The Stacks project

Lemma 73.7.2. Let $S$ be a scheme. Let $\{ T_ i \to T\} $ be an fpqc covering of algebraic spaces over $S$, see Topologies on Spaces, Definition 72.9.1. 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 65.17.1. 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 72.9.3 and 72.9.4. 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 (0)

There are also:

  • 1 comment(s) on Section 73.7: Fpqc coverings

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04P2. Beware of the difference between the letter 'O' and the digit '0'.