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 definition 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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