The Stacks project

Lemma 48.30.3. Let $j : U \to X$ be an open immersion of Noetherian schemes.

  1. Let $(K_ n)$ and $(L_ n)$ be Deligne systems. Let $K$ and $L$ be the values of the constant systems $(K_ n|_ U)$ and $(L_ n|_ U)$. Given a morphism $\alpha : K \to L$ of $D(\mathcal{O}_ U)$ there is a unique morphism of pro-systems $(K_ n) \to (L_ n)$ of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ whose restriction to $U$ is $\alpha $.

  2. Given $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ U)$ there exists a Deligne system $(K_ n)$ such that $(K_ n|_ U)$ is constant with value $K$.

  3. The pro-object $(K_ n)$ of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ of (2) is unique up to unique isomorphism (as a pro-object).

Proof. Part (1) is an immediate consequence of Lemma 48.30.1 and the fact that morphisms between pro-systems are the same as morphisms between the functors they corepresent, see Categories, Remark 4.22.7.

Let $K$ be as in (2). We can choose $K' \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ whose restriction to $U$ is isomorphic to $K$, see Derived Categories of Schemes, Lemma 36.13.2. By Derived Categories of Schemes, Proposition 36.11.2 we can represent $K'$ by a bounded complex $\mathcal{F}^\bullet $ of coherent $\mathcal{O}_ X$-modules. Choose a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ whose vanishing locus is $X \setminus U$ (for example choose $\mathcal{I}$ to correspond to the reduced induced subscheme structure on $X \setminus U$). Then we can set $K_ n$ equal to the object represented by the complex $\mathcal{I}^ n\mathcal{F}^\bullet $ as in the introduction to this section.

Part (3) is immediate from parts (1) and (2). $\square$

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