96.16 Cohomology
Let S be a scheme and let \mathcal{X} be a category fibred in groupoids over (\mathit{Sch}/S)_{fppf}. For any \tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} the categories \textit{Ab}(\mathcal{X}_\tau ) and \textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X}) have enough injectives, see Injectives, Theorems 19.7.4 and 19.8.4. Thus we can use the machinery of Cohomology on Sites, Section 21.2 to define the cohomology groups
H^ p(\mathcal{X}_\tau , \mathcal{F}) = H^ p_\tau (\mathcal{X}, \mathcal{F}) \quad \text{and}\quad H^ p(x, \mathcal{F}) = H^ p_\tau (x, \mathcal{F})
for any x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}) and any object \mathcal{F} of \textit{Ab}(\mathcal{X}_\tau ) or \textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X}). Moreover, if f : \mathcal{X} \to \mathcal{Y} is a 1-morphism of categories fibred in groupoids over (\mathit{Sch}/S)_{fppf}, then we obtain the higher direct images R^ if_*\mathcal{F} in \textit{Ab}(\mathcal{Y}_\tau ) or \textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y}). Of course, as explained in Cohomology on Sites, Section 21.3 there are also derived versions of H^ p(-) and R^ if_*.
Lemma 96.16.1. Let S be a scheme. Let \mathcal{X} be a category fibred in groupoids over (\mathit{Sch}/S)_{fppf}. Let \tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} . Let x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}) be an object lying over the scheme U. Let \mathcal{F} be an object of \textit{Ab}(\mathcal{X}_\tau ) or \textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X}). Then
H^ p_\tau (x, \mathcal{F}) = H^ p((\mathit{Sch}/U)_\tau , x^{-1}\mathcal{F})
and if \tau = {\acute{e}tale}, then we also have
H^ p_{\acute{e}tale}(x, \mathcal{F}) = H^ p(U_{\acute{e}tale}, \mathcal{F}|_{U_{\acute{e}tale}}).
Proof.
The first statement follows from Cohomology on Sites, Lemma 21.7.1 and the equivalence of Lemma 96.9.4. The second statement follows from the first combined with Étale Cohomology, Lemma 59.20.3.
\square
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