Lemma 59.51.5. Let $A$ be a ring, $(I, \leq )$ a directed set and $(B_ i, \varphi _{ij})$ a system of $A$-algebras. Set $B = \mathop{\mathrm{colim}}\nolimits _{i\in I} B_ i$. Let $X \to \mathop{\mathrm{Spec}}(A)$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}$ an abelian sheaf on $X_{\acute{e}tale}$. Denote $Y_ i = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(B_ i)$, $Y = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(B)$, $\mathcal{G}_ i = (Y_ i \to X)^{-1}\mathcal{F}$ and $\mathcal{G} = (Y \to X)^{-1}\mathcal{F}$. Then

$H_{\acute{e}tale}^ p(Y, \mathcal{G}) = \mathop{\mathrm{colim}}\nolimits _{i\in I} H_{\acute{e}tale}^ p (Y_ i, \mathcal{G}_ i).$

Proof. This is a special case of Theorem 59.51.3. We also outline a direct proof as follows.

1. Given $V \to Y$ étale with $V$ quasi-compact and quasi-separated, there exist $i\in I$ and $V_ i \to Y_ i$ such that $V = V_ i \times _{Y_ i} Y$. If all the schemes considered were affine, this would correspond to the following algebra statement: if $B = \mathop{\mathrm{colim}}\nolimits B_ i$ and $B \to C$ is étale, then there exist $i \in I$ and $B_ i \to C_ i$ étale such that $C \cong B \otimes _{B_ i} C_ i$. This is proved in Algebra, Lemma 10.143.3.

2. In the situation of (1) show that $\mathcal{G}(V) = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \mathcal{G}_{i'}(V_{i'})$ where $V_{i'}$ is the base change of $V_ i$ to $Y_{i'}$.

3. By (1), we see that for every étale covering $\mathcal{V} = \{ V_ j \to Y\} _{j\in J}$ with $J$ finite and the $V_ j$s quasi-compact and quasi-separated, there exists $i \in I$ and an étale covering $\mathcal{V}_ i = \{ V_{ij} \to Y_ i\} _{j \in J}$ such that $\mathcal{V} \cong \mathcal{V}_ i \times _{Y_ i} Y$.

4. Show that (2) and (3) imply

$\check H^*(\mathcal{V}, \mathcal{G})= \mathop{\mathrm{colim}}\nolimits _{i\in I} \check H^*(\mathcal{V}_ i, \mathcal{G}_ i).$
5. Cleverly use the Čech-to-cohomology spectral sequence (Theorem 59.19.2).

$\square$

Comment #2355 by Yu-Liang Huang on

In the last sentence of the lemma, the right side should be $Y_i$, rather than $X_i$.

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