The Stacks project

Lemma 35.9.1. Let $S$ be a scheme. Let

1. $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $ and $\mathcal{C} = (\mathit{Sch}/S)_\tau $, or

2. let $\tau = {\acute{e}tale}$ and $\mathcal{C} = S_{\acute{e}tale}$, or

3. let $\tau = Zariski$ and $\mathcal{C} = S_{Zar}$.

Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be affine. Let $\mathcal{U} = \{ U_ i \to U\} _{i = 1, \ldots , n}$ be a standard affine $\tau $-covering in $\mathcal{C}$. Then

1. $\mathcal{V} = \{ \coprod _{i = 1, \ldots , n} U_ i \to U\} $ is a $\tau $-covering of $U$,

2. $\mathcal{U}$ is a refinement of $\mathcal{V}$, and

3. the induced map on Čech complexes (Cohomology on Sites, Equation (21.8.2.1))

   \[ \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \]

   is an isomorphism of complexes.