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*}

6.8 Abelian sheaves

Definition 6.8.1. Let $X$ be a topological space.

  1. An abelian sheaf on $X$ or sheaf of abelian groups on $X$ is an abelian presheaf on $X$ such that the underlying presheaf of sets is a sheaf.

  2. The category of sheaves of abelian groups is denoted $\textit{Ab}(X)$.

Let $X$ be a topological space. In the case of an abelian presheaf $\mathcal{F}$ the sheaf condition with regards to an open covering $U = \bigcup U_ i$ is often expressed by saying that the complex of abelian groups

\[ 0 \to \mathcal{F}(U) \to \prod \nolimits _ i \mathcal{F}(U_ i) \to \prod \nolimits _{(i_0, i_1)} \mathcal{F}(U_{i_0} \cap U_{i_1}) \]

is exact. The first map is the usual one, whereas the second maps the element $(s_ i)_{i \in I}$ to the element

\[ ( s_{i_0}|_{U_{i_0} \cap U_{i_1}} - s_{i_1}|_{U_{i_0} \cap U_{i_1}} )_{(i_0, i_1)} \in \prod \nolimits _{(i_0, i_1)} \mathcal{F}(U_{i_0} \cap U_{i_1}) \]

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