The Stacks project

64.9 Presentations of algebraic spaces

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

Lemma 64.9.1. 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 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 64.9.2. 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 64.9.3. 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 64.10.5.

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