Lemma 35.39.1. Let \tau \in \{ Zariski, fppf, {\acute{e}tale}, smooth, syntomic\} 1. Let \mathit{Sch}_\tau be a big \tau -site. Let S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\tau ). Let \{ S_ i \to S\} _{i \in I} be a covering in the site (\mathit{Sch}/S)_\tau . There is an equivalence of categories
\left\{ \begin{matrix} \text{descent data }(X_ i, \varphi _{ii'})\text{ such that}
\\ \text{each }X_ i \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_\tau )
\end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{sheaves }F\text{ on }(\mathit{Sch}/S)_\tau \text{ such that}
\\ \text{each }h_{S_ i} \times F\text{ is representable}
\end{matrix} \right\} .
Moreover,
the objects representing h_{S_ i} \times F on the right hand side correspond to the schemes X_ i on the left hand side, and
the sheaf F is representable if and only if the corresponding descent datum (X_ i, \varphi _{ii'}) is effective.
Proof.
We have seen in Section 35.13 that representable presheaves are sheaves on the site (\mathit{Sch}/S)_\tau . Moreover, the Yoneda lemma (Categories, Lemma 4.3.5) guarantees that maps between representable sheaves correspond one to one with maps between the representing objects. We will use these remarks without further mention during the proof.
Let us construct the functor from right to left. Let F be a sheaf on (\mathit{Sch}/S)_\tau such that each h_{S_ i} \times F is representable. In this case let X_ i be a representing object in (\mathit{Sch}/S)_\tau . It comes equipped with a morphism X_ i \to S_ i. Then both X_ i \times _ S S_{i'} and S_ i \times _ S X_{i'} represent the sheaf h_{S_ i} \times F \times h_{S_{i'}} and hence we obtain an isomorphism
\varphi _{ii'} : X_ i \times _ S S_{i'} \to S_ i \times _ S X_{i'}
It is straightforward to see that the maps \varphi _{ii'} are morphisms over S_ i \times _ S S_{i'} and satisfy the cocycle condition. The functor from right to left is given by this construction F \mapsto (X_ i, \varphi _{ii'}).
Let us construct a functor from left to right. For each i denote F_ i the sheaf h_{X_ i}. The isomorphisms \varphi _{ii'} give isomorphisms
\varphi _{ii'} : F_ i \times h_{S_{i'}} \longrightarrow h_{S_ i} \times F_{i'}
over h_{S_ i} \times h_{S_{i'}}. Set F equal to the coequalizer in the following diagram
\xymatrix{ \coprod _{i, i'} F_ i \times h_{S_{i'}} \ar@<1ex>[rr]^-{\text{pr}_0} \ar@<-1ex>[rr]_-{\text{pr}_1 \circ \varphi _{ii'}} & & \coprod _ i F_ i \ar[r] & F }
The cocycle condition guarantees that h_{S_ i} \times F is isomorphic to F_ i and hence representable. The functor from left to right is given by this construction (X_ i, \varphi _{ii'}) \mapsto F.
We omit the verification that these constructions are mutually quasi-inverse functors. The final statements (1) and (2) follow from the constructions.
\square
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