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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