Lemma 18.36.2 (04EP)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.36: Stalks of modules
Lemma 18.36.2 (cite)

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

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