## 63.9 Presentations of algebraic spaces

Given an algebraic space we can find a “presentation” of it.

Lemma 63.9.1. Let $F$ be an algebraic space over $S$. Let $f : U \to F$ be a surjective étale morphism from a scheme to $F$. Set $R = U \times _ F U$. Then

1. $j : R \to U \times _ S U$ defines an equivalence relation on $U$ over $S$ (see Groupoids, Definition 39.3.1).

2. the morphisms $s, t : R \to U$ are étale, and

3. the diagram

$\xymatrix{ R \ar@<1ex>[r] \ar@<-1ex>[r] & U \ar[r] & F }$

is a coequalizer diagram in $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})$.

Proof. Let $T/S$ be an object of $(\mathit{Sch}/S)_{fppf}$. Then $R(T) = \{ (a, b) \in U(T) \times U(T) \mid f \circ a = f \circ b\}$ which is clearly defines an equivalence relation on $U(T)$. The morphisms $s, t : R \to U$ are étale because the morphism $U \to F$ is étale.

To prove (3) we first show that $U \to F$ is a surjection of sheaves, see Sites, Definition 7.11.1. Let $\xi \in F(T)$ with $T$ as above. Let $V = T \times _{\xi , F, f}U$. By assumption $V$ is a scheme and $V \to T$ is surjective étale. Hence $\{ V \to T\}$ is a covering for the fppf topology. Since $\xi |_ V$ factors through $U$ by construction we conclude $U \to F$ is surjective. Surjectivity implies that $F$ is the coequalizer of the diagram by Sites, Lemma 7.11.3. $\square$

This lemma suggests the following definitions.

Definition 63.9.2. Let $S$ be a scheme. Let $U$ be a scheme over $S$. An étale equivalence relation on $U$ over $S$ is an equivalence relation $j : R \to U \times _ S U$ such that $s, t : R \to U$ are étale morphisms of schemes.

Definition 63.9.3. Let $F$ be an algebraic space over $S$. A presentation of $F$ is given by a scheme $U$ over $S$ and an étale equivalence relation $R$ on $U$ over $S$, and a surjective étale morphism $U \to F$ such that $R = U \times _ F U$.

Equivalently we could ask for the existence of an isomorphism

$U/R \cong F$

where the quotient $U/R$ is as defined in Groupoids, Section 39.20. To construct algebraic spaces we will study the converse question, namely, for which equivalence relations the quotient sheaf $U/R$ is an algebraic space. It will finally turn out this is always the case if $R$ is an étale equivalence relation on $U$ over $S$, see Theorem 63.10.5.

## 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Your email address will not be published.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 0261. Beware of the difference between the letter 'O' and the digit '0'.