Lemma 36.14.3 (08EP)—The Stacks project

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Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.14: Approximation by perfect complexes
Lemma 36.14.3 (cite)

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Lemma 36.14.3. Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. Let $(T, E, m)$ be a triple as in Definition 36.14.1. If

$T \subset U$ ,
approximation holds for $(T, E|_ U, m)$ , and
the sheaves $H^ i(E)$ for $i \geq m$ are supported on $T$ ,

then approximation holds for $(T, E, m)$ .

Proof. Let $j : U \to X$ be the inclusion morphism. If $P \to E|_ U$ is an approximation of the triple $(T, E|_ U, m)$ over $U$ , then $j_!P = Rj_*P \to j_!(E|_ U) \to E$ is an approximation of $(T, E, m)$ over $X$ . See Cohomology, Lemmas 20.33.6 and 20.49.10. $\square$

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