Lemma 74.7.1. Let $S$ be a scheme. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be an fpqc covering of algebraic spaces over $S$. Suppose that for each $i$ we have an open subspace $W_ i \subset T_ i$ such that for all $i, j \in I$ we have $\text{pr}_0^{-1}(W_ i) = \text{pr}_1^{-1}(W_ j)$ as open subspaces of $T_ i \times _ T T_ j$. Then there exists a unique open subspace $W \subset T$ such that $W_ i = f_ i^{-1}(W)$ for each $i$.
74.7 Fpqc coverings
This section is the analogue of Descent, Section 35.13. At the moment we do not know if all of the material for fpqc coverings of schemes holds also for algebraic spaces.
Proof. By Topologies on Spaces, Lemma 73.9.5 we may assume each $T_ i$ is a scheme. Choose a scheme $U$ and a surjective étale morphism $U \to T$. Then $\{ T_ i \times _ T U \to U\} $ is an fpqc covering of $U$ and $T_ i \times _ T U$ is a scheme for each $i$. Hence we see that the collection of opens $W_ i \times _ T U$ comes from a unique open subscheme $W' \subset U$ by Descent, Lemma 35.13.6. As $U \to X$ is open we can define $W \subset X$ the Zariski open which is the image of $W'$, see Properties of Spaces, Section 66.4. We omit the verification that this works, i.e., that $W_ i$ is the inverse image of $W$ for each $i$. $\square$
Lemma 74.7.2. Let $S$ be a scheme. Let $\{ T_ i \to T\} $ be an fpqc covering of algebraic spaces over $S$, see Topologies on Spaces, Definition 73.9.1. Then given an algebraic space $B$ over $S$ the sequence
\[ \xymatrix{ \mathop{\mathrm{Mor}}\nolimits _ S(T, B) \ar[r] & \prod \nolimits _ i \mathop{\mathrm{Mor}}\nolimits _ S(T_ i, B) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \nolimits _{i, j} \mathop{\mathrm{Mor}}\nolimits _ S(T_ i \times _ T T_ j, B) } \]
is an equalizer diagram. In other words, every representable functor on the category of algebraic spaces over $S$ satisfies the sheaf condition for fpqc coverings.
Proof. We know this is true if $\{ T_ i \to T\} $ is an fpqc covering of schemes, see Properties of Spaces, Proposition 66.17.1. This is the key fact and we encourage the reader to skip the rest of the proof which is formal. Choose a scheme $U$ and a surjective étale morphism $U \to T$. Let $U_ i$ be a scheme and let $U_ i \to T_ i \times _ T U$ be a surjective étale morphism. Then $\{ U_ i \to U\} $ is an fpqc covering. This follows from Topologies on Spaces, Lemmas 73.9.3 and 73.9.4. By the above we have the result for $\{ U_ i \to U\} $.
What this means is the following: Suppose that $b_ i : T_ i \to B$ is a family of morphisms with $b_ i \circ \text{pr}_0 = b_ j \circ \text{pr}_1$ as morphisms $T_ i \times _ T T_ j \to B$. Then we let $a_ i : U_ i \to B$ be the composition of $U_ i \to T_ i$ with $b_ i$. By what was said above we find a unique morphism $a : U \to B$ such that $a_ i$ is the composition of $a$ with $U_ i \to U$. The uniqueness guarantees that $a \circ \text{pr}_0 = a \circ \text{pr}_1$ as morphisms $U \times _ T U \to B$. Then since $T = U/(U \times _ T U)$ as a sheaf, we find that $a$ comes from a unique morphism $b : T \to B$. Chasing diagrams we find that $b$ is the morphism we are looking for. $\square$
Comments (1)
Comment #1780 by Matthieu Romagny on