Definition 21.8.1. 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 $\check H^ i(\mathcal{U}, \mathcal{F})$.
21.8 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
21.8.0.1
\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 + 1}} |_{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.
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 21.8.2. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be an abelian presheaf on $\mathcal{C}$. The following are equivalent
$\mathcal{F}$ is an abelian sheaf on $\mathcal{C}$ and
for every covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ of the site $\mathcal{C}$ the natural map
(see Sites, Section 7.10) is bijective.
\[ \mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F}) \]
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
21.8.2.1
\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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