The Stacks project

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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