Lemma 36.13.11. Let X be a quasi-compact and quasi-separated scheme. Let U \subset X be a quasi-compact open. Let T \subset X be a closed subset with X \setminus T retro-compact in X. Let E be an object of D_\mathit{QCoh}(\mathcal{O}_ X). Let \alpha : P \to E|_ U be a map where P is a perfect object of D(\mathcal{O}_ U) supported on T \cap U. 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|_ U in D(\mathcal{O}_ U) compatible \alpha and \beta |_ U.
Proof. Since X is quasi-compact there exists an integer m such that X = U \cup V_1 \cup \ldots \cup V_ m for some affine opens V_ j of X. Arguing by induction on m we see that we may assume m = 1. In other words, we may assume that X = U \cup V with V affine. By Lemma 36.13.10 we can choose a perfect object Q in D(\mathcal{O}_ V) supported on T \cap V and an isomorphism Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}. By Lemma 36.13.7 we can replace Q by Q \otimes ^\mathbf {L} I (still supported on T \cap V) and assume that the map
Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V} \longrightarrow P|_{U \cap V} \longrightarrow E|_{U \cap V}
lifts to Q \to E|_ V. By Cohomology, Lemma 20.45.1 we find an morphism a : R \to E of D(\mathcal{O}_ X) such that a|_ U is isomorphic to P \oplus P[1] \to E|_ U and a|_ V isomorphic to Q \to E|_ V. Thus R is perfect and supported on T as desired. \square
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