The Stacks project

Proof. Let $\{ x_ i \to x\} $ be an fppf covering of $\mathcal{X}$ lying over the fppf covering $\{ f_ i : U_ i \to U\} $ of schemes over $S$. By assumption the restriction $\mathcal{G} = \mathcal{F}|_{U_{\acute{e}tale}}$ is quasi-coherent and the comparison maps $f_{i, small}^*\mathcal{G} \to \mathcal{F}|_{U_{i, {\acute{e}tale}}}$ are isomorphisms. Hence the sheaf condition for $\mathcal{F}$ and the covering $\{ x_ i \to x\} $ is equivalent to the sheaf condition for $\mathcal{G}^ a$ on $(\mathit{Sch}/U)_{fppf}$ and the covering $\{ U_ i \to U\} $ which holds by Descent, Lemma 35.8.1. $\square$

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

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). $\square$

Proof. Note that assumption (1) implies that if $\{ x_ i \to x\} $ is a covering of $\mathcal{X}_\tau $ and $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}')$, then we have $x_ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}')$. Hence we see that $\mathcal{X}' \to \mathcal{X}$ is continuous and cocontinuous as the coverings of objects of $\mathcal{X}'_\tau $ agree with their coverings seen as objects of $\mathcal{X}_\tau $. We obtain the morphism $g$ and the functor $g^{-1}$ is identified with the restriction functor, see Sites, Lemma 7.21.5.

In particular, if $\{ x_ i \to x\} $ is a covering in $\mathcal{X}'_\tau $, then for any abelian sheaf $\mathcal{F}$ on $\mathcal{X}$ then

Thus if $\mathcal{I}$ is an injective abelian sheaf on $\mathcal{X}_\tau $ then we see that the higher Čech cohomology groups are zero (Cohomology on Sites, Lemma 21.10.2). Hence $H^ p(x, g^{-1}\mathcal{I}) = 0$ for all objects $x$ of $\mathcal{X}'$ (Cohomology on Sites, Lemma 21.10.9). In other words injective abelian sheaves on $\mathcal{X}_\tau $ are right acyclic for the functor $H^0(x, g^{-1}-)$. It follows that $H^ p(x, g^{-1}\mathcal{F}) = H^ p(x, \mathcal{F})$ for all $\mathcal{F} \in \textit{Ab}(\mathcal{X})$ and all $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}')$.

Choose an object $x \in \mathcal{X}'$ lying over a scheme $U$ as in assumption (2). In particular $\mathcal{X}/x \to \mathcal{X}$ is a morphism of algebraic stacks which representable by algebraic spaces, surjective, and smooth. (Note that $\mathcal{X}/x$ is equivalent to $(\mathit{Sch}/U)_{fppf}$, see Lemma 96.9.1.) The map of sheaves

in $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau )$ is surjective. Namely, for any object $x'$ of $\mathcal{X}$ there exists a $\tau $-covering $\{ x'_ i \to x'\} $ such that there exist morphisms $x'_ i \to x$, see Lemma 96.19.10. Since $g$ is exact, the map of sheaves

in $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}'_\tau )$ is surjective also. Let $h_{x, n}$ be the $(n + 1)$-fold product $h_ x \times \ldots \times h_ x$. Then we have spectral sequences

and

see Cohomology on Sites, Lemma 21.13.2.

Case I: $\mathcal{X}$ has a final object $x$ which is also an object of $\mathcal{X}'$. This case follows immediately from the discussion in the second paragraph above.

Case II: $\mathcal{X}$ is representable by an algebraic space $F$. In this case the sheaves $h_{x, n}$ are representable by an object $x_ n$ in $\mathcal{X}$. (Namely, if $\mathcal{S}_ F = \mathcal{X}$ and $x : U \to F$ is the given object, then $h_{x, n}$ is representable by the object $U \times _ F \ldots \times _ F U \to F$ of $\mathcal{S}_ F$.) It follows that $H^ q(h_{x, p}, \mathcal{F}) = H^ q(x_ p, \mathcal{F})$. The morphisms $x_ n \to x$ lie over smooth morphisms of schemes, hence $x_ n \in \mathcal{X}'$ for all $n$. Hence $H^ q(g^{-1}h_{x, p}, g^{-1}\mathcal{F}) = H^ q(x_ p, g^{-1}\mathcal{F})$. Thus in the two spectral sequences (96.23.3.1) and (96.23.3.2) above the $E_1^{p, q}$ terms agree by the discussion in the second paragraph. The lemma follows in Case II as well.

Case III: $\mathcal{X}$ is an algebraic stack. We claim that in this case the cohomology groups $H^ q(h_{x, p}, \mathcal{F})$ and $H^ q(g^{-1}h_{x, n}, g^{-1}\mathcal{F})$ agree by Case II above. Once we have proved this the result will follow as before.

Namely, consider the category $\mathcal{X}/h_{x, n}$, see Sites, Lemma 7.30.3. Since $h_{x, n}$ is the $(n + 1)$-fold product of $h_ x$ an object of this category is an $(n + 2)$-tuple $(y, s_0, \ldots , s_ n)$ where $y$ is an object of $\mathcal{X}$ and each $s_ i : y \to x$ is a morphism of $\mathcal{X}$. This is a category over $(\mathit{Sch}/S)_{fppf}$. There is an equivalence

over $(\mathit{Sch}/S)_{fppf}$. Namely, if $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ also denotes the $1$-morphism associated with $x$ and $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ the structure functor, then we can think of $(y, s_0, \ldots , s_ n)$ as $(y, f_0, \ldots , f_ n, \alpha _0, \ldots , \alpha _ n)$ where $y$ is an object of $\mathcal{X}$, $f_ i : p(y) \to p(x)$ is a morphism of schemes, and $\alpha _ i : y \to x(f_ i)$ an isomorphism. The category of $2n+3$-tuples $(y, f_0, \ldots , f_ n, \alpha _0, \ldots , \alpha _ n)$ is an incarnation of the $(n + 1)$-fold fibred product $\mathcal{U}_ n$ of algebraic stacks displayed above, as we discussed in Section 96.18. By Cohomology on Sites, Lemma 21.13.3 we have

Finally, we discuss the “primed” analogue of this. Namely, $\mathcal{X}'/h_{x, n}$ corresponds, via the equivalence above to the full subcategory $\mathcal{U}'_ n \subset \mathcal{U}_ n$ consisting of those tuples $(y, f_0, \ldots , f_ n, \alpha _0, \ldots , \alpha _ n)$ with $y \in \mathcal{X}'$. Hence certainly property (1) of the statement of the lemma holds for the inclusion $\mathcal{U}'_ n \subset \mathcal{U}_ n$. To see property (2) choose an object $\xi = (y, s_0, \ldots , s_ n)$ which lies over a scheme $W$ such that $(\mathit{Sch}/W)_{fppf} \to \mathcal{U}_ n$ is smooth and surjective (this is possible as $\mathcal{U}_ n$ is an algebraic stack). Then $(\mathit{Sch}/W)_{fppf} \to \mathcal{U}_ n \to (\mathit{Sch}/U)_{fppf}$ is smooth as a composition of base changes of the morphism $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$, see Algebraic Stacks, Lemmas 94.10.6 and 94.10.5. Thus axiom (1) for $\mathcal{X}$ implies that $y$ is an object of $\mathcal{X}'$ whence $\xi $ is an object of $\mathcal{U}'_ n$. Using again

we now can use Case II for $\mathcal{U}'_ n \subset \mathcal{U}_ n$ to conclude. $\square$