## 18.3 Abelian sheaves

Let $\mathcal{C}$ be a site. The category of abelian sheaves on $\mathcal{C}$ is denoted $\textit{Ab}(\mathcal{C})$. It is the full subcategory of $\textit{PAb}(\mathcal{C})$ consisting of those abelian presheaves whose underlying presheaves of sets are sheaves. Properties ($\alpha$) – ($\zeta$) of Sites, Section 7.44 hold, see Sites, Proposition 7.44.3. In particular the inclusion functor $\textit{Ab}(\mathcal{C}) \to \textit{PAb}(\mathcal{C})$ has a left adjoint, namely the sheafification functor $\mathcal{G} \mapsto \mathcal{G}^\#$.

We suggest the reader prove the lemma on a piece of scratch paper rather than reading the proof.

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 $\mathop{\mathrm{Im}}(\mathcal{F} \to \mathcal{G}) = \mathop{\mathrm{Ker}}(\mathcal{G} \to \mathcal{H})$. $\square$

Another way to say part (6) of the lemma is that a sequence of abelian sheaves

$\mathcal{F}_1 \longrightarrow \mathcal{F}_2 \longrightarrow \mathcal{F}_3$

is exact if and only if the sheafification of $U \mapsto \mathop{\mathrm{Im}}(\mathcal{F}_1(U) \to \mathcal{F}_2(U))$ is equal to the kernel of $\mathcal{F}_2 \to \mathcal{F}_3$.

Lemma 18.3.2. Let $\mathcal{C}$ be a site.

1. All limits and colimits exist in $\textit{Ab}(\mathcal{C})$.

2. Limits are the same as the corresponding limits of abelian presheaves over $\mathcal{C}$ (i.e., commute with taking sections over objects of $\mathcal{C}$).

3. Finite direct sums are the same as the corresponding finite direct sums in the category of abelian pre-sheaves over $\mathcal{C}$.

4. A colimit is the sheafification of the corresponding colimit in the category of abelian presheaves.

5. Filtered colimits are exact.

Proof. By Lemma 18.2.1 limits and colimits of abelian presheaves exist, and are described by taking limits and colimits on the level of sections over objects.

Let $\mathcal{I} \to \textit{Ab}(\mathcal{C})$, $i \mapsto \mathcal{F}_ i$ be a diagram. Let $\mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i$ be the limit of the diagram as an abelian presheaf. By Sites, Lemma 7.10.1 this is an abelian sheaf. Then it is quite easy to see that $\mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i$ is the limit of the diagram in $\textit{Ab}(\mathcal{C})$. This proves limits exist and (2) holds.

By Categories, Lemma 4.24.5, and because sheafification is left adjoint to the inclusion functor we see that $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}$ exists and is the sheafification of the colimit in $\textit{PAb}(\mathcal{C})$. This proves colimits exist and (4) holds.

Finite direct sums are the same thing as finite products in any abelian category. Hence (3) follows from (2).

Proof of (5). The statement means that given a system $0 \to \mathcal{F}_ i \to \mathcal{G}_ i \to \mathcal{H}_ i \to 0$ of exact sequences of abelian sheaves over a directed set $I$ the sequence $0 \to \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i \to 0$ is exact as well. A formal argument using Homology, Lemma 12.5.8 and the definition of colimits shows that the sequence $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i \to 0$ is exact. Note that $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ is the sheafification of the map of presheaf colimits which is injective as each of the maps $\mathcal{F}_ i \to \mathcal{G}_ i$ is injective. Since sheafification is exact we conclude. $\square$

Comment #1799 by Keenan Kidwell on

Is the remark after the proof of 03CN really written as intended? It doesn't seem to make sense. Shouldn't it be the sheafification of $U\mapsto\mathrm{im}(\mathcal{F}_1(U)\to\mathcal{F}_2(U))$? But in that case isn't it somewhat tautological? Perhaps I'm missing the point.

Comment #1825 by on

OK, I fixed this and yes it is tautological.

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