Proposition 96.19.7 (06XE)—The Stacks project

Proposition 96.19.7. 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\} $. If

$\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau $, and
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 $\mathcal{U}$,

then the extended relative Čech complex

\[ \ldots \to 0 \to \mathcal{F} \to f_{0, *}f_0^{-1}\mathcal{F} \to f_{1, *}f_1^{-1}\mathcal{F} \to f_{2, *}f_2^{-1}\mathcal{F} \to \ldots \]

is exact in $\textit{Ab}(\mathcal{X}_\tau )$.

Proof. By Lemma 96.19.6 it suffices to check exactness after pulling back to $\mathcal{U}$. By Lemma 96.19.5 the pullback of the extended relative Čech complex is isomorphic to the extend relative Čech complex for the morphism $\mathcal{U} \times _\mathcal {X} \mathcal{U} \to \mathcal{U}$ and an abelian sheaf on $\mathcal{U}_\tau $. Since there is a section $\Delta _{\mathcal{U}/\mathcal{X}} : \mathcal{U} \to \mathcal{U} \times _\mathcal {X} \mathcal{U}$ exactness follows from Lemma 96.19.3. $\square$

The Stacks project

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