Lemma 75.14.3 (08HK)—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 75.14.3 (cite)

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Lemma 75.14.3. Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic space over $S$ . Let $E$ be a perfect object of $D(\mathcal{O}_ V)$ supported on $j^{-1}(T)$ where $T = |X| \setminus |U|$ . Then $Rj_*E$ is a perfect object of $D(\mathcal{O}_ X)$ .

Proof. Being perfect is local on $X_{\acute{e}tale}$ . Thus it suffices to check that $Rj_*E$ is perfect when restricted to $U$ and $V$ . We have $Rj_*E|_ V = E$ by Lemma 75.10.7 which is perfect. We have $Rj_*E|_ U = 0$ because $E|_{V \setminus j^{-1}(T)} = 0$ (use Lemma 75.3.1). $\square$

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