$ and $Y$ are effective Cartier divisors and $D = Z + Y$.

Proof. By Lemma 71.6.2 this reduces to the case of schemes which is Divisors, Lemma 31.13.9. $\square$

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