Lemma 59.39.2. Let S be a scheme. Let \tau \in \{ Zariski, {\acute{e}tale}\} . Consider the morphism
\pi _ S : (\mathit{Sch}/S)_\tau \longrightarrow S_\tau
of Topologies, Lemma 34.3.14 or 34.4.14. Let \mathcal{F} be a sheaf on S_\tau . Then \pi _ S^{-1}\mathcal{F} is given by the rule
(\pi _ S^{-1}\mathcal{F})(T) = \Gamma (T_\tau , f_{small}^{-1}\mathcal{F})
where f : T \to S. Moreover, \pi _ S^{-1}\mathcal{F} satisfies the sheaf condition with respect to fpqc coverings.
Proof.
Observe that we have a morphism i_ f : \mathop{\mathit{Sh}}\nolimits (T_\tau ) \to \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}/S)_\tau ) such that \pi _ S \circ i_ f = f_{small} as morphisms T_\tau \to S_\tau , see Topologies, Lemmas 34.3.13, 34.3.17, 34.4.13, and 34.4.17. Since pullback is transitive we see that i_ f^{-1} \pi _ S^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F} as desired.
Let \{ g_ i : T_ i \to T\} _{i \in I} be an fpqc covering. The final statement means the following: Given a sheaf \mathcal{G} on T_\tau and given sections s_ i \in \Gamma (T_ i, g_{i, small}^{-1}\mathcal{G}) whose pullbacks to T_ i \times _ T T_ j agree, there is a unique section s of \mathcal{G} over T whose pullback to T_ i agrees with s_ i.
Let V \to T be an object of T_\tau and let t \in \mathcal{G}(V). For every i there is a largest open W_ i \subset T_ i \times _ T V such that the pullbacks of s_ i and t agree as sections of the pullback of \mathcal{G} to W_ i \subset T_ i \times _ T V, see Lemma 59.39.1. Because s_ i and s_ j agree over T_ i \times _ T T_ j we find that W_ i and W_ j pullback to the same open over T_ i \times _ T T_ j \times _ T V. By Descent, Lemma 35.13.6 we find an open W \subset V whose inverse image to T_ i \times _ T V recovers W_ i.
By construction of g_{i, small}^{-1}\mathcal{G} there exists a \tau -covering \{ T_{ij} \to T_ i\} _{j \in J_ i}, for each j an open immersion or étale morphism V_{ij} \to T, a section t_{ij} \in \mathcal{G}(V_{ij}), and commutative diagrams
\xymatrix{ T_{ij} \ar[r] \ar[d] & V_{ij} \ar[d] \\ T_ i \ar[r] & T }
such that s_ i|_{T_{ij}} is the pullback of t_{ij}. In other words, after replacing the covering \{ T_ i \to T\} by \{ T_{ij} \to T\} we may assume there are factorizations T_ i \to V_ i \to T with V_ i \in \mathop{\mathrm{Ob}}\nolimits (T_\tau ) and sections t_ i \in \mathcal{G}(V_ i) pulling back to s_ i over T_ i. By the result of the previous paragraph we find opens W_ i \subset V_ i such that t_ i|_{W_ i} “agrees with” every s_ j over T_ j \times _ T W_ i. Note that T_ i \to V_ i factors through W_ i. Hence \{ W_ i \to T\} is a \tau -covering and the lemma is proven.
\square
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