The Stacks project

Lemma 59.18.2. Notation and assumptions as in Definition 59.18.1. The functor $\check{\mathcal{C}}^\bullet (\mathcal{U}, -)$ is exact on the category $\textit{PAb}(\mathcal{C})$.

Proof. This follows at once from the definition of a short exact sequence of presheaves. Namely, as the category of abelian presheaves is the category of functors on some category with values in $\textit{Ab}$, it is automatically an abelian category: a sequence $\mathcal{F}_1\to \mathcal{F}_2\to \mathcal{F}_3$ is exact in $\textit{PAb}$ if and only if for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, the sequence $\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$ is exact in $\textit{Ab}$. So the complex above is merely a product of short exact sequences in each degree. See also Cohomology on Sites, Lemma 21.9.1. $\square$

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