Lemma 36.13.7. 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 complexes on X with E_1 perfect 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 20.46.9. Then \alpha determines an element s of H^0(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, E')), see Cohomology, Lemma 20.42.1. 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 20.42.1.
\square
Comments (1)
Comment #8629 by nkym on