The Stacks project

Lemma 34.13.4. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Let $S$ be a scheme contained in a big site $\mathit{Sch}_\tau $. Let $F : (\mathit{Sch}/S)_\tau ^{opp} \to \textit{Sets}$ be a $\tau $-sheaf satisfying property (b) of Lemma 34.13.1 with $\mathcal{C} = (\mathit{Sch}/S)_\tau $. Then the extension $F'$ of $F$ to the category of all schemes over $S$ satisfies the sheaf condition for all $\tau $-coverings.

Proof. This follows from Lemma 34.13.3 applied with $\mathcal{C} = (\mathit{Sch}/S)_\tau $. Conditions (1), (2), (a), and (b) of Lemma 34.13.1 hold; we omit the details. Thus we get our unique extension $F'$ to the category of all schemes over $S$. Finally, observe that any standard $\tau $-covering is tautologically equivalent to a covering in $(\mathit{Sch}/S)_\tau $, see Sets, Lemma 3.9.9 as well as Lemmas 34.3.6, 34.4.7, 34.5.7, 34.6.7, and 34.7.7. By Sites, Lemma 7.8.4 the sheaf property passes through tautological equivalence of coverings. Hence the fact that $F$ is a $\tau $-sheaf implies that property (c) of Lemma 34.13.3 holds and we conclude. $\square$


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