Lemma 96.23.2. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Let $\mathcal{F}$ be a presheaf $\mathcal{O}_\mathcal {X}$-module such that
$\mathcal{F}$ is locally quasi-coherent, and
for any morphism $\varphi : x \to y$ of $\mathcal{X}$ which lies over a morphism of schemes $f : U \to V$ which is flat and locally of finite presentation, the comparison map $c_\varphi : f_{small}^*\mathcal{F}|_{V_{\acute{e}tale}} \to \mathcal{F}|_{U_{\acute{e}tale}}$ of (96.9.4.1) is an isomorphism.
Then $\mathcal{F}$ is an $\mathcal{O}_\mathcal {X}$-module and we have the following
If $\epsilon : \mathcal{X}_{fppf} \to \mathcal{X}_{\acute{e}tale}$ is the comparison morphism, then $R\epsilon _*\mathcal{F} = \epsilon _*\mathcal{F}$.
The cohomology groups $H^ p_{fppf}(\mathcal{X}, \mathcal{F})$ are equal to the cohomology groups computed in the étale topology on $\mathcal{X}$. Similarly for the cohomology groups $H^ p_{fppf}(x, \mathcal{F})$ and the derived versions $R\Gamma (\mathcal{X}, \mathcal{F})$ and $R\Gamma (x, \mathcal{F})$.
If $f : \mathcal{X} \to \mathcal{Y}$ is a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ then $R^ if_*\mathcal{F}$ is equal to the fppf-sheafification of the higher direct image computed in the étale cohomology. Similarly for derived pullback.
Proof.
The assertion that $\mathcal{F}$ is an $\mathcal{O}_\mathcal {X}$-module follows from Lemma 96.23.1. Note that $\epsilon $ is a morphism of sites given by the identity functor on $\mathcal{X}$. The sheaf $R^ p\epsilon _*\mathcal{F}$ is therefore the sheaf associated to the presheaf $x \mapsto H^ p_{fppf}(x, \mathcal{F})$, see Cohomology on Sites, Lemma 21.7.4. To prove (1) it suffices to show that $H^ p_{fppf}(x, \mathcal{F}) = 0$ for $p > 0$ whenever $x$ lies over an affine scheme $U$. By Lemma 96.16.1 we have $H^ p_{fppf}(x, \mathcal{F}) = H^ p((\mathit{Sch}/U)_{fppf}, x^{-1}\mathcal{F})$. Combining Descent, Lemma 35.12.4 with Cohomology of Schemes, Lemma 30.2.2 we see that these cohomology groups are zero.
We have seen above that $\epsilon _*\mathcal{F}$ and $\mathcal{F}$ are the sheaves on $\mathcal{X}_{\acute{e}tale}$ and $\mathcal{X}_{fppf}$ corresponding to the same presheaf on $\mathcal{X}$ (and this is true more generally for any sheaf in the fppf topology on $\mathcal{X}$). We often abusively identify $\mathcal{F}$ and $\epsilon _*\mathcal{F}$ and this is the sense in which parts (2) and (3) of the lemma should be understood. Thus part (2) follows formally from (1) and the Leray spectral sequence, see Cohomology on Sites, Lemma 21.14.6.
Finally we prove (3). The sheaf $R^ if_*\mathcal{F}$ (resp. $Rf_{{\acute{e}tale}, *}\mathcal{F}$) is the sheaf associated to the presheaf
\[ y \longmapsto H^ i_\tau \Big((\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big) \]
where $\tau $ is $fppf$ (resp. ${\acute{e}tale}$), see Lemma 96.21.2. Note that $\text{pr}^{-1}\mathcal{F}$ satisfies properties (a) and (b) also (by Lemmas 96.12.3 and 96.9.3), hence these two presheaves are equal by (2). This immediately implies (3).
$\square$
Comments (2)
Comment #3180 by anonymous on
Comment #3293 by Johan on