The Stacks project

Lemma 100.8.3. Let $\tau \in \{ {\acute{e}tale}, fppf\} $. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}$ be a parasitic object of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$.

  1. $H^ i_\tau (\mathcal{X}, \mathcal{F}) = 0$ for all $i$.

  2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then $R^ if_*\mathcal{F}$ (computed in $\tau $-topology) is a parasitic object of $\textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y})$.

Proof. We first reduce (2) to (1). By Sheaves on Stacks, Lemma 93.20.2 we see that $R^ if_*\mathcal{F}$ is the sheaf associated to the presheaf

\[ y \longmapsto H^ i_\tau \Big(V \times _{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big) \]

Here $y$ is a typical object of $\mathcal{Y}$ lying over the scheme $V$. By Lemma 100.8.2 it suffices to show that these cohomology groups are zero when $y : V \to \mathcal{Y}$ is flat. Note that $\text{pr} : V \times _{y, \mathcal{Y}} \mathcal{X} \to \mathcal{X}$ is flat as a base change of $y$. Hence by Lemma 100.8.2 we see that $\text{pr}^{-1}\mathcal{F}$ is parasitic. Thus it suffices to prove (1).

To see (1) we can use the spectral sequence of Sheaves on Stacks, Proposition 93.19.1 to reduce this to the case where $\mathcal{X}$ is an algebraic stack representable by an algebraic space. Note that in the spectral sequence each $f_ p^{-1}\mathcal{F} = f_ p^*\mathcal{F}$ is a parasitic module by Lemma 100.8.2 because the morphisms $f_ p : \mathcal{U}_ p = \mathcal{U} \times _\mathcal {X} \ldots \times _\mathcal {X} \mathcal{U} \to \mathcal{X}$ are flat. Reusing this spectral sequence one more time (as in the proof of the key Lemma 100.5.1) we reduce to the case where the algebraic stack $\mathcal{X}$ is representable by a scheme $X$. Then $H^ i_\tau (\mathcal{X}, \mathcal{F}) = H^ i((\mathit{Sch}/X)_\tau , \mathcal{F})$. In this case the vanishing follows easily from an argument with Čech coverings, see Descent, Lemma 35.9.2. $\square$


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Your email address will not be published.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 0775. Beware of the difference between the letter 'O' and the digit '0'.