Lemma 71.6.2. 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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