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*}

21.9 The Čech complex and Čech cohomology

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a family of morphisms with fixed target, see Sites, Definition 7.6.1. Assume that all fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exist in $\mathcal{C}$. Let $\mathcal{F}$ be an abelian presheaf on $\mathcal{C}$. Set

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

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 the factor $\mathcal{F}(U_{i_0} \times _ U \ldots \times _ U U_{i_ p})$. We define

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

by the formula

21.9.0.1
\begin{equation} \label{sites-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} |_{U_{i_0} \times _ U \ldots \times _ U U_{i_{p + 1}}} \end{equation}

where the restriction is via the projection map

\[ U_{i_0} \times _ U \ldots \times _ U U_{i_{p + 1}} \longrightarrow U_{i_0} \times _ U \ldots \times _ U \widehat{U_{i_ j}} \times _ U \ldots \times _ U U_{i_{p + 1}}. \]

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 21.9.1. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exist in $\mathcal{C}$. Let $\mathcal{F}$ be an abelian presheaf on $\mathcal{C}$. The complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is the Čech complex associated to $\mathcal{F}$ and the family $\mathcal{U}$. Its cohomology groups $H^ i(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}))$ are called the Čech cohomology groups of $\mathcal{F}$ with respect to $\mathcal{U}$. They are denoted $\check H^ i(\mathcal{U}, \mathcal{F})$.

We observe that any covering $\{ U_ i \to U\} $ of a site $\mathcal{C}$ is a family of morphisms with fixed target to which the definition applies.

Lemma 21.9.2. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be an abelian presheaf on $\mathcal{C}$. The following are equivalent

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

  2. for every covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ of the site $\mathcal{C}$ the natural map

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

    (see Sites, Section 7.10) 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 covering of $\mathcal{C}$. $\square$

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ U_ i \to U\} _{i\in I}$ be a family of morphisms of $\mathcal{C}$ with fixed target such that all fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exist in $\mathcal{C}$. Let $\mathcal{V} = \{ V_ j \to V\} _{j\in J}$ be another. Let $f : U \to V$, $\alpha : I \to J$ and $f_ i : U_ i \to V_{\alpha (i)}$ be a morphism of families of morphisms with fixed target, see Sites, Section 7.8. In this case we get a map of Čech complexes

21.9.2.1
\begin{equation} \label{sites-cohomology-equation-map-cech-complexes} \varphi : \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \end{equation}

which in degree $p$ is given by

\[ \varphi (s)_{i_0 \ldots i_ p} = (f_{i_0} \times \ldots \times f_{i_ p})^*s_{\alpha (i_0) \ldots \alpha (i_ p)} \]

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