The Stacks project

Lemma 96.19.12. Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, \linebreak[0] fppf\} $. Assume

1. $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$,

2. 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 $\mathcal{U}$,

3. the category $\mathcal{U}$ has equalizers, and

4. the functor $f$ is faithful.

Then there is a first quadrant spectral sequence in $\textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y})$

where all higher direct images are computed in the $\tau $-topology.