The Stacks project

Lemma 31.14.7. Let $X$ be a locally ringed space. 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 $f \in \mathcal{O}_{X, x}$ is a nonzerodivisor.

If $X$ is a scheme these are also equivalent to

  1. for any affine open $\mathop{\mathrm{Spec}}(A) = U \subset X$ the image $f \in A$ is a nonzerodivisor,

  2. there exists an affine open covering $X = \bigcup \mathop{\mathrm{Spec}}(A_ i)$ such that the image of $f$ in $A_ i$ is a nonzerodivisor for all $i$.

Proof. Omitted. $\square$

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