Lemma 74.23.1. Let S be a scheme. Let \{ X_ i \to X\} _{i \in I} be an fppf covering of algebraic spaces over S (Topologies on Spaces, Definition 73.7.1). There is an equivalence of categories
\left\{ \begin{matrix} \text{descent data }(V_ i, \varphi _{ij})
\\ \text{relative to }\{ X_ i \to X\}
\end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{sheaves }F\text{ on }(\mathit{Sch}/S)_{fppf}\text{ endowed}
\\ \text{with a map }F \to X\text{ such that each}
\\ X_ i \times _ X F\text{ is an algebraic space}
\end{matrix} \right\} .
Moreover,
the algebraic space X_ i \times _ X F on the right hand side corresponds to V_ i on the left hand side, and
the sheaf F is an algebraic space1 if and only if the corresponding descent datum (X_ i, \varphi _{ij}) is effective.
Proof.
Let us construct the functor from right to left. Let F \to X be a map of sheaves on (\mathit{Sch}/S)_{fppf} such that each V_ i = X_ i \times _ X F is an algebraic space. We have the projection V_ i \to X_ i. Then both V_ i \times _ X X_ j and X_ i \times _ X V_ j represent the sheaf X_ i \times _ X F \times _ X X_ j and hence we obtain an isomorphism
\varphi _{ii'} : V_ i \times _ X X_ j \to X_ i \times _ X V_ j
It is straightforward to see that the maps \varphi _{ij} are morphisms over X_ i \times _ X X_ j and satisfy the cocycle condition. The functor from right to left is given by this construction F \mapsto (V_ i, \varphi _{ij}).
Let us construct a functor from left to right. The isomorphisms \varphi _{ij} give isomorphisms
\varphi _{ij} : V_ i \times _ X X_ j \longrightarrow X_ i \times _ X V_ j
over X_ i \times X_ j. Set F equal to the coequalizer in the following diagram
\xymatrix{ \coprod _{i, i'} V_ i \times _ X X_ j \ar@<1ex>[rr]^-{\text{pr}_0} \ar@<-1ex>[rr]_-{\text{pr}_1 \circ \varphi _{ij}} & & \coprod _ i V_ i \ar[r] & F }
The cocycle condition guarantees that F comes with a map F \to X and that X_ i \times _ X F is isomorphic to V_ i. The functor from left to right is given by this construction (V_ i, \varphi _{ij}) \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
Comments (1)
Comment #1591 by S. Carnahan on
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