Lemma 75.15.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $W \subset X$ be a quasi-compact open. Let $T \subset |X|$ be a closed subset such that $X \setminus T \to X$ is a quasi-compact morphism. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $\alpha : P \to E|_ W$ be a map where $P$ is a perfect object of $D(\mathcal{O}_ W)$ supported on $T \cap W$. Then there exists a map $\beta : R \to E$ where $R$ is a perfect object of $D(\mathcal{O}_ X)$ supported on $T$ such that $P$ is a direct summand of $R|_ W$ in $D(\mathcal{O}_ W)$ compatible $\alpha $ and $\beta |_ W$.
Proof. We will use the induction principle of Lemma 75.9.6 to prove this. Thus we immediately reduce to the case where we have an elementary distinguished square $(W \subset X, f : V \to X)$ with $V$ affine and $P \to E|_ W$ as in the statement of the lemma. In the rest of the proof we will use Lemma 75.4.2 (and the compatibilities of Remark 75.6.3) for the representable algebraic spaces $V$ and $W \times _ X V$. We will also use the fact that perfectness on the Zariski site and étale site agree, see Lemma 75.13.5.
By Derived Categories of Schemes, Lemma 36.13.10 we can choose a perfect object $Q$ in $D(\mathcal{O}_ V)$ supported on $f^{-1}T$ and an isomorphism $Q|_{W \times _ X V} \to (P \oplus P[1])|_{W \times _ X V}$. By Derived Categories of Schemes, Lemma 36.13.7 we can replace $Q$ by $Q \otimes ^\mathbf {L} I$ (still supported on $f^{-1}T$) and assume that the map
\[ Q|_{W \times _ X V} \to (P \oplus P[1])|_{W \times V} \longrightarrow P|_{W \times _ X V} \longrightarrow E|_{W \times _ X V} \]
lifts to $Q \to E|_ V$. By Lemma 75.10.8 we find an morphism $a : R \to E$ of $D(\mathcal{O}_ X)$ such that $a|_ W$ is isomorphic to $P \oplus P[1] \to E|_ W$ and $a|_ V$ isomorphic to $Q \to E|_ V$. Thus $R$ is perfect and supported on $T$ as desired. $\square$
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