The Stacks project

Lemma 61.26.2. Let $j : U \to X$ be an étale morphism of schemes. Let $\mathcal{G}$ be an abelian sheaf on $U_{\acute{e}tale}$. Then $\epsilon ^{-1} j_!\mathcal{G} = j_!\epsilon ^{-1}\mathcal{G}$ as sheaves on $X_{pro\text{-}\acute{e}tale}$.

Proof. This is true because both functors are left adjoint to $j_{pro\text{-}\acute{e}tale}^{-1} \epsilon _* = \epsilon _* j_{\acute{e}tale}^{-1}$. The equality holds by the discussion in Section 61.23. $\square$

Comments (2)

Comment #10174 by Adrien Morin on

There is an error in the proof, both functors are adjoint to , which are equal since both are just restrictions as is étale.

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View Lemma 61.26.2 as pdf