Remark 75.19.3. Let $S$ be a scheme. Let $(U \subset X, f : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Assume $X$, $U$, $V$ are quasi-compact and quasi-separated. By Lemma 75.19.1 the functors $DQ_ X$, $DQ_ U$, $DQ_ V$, $DQ_{U \times _ X V}$ exist. Moreover, there is a canonical distinguished triangle
\[ DQ_ X(K) \to Rj_{U, *}DQ_ U(K|_ U) \oplus Rj_{V, *}DQ_ V(K|_ V) \to Rj_{U \times _ X V, *}DQ_{U \times _ X V}(K|_{U \times _ X V}) \to \]
for any $K \in D(\mathcal{O}_ X)$. This follows by applying the exact functor $DQ_ X$ to the distinguished triangle of Lemma 75.10.2 and using Lemma 75.19.2 three times.
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