Lemma 75.14.4 (08HL)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 75: Derived Categories of Spaces
Section 75.14: Approximation by perfect complexes
Lemma 75.14.4 (cite)

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Lemma 75.14.4. Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$ . Let $T$ be a closed subset of $|X| \setminus |U|$ and let $(T, E, m)$ be a triple as in Definition 75.14.1. If

approximation holds for $(j^{-1}T, E|_ V, m)$ , and
the sheaves $H^ i(E)$ for $i \geq m$ are supported on $T$ ,

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

Proof. Let $P \to E|_ V$ be an approximation of the triple $(j^{-1}T, E|_ V, m)$ over $V$ . Then $Rj_*P$ is a perfect object of $D(\mathcal{O}_ X)$ by Lemma 75.14.3. On the other hand, $Rj_*P = j_!P$ by Lemma 75.10.7. We see that $j_!P$ is supported on $T$ for example by (75.10.0.2). Hence we obtain an approximation $Rj_*P = j_!P \to j_!(E|_ V) \to E$ . $\square$

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