The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

18.2 Abelian presheaves

Let $\mathcal{C}$ be a category. Abelian presheaves were introduced in Sites, Sections 7.2 and 7.7 and discussed a bit more in Sites, Section 7.44. We will follow the convention of this last reference, in that we think of an abelian presheaf as a presheaf of sets endowed with addition rules on all sets of sections compatible with the restriction mappings. Recall that the category of abelian presheaves on $\mathcal{C}$ is denoted $\textit{PAb}(\mathcal{C})$.

The category $\textit{PAb}(\mathcal{C})$ is abelian as defined in Homology, Definition 12.5.1. Given a map of presheaves $\varphi : \mathcal{G}_1 \to \mathcal{G}_2$ the kernel of $\varphi $ is the abelian presheaf $U \mapsto \mathop{\mathrm{Ker}}(\mathcal{G}_1(U) \to \mathcal{G}_2(U))$ and the cokernel of $\varphi $ is the presheaf $U \mapsto \mathop{\mathrm{Coker}}(\mathcal{G}_1(U) \to \mathcal{G}_2(U))$. Since the category of abelian groups is abelian it follows that $\mathop{\mathrm{Coim}}= \mathop{\mathrm{Im}}$ because this holds over each $U$. A sequence of abelian presheaves

\[ \mathcal{G}_1 \longrightarrow \mathcal{G}_2 \longrightarrow \mathcal{G}_3 \]

is exact if and only if $\mathcal{G}_1(U) \to \mathcal{G}_2(U) \to \mathcal{G}_3(U)$ is an exact sequence of abelian groups for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. We leave the verifications to the reader.

Lemma 18.2.1. Let $\mathcal{C}$ be a category.

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

  2. All limits and colimits commute with taking sections over objects of $\mathcal{C}$.

Proof. Let $\mathcal{I} \to \textit{PAb}(\mathcal{C})$, $i \mapsto \mathcal{F}_ i$ be a diagram. We can simply define abelian presheaves $L$ and $C$ by the rules

\[ L : U \longmapsto \mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i(U) \]

and

\[ C : U \longmapsto \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i(U). \]

It is clear that there are maps of abelian presheaves $L \to \mathcal{F}_ i$ and $\mathcal{F}_ i \to C$, by using the corresponding maps on groups of sections over each $U$. It is straightforward to check that $L$ and $C$ endowed with these maps are the limit and colimit of the diagram in $\textit{PAb}(\mathcal{C})$. This proves (1) and (2). Details omitted. $\square$


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