The Stacks project

Lemma 63.10.5. Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $U$ be a quasi-compact open of $X$ with complement $Z \subset X$. Denote $g : U \to Y$ and $h : Z \to Y$ 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

\[ Rg_!(K|_ U) \to Rf_!K \to Rh_!(K|_ Z) \to Rg_!(K|_ U)[1] \]

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

Proof. This follows from Lemma 63.7.4, the fact that $Rf_! \circ Rj_! = Rg_!$ and $Rf_! \circ Ri_!$ by Lemma 63.9.2, and the fact that $Rj_! = j_!$ and $Ri_! = i_! = i_*$ by Lemma 63.10.3. $\square$

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