The Stacks project

Remark 36.21.3. Let $X$ be a quasi-compact and quasi-separated scheme. Let $X = U \cup V$ with $U$ and $V$ quasi-compact open. By Lemma 36.21.1 the functors $DQ_ X$, $DQ_ U$, $DQ_ V$, $DQ_{U \cap V}$ exist. Moreover, there is a canonical distinguished triangle

\[ DQ_ X(K) \to Rj_{U, *}DQ_ U(K|_ U) \oplus Rj_{V, *}DQ_ V(K|_ V) \to Rj_{U \cap V, *}DQ_{U \cap V}(K|_{U \cap V}) \to \]

for any $K \in D(\mathcal{O}_ X)$. This follows by applying the exact functor $DQ_ X$ to the distinguished triangle of Cohomology, Lemma 20.33.2 and using Lemma 36.21.2 three times.

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