The Stacks project

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-├ętale 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 $U_{lisse, {\acute{e}tale}}$. $\square$

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