The Stacks project

Lemma 36.13.6. Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open. Let $E, E'$ be objects of $D_\mathit{QCoh}(\mathcal{O}_ X)$ with $E$ perfect. For every map $\alpha : E|_ U \to E'|_ U$ there exist maps

\[ E \xleftarrow {\beta } E_1 \xrightarrow {\gamma } E' \]

of perfect complexes on $X$ such that $\beta : E_1 \to E$ restricts to an isomorphism on $U$ and such that $\alpha = \gamma |_ U \circ \beta |_ U^{-1}$. Moreover we can assume $E_1 = E \otimes _{\mathcal{O}_ X}^\mathbf {L} I$ for some perfect complex $I$ on $X$.

Proof. Write $X = \mathop{\mathrm{Spec}}(A)$. Write $U = D(f_1) \cup \ldots \cup D(f_ r)$. Choose finite complex of finite projective $A$-modules $M^\bullet $ representing $E$ (Lemma 36.10.7). Choose a complex of $A$-modules $(M')^\bullet $ representing $E'$ (Lemma 36.3.5). In this case the complex $H^\bullet = \mathop{\mathrm{Hom}}\nolimits _ A(M^\bullet , (M')^\bullet )$ is a complex of $A$-modules whose associated complex of quasi-coherent $\mathcal{O}_ X$-modules represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, E')$, see Cohomology, Lemma 20.44.9. Then $\alpha $ determines an element $s$ of $H^0(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, E'))$, see Cohomology, Lemma 20.40.1. There exists an $e$ and a map

\[ \xi : I^\bullet (f_1^ e, \ldots , f_ r^ e) \to \mathop{\mathrm{Hom}}\nolimits _ A(M^\bullet , (M')^\bullet ) \]

corresponding to $s$, see Proposition 36.9.5. Letting $E_1$ be the object corresponding to complex of quasi-coherent $\mathcal{O}_ X$-modules associated to

\[ \text{Tot}(I^\bullet (f_1^ e, \ldots , f_ r^ e) \otimes _ A M^\bullet ) \]

we obtain $E_1 \to E$ using the canonical map $I^\bullet (f_1^ e, \ldots , f_ r^ e) \to A$ and $E_1 \to E'$ using $\xi $ and Cohomology, Lemma 20.40.1. $\square$

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