The Stacks project

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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