96.20 Cohomology on algebraic stacks
Let $\mathcal{X}$ be an algebraic stack over $S$. In the sections above we have seen how to define sheaves for the étale, ..., fppf topologies on $\mathcal{X}$. In fact, we have constructed a site $\mathcal{X}_\tau $ for each $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. There is a notion of an abelian sheaf $\mathcal{F}$ on these sites. In the chapter on cohomology of sites we have explained how to define cohomology. Putting all of this together, let's define the derived global sections or total cohomology
\[ R\Gamma _{Zar}(\mathcal{X}, \mathcal{F}), R\Gamma _{\acute{e}tale}(\mathcal{X}, \mathcal{F}), \ldots , R\Gamma _{fppf}(\mathcal{X}, \mathcal{F}) \]
as $\Gamma (\mathcal{X}_\tau , \mathcal{I}^\bullet )$ where $\mathcal{F} \to \mathcal{I}^\bullet $ is an injective resolution in $\textit{Ab}(\mathcal{X}_\tau )$. The $i$th cohomology group of $\mathcal{F}$ is the $i$th cohomology of the total cohomology. We will denote this
\[ H^ i_{Zar}(\mathcal{X}, \mathcal{F}), H^ i_{\acute{e}tale}(\mathcal{X}, \mathcal{F}), \ldots , H^ i_{fppf}(\mathcal{X}, \mathcal{F}). \]
It will turn out that $H^ i_{\acute{e}tale}= H^ i_{smooth}$ because of More on Morphisms, Lemma 37.38.7.
If $\mathcal{F}$ is a presheaf of $\mathcal{O}_\mathcal {X}$-modules which is a sheaf in the $\tau $-topology, then we use injective resolutions in $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$ to compute its total cohomology, resp. cohomology groups; the end result is quasi-isomorphic, resp. isomorphic to the cohomology of $\mathcal{F}$ viewed as a sheaf of abelian groups by the very general Cohomology on Sites, Lemma 21.12.4.
So far our only tool to compute cohomology groups is the result on Čech complexes proved above. We rephrase it here in the language of algebraic stacks for the étale and the fppf topology. Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of algebraic stacks. Recall that
\[ f_ p : \mathcal{U}_ p = \mathcal{U} \times _\mathcal {X} \ldots \times _\mathcal {X} \mathcal{U} \longrightarrow \mathcal{X} \]
is the structure morphism where there are $(p + 1)$-factors. Also, recall that a sheaf on $\mathcal{X}$ is a sheaf for the fppf topology. Note that if $\mathcal{U}$ is an algebraic space, then $f : \mathcal{U} \to \mathcal{X}$ is representable by algebraic spaces, see Algebraic Stacks, Lemma 94.10.11. Thus the proposition applies in particular to a smooth cover of the algebraic stack $\mathcal{X}$ by a scheme.
Proposition 96.20.1. Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of algebraic stacks.
Let $\mathcal{F}$ be an abelian étale sheaf on $\mathcal{X}$. Assume that $f$ is representable by algebraic spaces, surjective, and smooth. Then there is a spectral sequence
\[ E_1^{p, q} = H^ q_{\acute{e}tale}(\mathcal{U}_ p, f_ p^{-1}\mathcal{F}) \Rightarrow H^{p + q}_{\acute{e}tale}(\mathcal{X}, \mathcal{F}) \]
Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{X}$. Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then there is a spectral sequence
\[ E_1^{p, q} = H^ q_{fppf}(\mathcal{U}_ p, f_ p^{-1}\mathcal{F}) \Rightarrow H^{p + q}_{fppf}(\mathcal{X}, \mathcal{F}) \]
Proof.
To see this we will check the hypotheses (1) – (4) of Lemma 96.19.8. The $1$-morphism $f$ is faithful by Algebraic Stacks, Lemma 94.15.2. This proves (4). Hypothesis (3) follows from the fact that $\mathcal{U}$ is an algebraic stack, see Lemma 96.17.2. To see (2) apply Lemma 96.19.10. Condition (1) is satisfied by fiat.
$\square$
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