The Stacks project

Lemma 103.8.2. 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 103.3. By Lemma 103.4.2 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 103.4.2 part (2)(a). $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0H14. Beware of the difference between the letter 'O' and the digit '0'.