The Stacks project

Lemma 71.7.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f \in \Gamma (X, \mathcal{O}_ X)$. The following are equivalent:

  1. $f$ is a regular section, and

  2. for any $x \in X$ the image $f \in \mathcal{O}_{X, \overline{x}}$ is not a zerodivisor.

  3. for any affine $U = \mathop{\mathrm{Spec}}(A)$ ├ętale over $X$ the restriction $f|_ U$ is a nonzerodivisor of $A$, and

  4. there exists a scheme $U$ and a surjective ├ętale morphism $U \to X$ such that $f|_ U$ is a regular section of $\mathcal{O}_ U$.

Proof. Omitted. $\square$

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