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74.23 Descent data in terms of sheaves

This section is the analogue of Descent, Section 35.39. It is slightly different as algebraic spaces are already sheaves.

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,

  1. the algebraic space $X_ i \times _ X F$ on the right hand side corresponds to $V_ i$ on the left hand side, and

  2. 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. $\square$

[1] We will see later that this is always the case if $I$ is not too large, see Bootstrap, Lemma 80.11.3.

Comments (2)

Comment #978 by David Holmes on

The link in the footnote to tag 0ADV (`Bootstrap, lemma 62.11.2') seems to be broken.

Comment #998 by on

OK, thanks. This is now fixed.

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