The Stacks project

Lemma 34.36.1. 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,

  1. 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

  2. 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 34.10 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. $\square$

[1] The fact that fpqc is missing is not a typo. See discussion in Topologies, Section 33.9.

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