Lemma 18.3.1. Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of abelian sheaves on $\mathcal{C}$.

1. The category $\textit{Ab}(\mathcal{C})$ is an abelian category.

2. The kernel $\mathop{\mathrm{Ker}}(\varphi )$ of $\varphi$ is the same as the kernel of $\varphi$ as a morphism of presheaves.

3. The morphism $\varphi$ is injective (Homology, Definition 12.5.3) if and only if $\varphi$ is injective as a map of presheaves (Sites, Definition 7.3.1), if and only if $\varphi$ is injective as a map of sheaves (Sites, Definition 7.11.1).

4. The cokernel $\mathop{\mathrm{Coker}}(\varphi )$ of $\varphi$ is the sheafification of the cokernel of $\varphi$ as a morphism of presheaves.

5. The morphism $\varphi$ is surjective (Homology, Definition 12.5.3) if and only if $\varphi$ is surjective as a map of sheaves (Sites, Definition 7.11.1).

6. A complex of abelian sheaves

$\mathcal{F} \to \mathcal{G} \to \mathcal{H}$

is exact at $\mathcal{G}$ if and only if for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and all $s \in \mathcal{G}(U)$ mapping to zero in $\mathcal{H}(U)$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ in $\mathcal{C}$ such that each $s|_{U_ i}$ is in the image of $\mathcal{F}(U_ i) \to \mathcal{G}(U_ i)$.

Proof. We claim that Homology, Lemma 12.7.4 applies to the categories $\mathcal{A} = \textit{Ab}(\mathcal{C})$ and $\mathcal{B} = \textit{PAb}(\mathcal{C})$, and the functors $a : \mathcal{A} \to \mathcal{B}$ (inclusion), and $b : \mathcal{B} \to \mathcal{A}$ (sheafification). Let us check the assumptions of Homology, Lemma 12.7.4. Assumption (1) is that $\mathcal{A}$, $\mathcal{B}$ are additive categories, $a$, $b$ are additive functors, and $a$ is right adjoint to $b$. The first two statements are clear and adjointness is Sites, Section 7.44 ($\epsilon$). Assumption (2) says that $\textit{PAb}(\mathcal{C})$ is abelian which we saw in Section 18.2 and that sheafification is left exact, which is Sites, Section 7.44 ($\zeta$). The final assumption is that $ba \cong \text{id}_\mathcal {A}$ which is Sites, Section 7.44 ($\delta$). Hence Homology, Lemma 12.7.4 applies and we conclude that $\textit{Ab}(\mathcal{C})$ is abelian.

In the proof of Homology, Lemma 12.7.4 it is shown that $\mathop{\mathrm{Ker}}(\varphi )$ and $\mathop{\mathrm{Coker}}(\varphi )$ are equal to the sheafification of the kernel and cokernel of $\varphi$ as a morphism of abelian presheaves. This proves (4). Since the kernel is a equalizer (i.e., a limit) and since sheafification commutes with finite limits, we conclude that (2) holds.

Statement (2) implies (3). Statement (4) implies (5) by our description of sheafification. The characterization of exactness in (6) follows from (2) and (5), and the fact that the sequence is exact if and only if $\mathop{\mathrm{Im}}(\mathcal{F} \to \mathcal{G}) = \mathop{\mathrm{Ker}}(\mathcal{G} \to \mathcal{H})$. $\square$

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