Lemma 104.8.2 (0H14)—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 104.8.2 (cite)

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Lemma 104.8.2. Let $\mathcal{X}$ be an algebraic stack. Let $K$ be an object of $D(\mathcal{X}_{fppf})$ such that $R\Gamma (x, K) = 0$ for all objects $x$ of $\mathcal{X}$ lying over an affine scheme $U$ such that $U \to \mathcal{X}$ is flat. Then $H^ i(\mathcal{X}, K) = 0$ for all $i$ .

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. By Lemma 104.4.2 part (2)(b) our assumption means that $g^{-1}K$ has vanishing cohomology over every object of $\mathcal{X}_{flat, fppf}$ which lies over an affine scheme. Since every object $x$ of $\mathcal{X}_{flat, fppf}$ has a covering by such objects, we conclude that $g^{-1}K$ has vanishing cohomology sheaves, i.e., we conclude $g^{-1}K = 0$ . Then of course $R\Gamma (\mathcal{X}_{flat, fppf}, g^{-1}K) = 0$ which in turn implies what we want by Lemma 104.4.2 part (2)(a). $\square$

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