The Stacks project

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 ├ętale morphism $U \to T$. Let $U_ i$ be a scheme and let $U_ i \to T_ i \times _ T U$ be a surjective ├ętale 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. $\square$

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