Lemma 36.13.8 (08EI)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.13: Lifting complexes
Lemma 36.13.8 (cite)

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Lemma 36.13.8. Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open. For every perfect object $F$ of $D(\mathcal{O}_ U)$ the object $F \oplus F[1]$ is the restriction of a perfect object of $D(\mathcal{O}_ X)$ .

Proof. By Lemma 36.13.6 we can find a perfect object $E$ of $D(\mathcal{O}_ X)$ such that $E|_ U = \mathcal{F}[r] \oplus F$ for some finite locally free $\mathcal{O}_ U$ -module $\mathcal{F}$ . By Lemma 36.13.7 we can find a morphism of perfect complexes $\alpha : E_1 \to E$ such that $(E_1)|_ U \cong E|_ U$ and such that $\alpha |_ U$ is the map

$\left( \begin{matrix} \text{id}_{\mathcal{F}[r]} & 0 \\ 0 & 0 \end{matrix} \right) : \mathcal{F}[r] \oplus F \to \mathcal{F}[r] \oplus F$

Then the cone on $\alpha$ is a solution. $\square$

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