Lemma 18.35.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $p$ be a point of $\mathcal{C}$.

1. The functor $\textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ p)$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact.

2. The stalk functor $\textit{PMod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ p)$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact.

3. For $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{PMod}(\mathcal{O}))$ we have $\mathcal{F}_ p = \mathcal{F}^\# _ p$.

Proof. This is formal from the results of Lemma 18.35.2, the construction of the stalk functor above, and Lemma 18.14.1. $\square$

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