Lemma 59.22.1. Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $(\mathit{Sch}/S)_\tau $, or on $S_\tau $ in case $\tau = {\acute{e}tale}$, and let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a standard $\tau $-covering of this site. Let $V = \coprod _{i \in I} U_ i$. Then
$V$ is an affine scheme,
$\mathcal{V} = \{ V \to U\} $ is an fpqc covering and also a $\tau $-covering unless $\tau = Zariski$,
the Čech complexes $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ and $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F})$ agree.
Proof.
The defintion of a standard $\tau $-covering is given in Topologies, Definition 34.3.4, 34.4.5, 34.5.5, 34.6.5, and 34.7.5. By definition each of the schemes $U_ i$ is affine and $I$ is a finite set. Hence $V$ is an affine scheme. It is clear that $V \to U$ is flat and surjective, hence $\mathcal{V}$ is an fpqc covering, see Example 59.15.3. Excepting the Zariski case, the covering $\mathcal{V}$ is also a $\tau $-covering, see Topologies, Definition 34.4.1, 34.5.1, 34.6.1, and 34.7.1.
Note that $\mathcal{U}$ is a refinement of $\mathcal{V}$ and hence there is a map of Čech complexes $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$, see Cohomology on Sites, Equation (21.8.2.1). Next, we observe that if $T = \coprod _{j \in J} T_ j$ is a disjoint union of schemes in the site on which $\mathcal{F}$ is defined then the family of morphisms with fixed target $\{ T_ j \to T\} _{j \in J}$ is a Zariski covering, and so
59.22.1.1
\begin{equation} \label{etale-cohomology-equation-sheaf-coprod} \mathcal{F}(T) = \mathcal{F}(\coprod \nolimits _{j \in J} T_ j) = \prod \nolimits _{j \in J} \mathcal{F}(T_ j) \end{equation}
by the sheaf condition of $\mathcal{F}$. This implies the map of Čech complexes above is an isomorphism in each degree because
\[ V \times _ U \ldots \times _ U V = \coprod \nolimits _{i_0, \ldots i_ p} U_{i_0} \times _ U \ldots \times _ U U_{i_ p} \]
as schemes.
$\square$
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Comment #1472 by Xiaowen Hu on
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