Lemma 48.30.11 (0G4U)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.30: Extension by zero for coherent modules
Lemma 48.30.11 (cite)

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Lemma 48.30.11. Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals with $V(\mathcal{I}) = X \setminus U$ . Let $K$ be in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ . Then

$K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{I}^ n$

is pro-isomorphic to a Deligne system with constant value $K|_ U$ over $U$ .

Proof. By Lemma 48.30.10 the question is local on $X$ . Thus we may assume $X$ is the spectrum of a Noetherian ring. In this case the statement follows from the algebra version which is More on Algebra, Lemma 15.101.6. $\square$

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