Lemma 48.30.4. Let j : U \to X be an open immersion of Noetherian schemes. Let
K \to L \to M \to K[1]
be a distinguished triangle of D^ b_{\textit{Coh}}(\mathcal{O}_ U). Then there exists an inverse system of distinguished triangles
K_ n \to L_ n \to M_ n \to K_ n[1]
in D^ b_{\textit{Coh}}(\mathcal{O}_ X) such that (K_ n), (L_ n), (M_ n) are Deligne systems and such that the restriction of these distinguished triangles to U is isomorphic to the distinguished triangle we started out with.
Proof.
Let (K_ n) be as in Lemma 48.30.3 part (2). Choose an object L' of D^ b_{\textit{Coh}}(\mathcal{O}_ X) whose restriction to U is L (we can do this as the lemma shows). By Lemma 48.30.1 we can find an n and a morphism K_ n \to L' on X whose restriction to U is the given arrow K \to L. We conclude there is a morphism K' \to L' of D^ b_{\textit{Coh}}(\mathcal{O}_ X) whose restriction to U is the given arrow K \to L.
By Derived Categories of Schemes, Proposition 36.11.2 we can find a morphism \alpha ^\bullet : \mathcal{F}^\bullet \to \mathcal{G}^\bullet of bounded complexes of coherent \mathcal{O}_ X-modules representing K' \to L'. Choose a quasi-coherent sheaf of ideals \mathcal{I} \subset \mathcal{O}_ X whose vanishing locus is X \setminus U. Then we let K_ n = \mathcal{I}^ n\mathcal{F}^\bullet and L_ n = \mathcal{I}^ n\mathcal{G}^\bullet . Observe that \alpha ^\bullet induces a morphism of complexes \alpha _ n^\bullet : \mathcal{I}^ n\mathcal{F}^\bullet \to \mathcal{I}^ n\mathcal{G}^\bullet . From the construction of cones in Derived Categories, Section 13.9 it is clear that
C(\alpha _ n)^\bullet = \mathcal{I}^ nC(\alpha ^\bullet )
and hence we can set M_ n = C(\alpha _ n)^\bullet . Namely, we have a compatible system of distinguished triangles (see discussion in Derived Categories, Section 13.12)
K_ n \to L_ n \to M_ n \to K_ n[1]
whose restriction to U is isomorphic to the distinguished triangle we started out with by axiom TR3 and Derived Categories, Lemma 13.4.3.
\square
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