The Stacks project

Lemma 59.51.7. Let $S$ be a scheme. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be a limit of a directed system of schemes over $S$ with affine transition morphisms $f_{i'i} : X_{i'} \to X_ i$. We assume the structure morphisms $g_ i : X_ i \to S$ and $g : X \to S$ are quasi-compact and quasi-separated. Let $(\mathcal{F}_ i, \varphi _{i'i})$ be a system of abelian sheaves on $(X_ i, f_{i'i})$. Denote $f_ i : X \to X_ i$ the projection and set $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i$. Then

\[ \mathop{\mathrm{colim}}\nolimits _{i\in I} R^ p g_{i, *} \mathcal{F}_ i = R^ p g_* \mathcal{F} \]

for all $p \geq 0$.

Proof. Recall (Lemma 59.51.6) that $R^ p g_{i, *} \mathcal{F}_ i$ is the sheaf associated to the presheaf $U \mapsto H^ p_{\acute{e}tale}(U \times _ S X_ i, \mathcal{F}_ i)$ and similarly for $R^ pg_*\mathcal{F}$. Moreover, the colimit of a system of sheaves is the sheafification of the colimit on the level of presheaves. Note that every object of $S_{\acute{e}tale}$ has a covering by quasi-compact and quasi-separated objects (e.g., affine schemes). Moreover, if $U$ is a quasi-compact and quasi-separated object, then we have

\[ \mathop{\mathrm{colim}}\nolimits H^ p_{\acute{e}tale}(U \times _ S X_ i, \mathcal{F}_ i) = H^ p_{\acute{e}tale}(U \times _ S X, \mathcal{F}) \]

by Theorem 59.51.3. Thus the lemma follows. $\square$

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