Lemma 70.6.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be a closed subspace. The following are equivalent:

The subspace $D$ is an effective Cartier divisor on $X$.

For some scheme $U$ and surjective étale morphism $U \to X$ the inverse image $D \times _ X U$ is an effective Cartier divisor on $U$.

For every scheme $U$ and every étale morphism $U \to X$ the inverse image $D \times _ X U$ is an effective Cartier divisor on $U$.

For every $x \in |D|$ there exists an étale morphism $(U, u) \to (X, x)$ of pointed algebraic spaces such that $U = \mathop{\mathrm{Spec}}(A)$ and $D \times _ X U = \mathop{\mathrm{Spec}}(A/(f))$ with $f \in A$ not a zerodivisor.

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