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

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

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

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

The same holds for sheaves of modules.

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

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

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

The same holds for sheaves of modules.

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