Lemma 59.76.2 (095X)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.76: Complexes with constructible cohomology
Lemma 59.76.2 (cite)

Lemma 59.76.2. Let $\Lambda $ be a Noetherian ring. If $j : U \to X$ is an étale morphism of schemes, then

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

Proof. The first assertion is clear. The second follows from the fact that $j_!$ is exact and Lemma 59.73.1. $\square$

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