Lemma 63.7.3 (0GKK)—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 63.7.3 (cite)

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Lemma 63.7.3. Let $X$ be a scheme. Let $X = U \cup V$ with $U$ and $V$ open. Let $\Lambda $ be a ring. Let $K \in D(X_{\acute{e}tale}, \Lambda )$. There is a distinguished triangle

\[ j_{U \cap V!}K|_{U \cap V} \to j_{U!}K|_ U \oplus j_{V!}K|_ V \to K \to j_{U \cap V!}K|_{U \cap V}[1] \]

in $D(X_{\acute{e}tale}, \Lambda )$ with obvious notation.

Proof. Since the restriction functors and the lower shriek functors we use are exact, it suffices to show for any abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ the sequence

\[ 0 \to j_{U \cap V!}\mathcal{F}|_{U \cap V} \to j_{U!}\mathcal{F}|_ U \oplus j_{V!}\mathcal{F}|_ V \to \mathcal{F} \to 0 \]

is exact. This can be seen by looking at stalks. $\square$

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