Lemma 63.7.4 (0GKL)—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 63.7.4 (cite)

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Lemma 63.7.4. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme and let $U \subset X$ be the complement. Denote $i : Z \to X$ and $j : U \to X$ the inclusion morphisms. Let $\Lambda $ be a ring. Let $K \in D(X_{\acute{e}tale}, \Lambda )$. There is a distinguished triangle

\[ j_!j^{-1}K \to K \to i_*i^{-1}K \to j_!j^{-1}K[1] \]

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

Proof. Immediate consequence of Étale Cohomology, Lemma 59.70.8 and the fact that the functors $j_!$, $j^{-1}$, $i_*$, $i^{-1}$ are exact and hence their derived versions are computed by applying these functors to any complex of sheaves representing $K$. $\square$

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