Lemma 96.19.8. Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. Assume
$\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau $,
for every object $x$ of $\mathcal{X}$ there exists a covering $\{ x_ i \to x\} $ in $\mathcal{X}_\tau $ such that each $x_ i$ is isomorphic to $f(u_ i)$ for some object $u_ i$ of $\mathcal{U}$,
the category $\mathcal{U}$ has equalizers, and
the functor $f$ is faithful.
Then there is a first quadrant spectral sequence of abelian groups
\[ E_1^{p, q} = H^ q((\mathcal{U}_ p)_\tau , f_ p^{-1}\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}_\tau , \mathcal{F}) \]
converging to the cohomology of $\mathcal{F}$ in the $\tau $-topology.
Proof.
Before we start the proof we make some remarks. By Lemma 96.17.4 (and induction) all of the categories fibred in groupoids $\mathcal{U}_ p$ have equalizers and all of the morphisms $f_ p : \mathcal{U}_ p \to \mathcal{X}$ are faithful. Let $\mathcal{I}$ be an injective object of $\textit{Ab}(\mathcal{X}_\tau )$. By Lemma 96.17.5 we see $f_ p^{-1}\mathcal{I}$ is an injective object of $\textit{Ab}((\mathcal{U}_ p)_\tau )$. Hence $f_{p, *}f_ p^{-1}\mathcal{I}$ is an injective object of $\textit{Ab}(\mathcal{X}_\tau )$ by Lemma 96.17.1. Hence Proposition 96.19.7 shows that the extended relative Čech complex
\[ \ldots \to 0 \to \mathcal{I} \to f_{0, *}f_0^{-1}\mathcal{I} \to f_{1, *}f_1^{-1}\mathcal{I} \to f_{2, *}f_2^{-1}\mathcal{I} \to \ldots \]
is an exact complex in $\textit{Ab}(\mathcal{X}_\tau )$ all of whose terms are injective. Taking global sections of this complex is exact and we see that the Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U} \to \mathcal{X}, \mathcal{I})$ is quasi-isomorphic to $\Gamma (\mathcal{X}_\tau , \mathcal{I})[0]$.
With these preliminaries out of the way consider the two spectral sequences associated to the double complex (see Homology, Section 12.25)
\[ \check{\mathcal{C}}^\bullet (\mathcal{U} \to \mathcal{X}, \mathcal{I}^\bullet ) \]
where $\mathcal{F} \to \mathcal{I}^\bullet $ is an injective resolution in $\textit{Ab}(\mathcal{X}_\tau )$. The discussion above shows that Homology, Lemma 12.25.4 applies which shows that $\Gamma (\mathcal{X}_\tau , \mathcal{I}^\bullet )$ is quasi-isomorphic to the total complex associated to the double complex. By our remarks above the complex $f_ p^{-1}\mathcal{I}^\bullet $ is an injective resolution of $f_ p^{-1}\mathcal{F}$. Hence the other spectral sequence is as indicated in the lemma.
$\square$
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