The Stacks project

Lemma 61.14.3. Let $X$ be a quasi-compact and quasi-separated scheme. The functor $R\Gamma (X, -) : D^+(X_{pro\text{-}\acute{e}tale}) \to D(\textit{Ab})$ commutes with direct sums and homotopy colimits.

Proof. The statement means the following: Suppose we have a family of objects $E_ i$ of $D^+(X_{pro\text{-}\acute{e}tale})$ such that $\bigoplus E_ i$ is an object of $D^+(X_{pro\text{-}\acute{e}tale})$. Then $R\Gamma (X, \bigoplus E_ i) = \bigoplus R\Gamma (X, E_ i)$. To see this choose a hypercovering $K$ of $X$ with $K_ n = \{ U_ n \to X\} $ where $U_ n$ is an affine and weakly contractible scheme, see Lemma 61.14.1. Let $N$ be an integer such that $H^ p(E_ i) = 0$ for $p < N$. Choose a complex of abelian sheaves $\mathcal{E}_ i^\bullet $ representing $E_ i$ with $\mathcal{E}_ i^ p = 0$ for $p < N$. The termwise direct sum $\bigoplus \mathcal{E}_ i^\bullet $ represents $\bigoplus E_ i$ in $D(X_{pro\text{-}\acute{e}tale})$, see Injectives, Lemma 19.13.4. By Lemma 61.14.2 we have

\[ R\Gamma (X, \bigoplus E_ i) = \text{Tot}(s((\bigoplus \mathcal{E}^\bullet _ i)(K))) \]


\[ R\Gamma (X, E_ i) = \text{Tot}(s(\mathcal{E}^\bullet _ i(K))) \]

Since each $U_ n$ is quasi-compact we see that

\[ \text{Tot}(s((\bigoplus \mathcal{E}^\bullet _ i)(K))) = \bigoplus \text{Tot}(s(\mathcal{E}^\bullet _ i(K))) \]

by Modules on Sites, Lemma 18.30.3. The statement on homotopy colimits is a formal consequence of the fact that $R\Gamma $ is an exact functor of triangulated categories and the fact (just proved) that it commutes with direct sums. $\square$

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