The Stacks project

Lemma 95.23.3. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Let $\tau = {\acute{e}tale}$ (resp. $\tau = fppf$). Let $\mathcal{X}' \subset \mathcal{X}$ be a full subcategory with the following properties

  1. if $x \to x'$ is a morphism of $\mathcal{X}$ which lies over a smooth (resp. flat and locally finitely presented) morphism of schemes and $x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}')$, then $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}')$, and

  2. there exists an object $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}')$ lying over a scheme $U$ such that the associated $1$-morphism $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ is smooth and surjective.

We get a site $\mathcal{X}'_\tau $ by declaring a covering of $\mathcal{X}'$ to be any family of morphisms $\{ x_ i \to x\} $ in $\mathcal{X}'$ which is a covering in $\mathcal{X}_\tau $. Then the inclusion functor $\mathcal{X}' \to \mathcal{X}_\tau $ is fully faithful, cocontinuous, and continuous, whence defines a morphism of topoi

\[ g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}'_\tau ) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau ) \]

and $H^ p(\mathcal{X}'_\tau , g^{-1}\mathcal{F}) = H^ p(\mathcal{X}_\tau , \mathcal{F})$ for all $p \geq 0$ and all $\mathcal{F} \in \textit{Ab}(\mathcal{X}_\tau )$.

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

\[ \check H^ p(\{ x_ i \to x\} , g^{-1}\mathcal{F}) = \check H^ p(\{ x_ i \to x\} , \mathcal{F}) \]

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 95.9.1.) The map of sheaves

\[ h_ x \longrightarrow * \]

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 95.19.10. Since $g$ is exact, the map of sheaves

\[ g^{-1}h_ x \longrightarrow * = g^{-1}* \]

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
\begin{equation} \label{stacks-sheaves-equation-spectral-sequence-one} E_1^{p, q} = H^ q(h_{x, p}, \mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}_\tau , \mathcal{F}) \end{equation}

\begin{equation} \label{stacks-sheaves-equation-spectral-sequence-two} E_1^{p, q} = H^ q(g^{-1}h_{x, p}, g^{-1}\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}'_\tau , g^{-1}\mathcal{F}) \end{equation}

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 ( and ( 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

\[ \mathcal{X}/h_{x, n} \longrightarrow (\mathit{Sch}/U)_{fppf} \times _\mathcal {X} \ldots \times _\mathcal {X} (\mathit{Sch}/U)_{fppf} =: \mathcal{U}_ n \]

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 95.18. By Cohomology on Sites, Lemma 21.13.3 we have

\[ H^ p(\mathcal{U}_ n, \mathcal{F}|_{\mathcal{U}_ n}) = H^ p(\mathcal{X}/h_{x, n}, \mathcal{F}|_{\mathcal{X}/h_{x, n}}) = H^ p(h_{x, n}, \mathcal{F}). \]

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 93.10.6 and 93.10.5. 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

\[ H^ p(\mathcal{U}'_ n, \mathcal{F}|_{\mathcal{U}'_ n}) = H^ p(\mathcal{X}'/h_{x, n}, \mathcal{F}|_{\mathcal{X}'/h_{x, n}}) = H^ p(g^{-1}h_{x, n}, g^{-1}\mathcal{F}). \]

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

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