Lemma 62.7.3. 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).