Lemma 48.30.9 (0G4S)—The Stacks project

Lemma 48.30.9. Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $a \leq b$ be integers. Let $(K_ n)$ be an inverse system of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ such that $H^ i(K_ n) = 0$ for $i \not\in [a, b]$ . The following are equivalent

$(K_ n)$ is pro-isomorphic to a Deligne system,
for every $p \in \mathbf{Z}$ there exists a coherent $\mathcal{O}_ X$ -module $\mathcal{F}$ such that the pro-systems $(H^ p(K_ n))$ and $(\mathcal{I}^ n\mathcal{F})$ are pro-isomorphic.

Proof. Assume (1). To prove (2) holds we may assume $(K_ n)$ is a Deligne system. By definition we may choose a bounded complex $\mathcal{F}^\bullet$ of coherent $\mathcal{O}_ X$ -modules and a quasi-coherent sheaf of ideals cutting out $X \setminus U$ such that $K_ n$ is represented by $\mathcal{I}^ n\mathcal{F}^\bullet$ . Thus the result follows from Lemma 48.30.8.

Assume (2). We will prove that $(K_ n)$ is as in (1) by induction on $b - a$ . If $a = b$ then (1) holds essentially by assumption. If $a < b$ then we consider the compatible system of distinguished triangles

$\tau _{\leq a}K_ n \to K_ n \to \tau _{\geq a + 1}K_ n \to (\tau _{\leq a}K_ n)[1]$

See Derived Categories, Remark 13.12.4. By induction on $b - a$ we know that $\tau _{\leq a}K_ n$ and $\tau _{\geq a + 1}K_ n$ are pro-isomorphic to Deligne systems. We conclude by Lemma 48.30.7. $\square$

The Stacks project

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