Lemma 103.4.2. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma 103.3.1.

1. For $K$ in $D(\mathcal{X}_{\acute{e}tale})$ we have

1. $R\Gamma (\mathcal{X}_{\acute{e}tale}, K) = R\Gamma (\mathcal{X}_{lisse,{\acute{e}tale}}, g^{-1}K)$, and

2. $R\Gamma (x, K) = R\Gamma (\mathcal{X}_{lisse,{\acute{e}tale}}/x, g^{-1}K)$ for any object $x$ of $\mathcal{X}_{lisse,{\acute{e}tale}}$.

2. For $K$ in $D(\mathcal{X}_{fppf})$ we have

1. $R\Gamma (\mathcal{X}_{fppf}, K) = R\Gamma (\mathcal{X}_{flat,fppf}, g^{-1}K)$, and

2. $H^ p(x, K) = R\Gamma (\mathcal{X}_{flat,fppf}/x, g^{-1}K)$ for any object $x$ of $\mathcal{X}_{flat,fppf}$.

In both cases, the same holds for modules, since we have $g^{-1} = g^*$ and there is no difference in computing cohomology by Cohomology on Sites, Lemma 21.20.7.

Proof. We prove this for the comparison between the flat-fppf site with the fppf site; the case of the lisse-étale site is exactly the same. By Lemma 103.4.1 we have $Lg_!\mathbf{Z} = \mathbf{Z}$. Then we obtain

\begin{align*} R\Gamma (\mathcal{X}_{fppf}, K) & = R\mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, K) \\ & = R\mathop{\mathrm{Hom}}\nolimits (Lg_!\mathbf{Z}, K) \\ & = R\mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, g^{-1}K) \\ & = R\Gamma (\mathcal{X}_{lisse,{\acute{e}tale}}, g^{-1}K) \end{align*}

This proves (1)(a). Part (1)(b) follows from part (1)(a). Namely, if $x$ lies over the scheme $U$, then the site $\mathcal{X}_{\acute{e}tale}/x$ is equivalent to $(\mathit{Sch}/U)_{\acute{e}tale}$ and $\mathcal{X}_{lisse,{\acute{e}tale}}$ is equivalent to $U_{lisse, {\acute{e}tale}}$. $\square$

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.

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 0H10. Beware of the difference between the letter 'O' and the digit '0'.