The Stacks project

Lemma 102.14.3. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma 102.14.2.

  1. For an abelian sheaf $\mathcal{F}$ on $\mathcal{X}_{\acute{e}tale}$ we have

    1. $H^ p(\mathcal{X}_{\acute{e}tale}, \mathcal{F}) = H^ p(\mathcal{X}_{lisse,{\acute{e}tale}}, g^{-1}\mathcal{F})$, and

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

    The same holds for sheaves of modules.

  2. For an abelian sheaf $\mathcal{F}$ on $\mathcal{X}_{fppf}$ we have

    1. $H^ p(\mathcal{X}_{fppf}, \mathcal{F}) = H^ p(\mathcal{X}_{flat,fppf}, g^{-1}\mathcal{F})$, and

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

    The same holds for sheaves of modules.

Proof. Part (1)(a) follows from Sheaves on Stacks, Lemma 95.23.3 applied to the inclusion functor $\mathcal{X}_{lisse,{\acute{e}tale}} \to \mathcal{X}_{\acute{e}tale}$. 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 $U_{lisse,{\acute{e}tale}}$. Part (2) is proved in the same manner. $\square$


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