The Stacks project

Lemma 18.36.2. Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$.

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

  2. The stalk functor $\textit{PAb}(\mathcal{C}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact.

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

Proof. This is formal from the results of Lemma 18.36.1 and the construction of the stalk functor above. $\square$

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