Lemma 63.7.1 (0F5N)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 63: More Étale Cohomology
Section 63.7: Derived upper shriek for locally quasi-finite morphisms
Lemma 63.7.1 (cite)

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Lemma 63.7.1. Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. Let $\Lambda $ be a ring. The functors $f_!$ and $f^!$ of Definition 63.4.4 and Lemma 63.6.1 induce adjoint functors $f_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$ and $Rf^! : D(Y_{\acute{e}tale}, \Lambda ) \to D(X_{\acute{e}tale}, \Lambda )$ on derived categories.

Proof. This follows immediately from Derived Categories, Lemma 13.30.3, the fact that $f_!$ is exact (Lemma 63.4.5) and hence $Lf_! = f_!$ and the fact that we have enough K-injective complexes of $\Lambda $-modules on $Y_{\acute{e}tale}$ so that $Rf^!$ is defined. $\square$

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