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The Stacks project

Lemma 59.52.5. Let I, g_ i : X_ i \to S_ i, g : X \to S, f_{ii'}, f_ i, g_ i, h_ i be as in Lemma 59.51.8. 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

R^ pg_*(\mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i^\bullet ) = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} h_ i^{-1}R^ pg_{i, *}\mathcal{F}_ i^\bullet

for all p \in \mathbf{Z}.

Proof. This is a consequence of Lemma 59.51.8. Set \mathcal{F}^\bullet = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i^\bullet . The lemma tells us that

\mathop{\mathrm{colim}}\nolimits _{i \in I} h_ i^{-1}R^ pg_{i, *}\mathcal{F}_ i^ n = R^ pg_*\mathcal{F}^ n

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

E_{1, i}^{s, t} = R^ tg_{i, *}\mathcal{F}_ i^ s \Rightarrow R^{s + t}g_{i, *}\mathcal{F}_ i^\bullet

and

E_1^{s, t} = R^ tg_*\mathcal{F}^ s \Rightarrow R^{s + t}g_*\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 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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