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*}

20.10 The Čech complex and Čech cohomology

Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ be an open covering, see Topology, Basic notion (13). As is customary we denote $U_{i_0\ldots i_ p} = U_{i_0} \cap \ldots \cap U_{i_ p}$ for the $(p + 1)$-fold intersection of members of $\mathcal{U}$. Let $\mathcal{F}$ be an abelian presheaf on $X$. Set

\[ \check{\mathcal{C}}^ p(\mathcal{U}, \mathcal{F}) = \prod \nolimits _{(i_0, \ldots , i_ p) \in I^{p + 1}} \mathcal{F}(U_{i_0\ldots i_ p}). \]

This is an abelian group. For $s \in \check{\mathcal{C}}^ p(\mathcal{U}, \mathcal{F})$ we denote $s_{i_0\ldots i_ p}$ its value in $\mathcal{F}(U_{i_0\ldots i_ p})$. Note that if $s \in \check{\mathcal{C}}^1(\mathcal{U}, \mathcal{F})$ and $i, j \in I$ then $s_{ij}$ and $s_{ji}$ are both elements of $\mathcal{F}(U_ i \cap U_ j)$ but there is no imposed relation between $s_{ij}$ and $s_{ji}$. In other words, we are not working with alternating cochains (these will be defined in Section 20.24). We define

\[ d : \check{\mathcal{C}}^ p(\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^{p + 1}(\mathcal{U}, \mathcal{F}) \]

by the formula

20.10.0.1
\begin{equation} \label{cohomology-equation-d-cech} d(s)_{i_0\ldots i_{p + 1}} = \sum \nolimits _{j = 0}^{p + 1} (-1)^ j s_{i_0\ldots \hat i_ j \ldots i_{p + 1}}|_{U_{i_0\ldots i_{p + 1}}} \end{equation}

It is straightforward to see that $d \circ d = 0$. In other words $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is a complex.

Definition 20.10.1. Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ be an open covering. Let $\mathcal{F}$ be an abelian presheaf on $X$. The complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is the Čech complex associated to $\mathcal{F}$ and the open covering $\mathcal{U}$. Its cohomology groups $H^ i(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}))$ are called the Čech cohomology groups associated to $\mathcal{F}$ and the covering $\mathcal{U}$. They are denoted $\check H^ i(\mathcal{U}, \mathcal{F})$.

Lemma 20.10.2. Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian presheaf on $X$. The following are equivalent

  1. $\mathcal{F}$ is an abelian sheaf and

  2. for every open covering $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ the natural map

    \[ \mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F}) \]

    is bijective.

Proof. This is true since the sheaf condition is exactly that $\mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F})$ is bijective for every open covering. $\square$


Comments (2)

Comment #783 by Anfang Zhou on

Typo. In the second paragraph, it should be .


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