Lemma 63.10.4 (0GKN)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 63: More Étale Cohomology
Section 63.10: Properties of derived lower shriek
Lemma 63.10.4 (cite)

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Lemma 63.10.4. Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $U$ and $V$ be quasi-compact opens of $X$ such that $X = U \cup V$ . Denote $a : U \to Y$ , $b : V \to Y$ and $c : U \cap V \to Y$ the restrictions of $f$ . Let $\Lambda$ be a ring. For $K$ in $D^+_{tors}(X_{\acute{e}tale}, \Lambda )$ or $K \in D(X_{\acute{e}tale}, \Lambda )$ if $\Lambda$ is torsion, we have a distinguished triangle

$Rc_!(K|_{U \cap V}) \to Ra_!(K|_ U) \oplus Rb_!(K|_ V) \to Rf_!K \to Rc_!(K|_{U \cap V})[1]$

in $D(Y_{\acute{e}tale}, \Lambda )$ .

Proof. This follows from Lemma 63.7.3, the fact that $Rf_! \circ Rj_{U!} = Ra_!$ by Lemma 63.9.2, and the fact that $Rj_{U!} = j_{U!}$ by Lemma 63.10.3. $\square$

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