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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