The Stacks project

Lemma 59.51.4. Let $X$ be a quasi-compact and quasi-separated scheme. Let $I$ be a directed set. Let $(\mathcal{F}_ i, \varphi _{ij})$ be a system of abelian sheaves on $X_{\acute{e}tale}$ over $I$. Then

\[ \mathop{\mathrm{colim}}\nolimits _{i\in I} H_{\acute{e}tale}^ p(X, \mathcal{F}_ i) = H_{\acute{e}tale}^ p(X, \mathop{\mathrm{colim}}\nolimits _{i\in I} \mathcal{F}_ i). \]

Proof. This is a special case of Theorem 59.51.3. We also sketch a direct proof. We prove it for all $X$ at the same time, by induction on $p$.

  1. For any quasi-compact and quasi-separated scheme $X$ and any étale covering $\mathcal{U}$ of $X$, show that there exists a refinement $\mathcal{V} = \{ V_ j \to X\} _{j\in J}$ with $J$ finite and each $V_ j$ quasi-compact and quasi-separated such that all $V_{j_0} \times _ X \ldots \times _ X V_{j_ p}$ are also quasi-compact and quasi-separated.

  2. Using the previous step and the definition of colimits in the category of sheaves, show that the theorem holds for $p = 0$ and all $X$.

  3. Using the locality of cohomology (Lemma 59.22.3), the Čech-to-cohomology spectral sequence (Theorem 59.19.2) and the fact that the induction hypothesis applies to all $V_{j_0} \times _ X \ldots \times _ X V_{j_ p}$ in the above situation, prove the induction step $p \to p + 1$.


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