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*}

6.12 Stalks of abelian presheaves

We first deal with the case of abelian groups as a model for the general case.

Lemma 6.12.1. Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf of abelian groups on $X$. There exists a unique structure of an abelian group on $\mathcal{F}_ x$ such that for every $U \subset X$ open, $x\in U$ the map $\mathcal{F}(U) \to \mathcal{F}_ x$ is a group homomorphism. Moreover,

\[ \mathcal{F}_ x = \mathop{\mathrm{colim}}\nolimits _{x\in U} \mathcal{F}(U) \]

holds in the category of abelian groups.

Proof. We define addition of a pair of elements $(U, s)$ and $(V, t)$ as the pair $(U \cap V, s|_{U\cap V} + t|_{U \cap V})$. The rest is easy to check. $\square$

What is crucial in the proof above is that the partially ordered set of open neighbourhoods is a directed set (Categories, Definition 4.21.1). Namely, the coproduct of two abelian groups $A, B$ is the direct sum $A \oplus B$, whereas the coproduct in the category of sets is the disjoint union $A \amalg B$, showing that colimits in the category of abelian groups do not agree with colimits in the category of sets in general.


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