The Stacks project

Lemma 36.13.7. 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 36.13.5 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 36.13.6 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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