The Stacks project

Proposition 115.8.27. Let $f : X \to Y$ be a locally quasi-finite morphism. There exist adjoint functors $f_! : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ and $f^! : \textit{Ab}(Y_{\acute{e}tale}) \to \textit{Ab}(X_{\acute{e}tale})$ with the following properties

  1. the functor $f^!$ is the one constructed in More Étale Cohomology, Lemma 63.6.1,

  2. for any open $j : U \to X$ with $f \circ j$ separated there is a canonical isomorphism $f_! \circ j_! = (f \circ j)_!$, and

  3. these isomorphisms for $U \subset U' \subset X$ are compatible with the isomorphisms in More Étale Cohomology, Lemma 63.3.13.

Proof. See More Étale Cohomology, Sections 63.4 and 63.6. $\square$

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