The Stacks project

Lemma 59.52.1. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be a limit of a directed system of schemes with affine transition morphisms $f_{i'i} : X_{i'} \to X_ i$. We assume that $X_ i$ is quasi-compact and quasi-separated for all $i \in I$. Let $\mathcal{F}_ i^\bullet $ be a complex of abelian sheaves on $X_{i, {\acute{e}tale}}$. Let $\varphi _{i'i} : f_{i'i}^{-1}\mathcal{F}_ i^\bullet \to \mathcal{F}_{i'}^\bullet $ be a map of complexes on $X_{i, {\acute{e}tale}}$ such that $\varphi _{i''i} = \varphi _{i''i'} \circ f_{i'' i'}^{-1}\varphi _{i'i}$ whenever $i'' \geq i' \geq i$. Assume there is an integer $a$ such that $\mathcal{F}_ i^ n = 0$ for $n < a$ and all $i \in I$. Then we have

\[ H^ p_{\acute{e}tale}(X, \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i^\bullet ) = \mathop{\mathrm{colim}}\nolimits H^ p_{\acute{e}tale}(X_ i, \mathcal{F}^\bullet _ i) \]

where $f_ i : X \to X_ i$ is the projection.

Proof. This is a consequence of Theorem 59.51.3. Set $\mathcal{F}^\bullet = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i^\bullet $. The theorem tells us that

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

for all $n, p \in \mathbf{Z}$. Let us use the spectral sequences

\[ E_{1, i}^{s, t} = H_{\acute{e}tale}^ t(X_ i, \mathcal{F}_ i^ s) \Rightarrow H_{\acute{e}tale}^{s + t}(X_ i, \mathcal{F}_ i^\bullet ) \]


\[ E_1^{s, t} = H_{\acute{e}tale}^ t(X, \mathcal{F}^ s) \Rightarrow H_{\acute{e}tale}^{s + t}(X, \mathcal{F}^\bullet ) \]

of Derived Categories, Lemma 13.21.3. Since $\mathcal{F}_ i^ n = 0$ for $n < a$ (with $a$ independent of $i$) we see that only a fixed finite number of terms $E_{1, i}^{s, t}$ (independent of $i$) and $E_1^{s, t}$ contribute to $H^ q_{\acute{e}tale}(X_ i, \mathcal{F}_ i^\bullet )$ and $H^ q_{\acute{e}tale}(X, \mathcal{F}^\bullet )$ and $E_1^{s, t} = \mathop{\mathrm{colim}}\nolimits E_{i, i}^{s, t}$. This implies what we want. Some details omitted. (There is an alternative argument using “stupid” truncations of complexes which avoids using spectral sequences.) $\square$

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