The Stacks project

Lemma 59.51.10. Let $I$ be a directed set. Let $g_ i : X_ i \to S_ i$ be an inverse system of morphisms of schemes over $I$. Assume $g_ i$ is quasi-compact and quasi-separated and for $i' \geq i$ the transition morphisms $X_{i'} \to X_ i$ and $S_{i'} \to S_ i$ are affine. Let $g : X \to S$ be the limit of the morphisms $g_ i$, see Limits, Section 32.2. Denote $f_ i : X \to X_ i$ and $h_ i : S \to S_ i$ the projections. Let $\mathcal{F}$ be an abelian sheaf on $X$. Then we have

\[ R^ pg_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in I} h_ i^{-1}R^ pg_{i, *}(f_{i, *}\mathcal{F}) \]

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