The Stacks project

Lemma 18.40.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Consider the statements

  1. (18.40.2.1) is an isomorphism, and

  2. for every point $p$ of $\mathcal{C}$ the stalk $\mathcal{O}_ p$ is not the zero ring.

We always have (1) $\Rightarrow $ (2) and if $\mathcal{C}$ has enough points then (1) $\Leftrightarrow $ (2).

Proof. Omitted. $\square$

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