Lemma 96.19.6. 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\} $. Let
\[ \mathcal{F} \to \mathcal{G} \to \mathcal{H} \]
be a complex in $\textit{Ab}(\mathcal{X}_\tau )$. Assume that
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 $\mathcal{U}$, and
$f^{-1}\mathcal{F} \to f^{-1}\mathcal{G} \to f^{-1}\mathcal{H}$ is exact.
Then the sequence $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ is exact.
Proof.
Let $x$ be an object of $\mathcal{X}$ lying over the scheme $T$. Consider the sequence $x^{-1}\mathcal{F} \to x^{-1}\mathcal{G} \to x^{-1}\mathcal{H}$ of abelian sheaves on $(\mathit{Sch}/T)_\tau $. It suffices to show this sequence is exact. By assumption there exists a $\tau $-covering $\{ T_ i \to T\} $ such that $x|_{T_ i}$ is isomorphic to $f(u_ i)$ for some object $u_ i$ of $\mathcal{U}$ over $T_ i$ and moreover the sequence $u_ i^{-1}f^{-1}\mathcal{F} \to u_ i^{-1}f^{-1}\mathcal{G} \to u_ i^{-1}f^{-1}\mathcal{H}$ of abelian sheaves on $(\mathit{Sch}/T_ i)_\tau $ is exact. Since $u_ i^{-1}f^{-1}\mathcal{F} = x^{-1}\mathcal{F}|_{(\mathit{Sch}/T_ i)_\tau }$ we conclude that the sequence $x^{-1}\mathcal{F} \to x^{-1}\mathcal{G} \to x^{-1}\mathcal{H}$ become exact after localizing at each of the members of a covering, hence the sequence is exact.
$\square$
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