The Stacks project

Lemma 61.26.6. Let $j : U \to X$ be a quasi-compact open immersion morphism of schemes. The functor $j_! : \textit{Ab}(U_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(X_{pro\text{-}\acute{e}tale})$ commutes with limits.

Proof. Since $j_!$ is exact it suffices to show that $j_!$ commutes with products. The question is local on $X$, hence we may assume $X$ affine. Let $\mathcal{G}$ be an abelian sheaf on $U_{pro\text{-}\acute{e}tale}$. We have $j^{-1}j_*\mathcal{G} = \mathcal{G}$. Hence applying the exact sequence of Lemma 61.26.5 we get

\[ 0 \to j_!\mathcal{G} \to j_*\mathcal{G} \to i_*i^{-1}j_*\mathcal{G} \to 0 \]

where $i : Z \to X$ is the inclusion of the reduced induced scheme structure on the complement $Z = X \setminus U$. The functors $j_*$ and $i_*$ commute with products as right adjoints. The functor $i^{-1}$ commutes with products by Lemma 61.25.3. Hence $j_!$ does because on the pro-├ętale site products are exact (Cohomology on Sites, Proposition 21.51.2). $\square$

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