Lemma 59.77.5 (0962)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.77: Tor finite with constructible cohomology
Lemma 59.77.5 (cite)

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Lemma 59.77.5. Let $\Lambda $ be a Noetherian ring. If $j : U \to X$ is an étale morphism of schemes, then

$K|_ U \in D_{ctf}(U_{\acute{e}tale}, \Lambda )$ if $K \in D_{ctf}(X_{\acute{e}tale}, \Lambda )$, and
$j_!M \in D_{ctf}(X_{\acute{e}tale}, \Lambda )$ if $M \in D_{ctf}(U_{\acute{e}tale}, \Lambda )$ and the morphism $j$ is quasi-compact and quasi-separated.

Proof. Perhaps the easiest way to prove this lemma is to reduce to the case where $X$ is affine and then apply Lemma 59.77.3 to translate it into a statement about finite complexes of flat constructible sheaves of $\Lambda $-modules where the result follows from Lemma 59.73.1. $\square$

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