The Stacks project

103.4 Cohomology and the lisse-étale and flat-fppf sites

We have already seen that cohomology of a sheaf on an algebraic stack $\mathcal{X}$ can be computed on flat-fppf site. In this section we prove the same is true for (possibly) unbounded objects of the direct category of $\mathcal{X}$.

Lemma 103.4.1. Let $\mathcal{X}$ be an algebraic stack. We have $Lg_!\mathbf{Z} = \mathbf{Z}$ for either $Lg_!$ as in Lemma 103.3.1 part (1) or $Lg_!$ as in Lemma 103.3.1 part (3).

Proof. We prove this for the comparison between the flat-fppf site with the fppf site; the case of the lisse-étale site is exactly the same. We have to show that $H^ i(Lg_!\mathbf{Z})$ is $0$ for $i \not= 0$ and that the canonical map $H^0(Lg_!\mathbf{Z}) \to \mathbf{Z}$ is an isomorphism. Let $f : \mathcal{U} \to \mathcal{X}$ be a surjective, flat morphism where $\mathcal{U}$ is a scheme such that $f$ is also locally of finite presentation. (For example, pick a presentation $U \to \mathcal{X}$ and let $\mathcal{U}$ be the algebraic stack corresponding to $U$.) By Sheaves on Stacks, Lemmas 95.19.6 and 95.19.10 it suffices to show that the pullback $f^{-1}H^ i(Lg_!\mathbf{Z})$ is $0$ for $i \not= 0$ and that the pullback $H^0(Lg_!\mathbf{Z}) \to f^{-1}\mathbf{Z}$ is an isomorphism. By Lemma 103.3.2 we find $f^{-1}Lg_!\mathbf{Z} = L(g')_!\mathbf{Z}$ where $g' : \mathop{\mathit{Sh}}\nolimits (\mathcal{U}_{flat, fppf}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{U}_{fppf})$ is the corresponding comparision morphism for $\mathcal{U}$. This reduces us to the case studied in the next paragraph.

Assume $\mathcal{X} = (\mathit{Sch}/X)_{fppf}$ for some scheme $X$. In this case the category $\mathcal{X}_{flat, fppf}$ has a final object $e$, namely $X/X$, and moreover the functor $u : \mathcal{X}_{flat, fppf} \to \mathcal{X}_{fppf}$ sends $e$ to the final object. Since $\mathbf{Z}$ is the free abelian sheaf on the final object (provided the final object exists) we find that $Lg_!\mathbf{Z} = \mathbf{Z}$ by the very construction of $Lg_!$ in Cohomology on Sites, Lemma 21.37.2. $\square$

Lemma 103.4.2. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma 103.3.1.

  1. For $K$ in $D(\mathcal{X}_{\acute{e}tale})$ we have

    1. $R\Gamma (\mathcal{X}_{\acute{e}tale}, K) = R\Gamma (\mathcal{X}_{lisse,{\acute{e}tale}}, g^{-1}K)$, and

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

  2. For $K$ in $D(\mathcal{X}_{fppf})$ we have

    1. $R\Gamma (\mathcal{X}_{fppf}, K) = R\Gamma (\mathcal{X}_{flat,fppf}, g^{-1}K)$, and

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

In both cases, the same holds for modules, since we have $g^{-1} = g^*$ and there is no difference in computing cohomology by Cohomology on Sites, Lemma 21.20.7.

Proof. We prove this for the comparison between the flat-fppf site with the fppf site; the case of the lisse-étale site is exactly the same. By Lemma 103.4.1 we have $Lg_!\mathbf{Z} = \mathbf{Z}$. Then we obtain

\begin{align*} R\Gamma (\mathcal{X}_{fppf}, K) & = R\mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, K) \\ & = R\mathop{\mathrm{Hom}}\nolimits (Lg_!\mathbf{Z}, K) \\ & = R\mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, g^{-1}K) \\ & = R\Gamma (\mathcal{X}_{lisse,{\acute{e}tale}}, g^{-1}K) \end{align*}

This proves (1)(a). 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 $U_{lisse, {\acute{e}tale}}$. $\square$


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