Lemma 104.8.1 (0H13)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 104: Derived Categories of Stacks
Section 104.8: Quasi-coherent objects in the derived category
Lemma 104.8.1 (cite)

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Lemma 104.8.1. Let $\mathcal{X}$ be an algebraic stack. Let $K$ be an object of $D(\mathcal{X}_{fppf})$ whose cohomology sheaves are parasitic. Then $R\Gamma (x, K) = 0$ for all objects $x$ of $\mathcal{X}$ lying over a scheme $U$ such that $U \to \mathcal{X}$ is flat.

Proof. Denote $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{flat, fppf}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$ the morphism of topoi discussed in Section 104.3. Let $x$ be an object of $\mathcal{X}$ lying over a scheme $U$ such that $U \to \mathcal{X}$ is flat, i.e., $x$ is an object of $\mathcal{X}_{flat, fppf}$ . By Lemma 104.4.2 part (2)(b) we have $R\Gamma (x, K) = R\Gamma (\mathcal{X}_{flat, fppf}/x, g^{-1}K)$ . However, our assumption means that the cohomology sheaves of the object $g^{-1}K$ of $D(\mathcal{X}_{flat, fppf})$ are zero, see Cohomology of Stacks, Definition 103.9.1. Hence $g^{-1}K = 0$ and we win. $\square$

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