Let \(S\) be a scheme, and regard \(S\)-schemes as sheaves on \((\Sch_{/S})_{\mathrm{et}}\) through their functors of points (D-ETFPPF).
(Étale equivalence relation)
Let \(U\) be an \(S\)-scheme. An étale equivalence relation on \(U\) is an \(S\)-scheme \(R\) with a monomorphism \((s, t) \colon R \to U \times_S U\) such that
for every \(S\)-scheme \(T\), the subset \(R(T) \subseteq U(T) \times U(T)\) is an equivalence relation on \(U(T)\), and
the two projections \(s, t \colon R \to U\) are étale.
Its quotient \(U/R\) is the sheafification, on \((\Sch_{/S})_{\mathrm{et}}\), of the presheaf \(T \mapsto U(T)/R(T)\).
(Algebraic space)
An algebraic space over \(S\) is a sheaf \(X\) on \((\Sch_{/S})_{\mathrm{et}}\) such that
the diagonal \(X \to X \times_S X\) is representable: for every \(S\)-scheme \(T\) and every \(T \to X \times_S X\), the fibre product \(X \times_{X \times_S X} T\) is a scheme, and
there are a scheme \(U\) and a morphism \(U \to X\) that is étale and surjective, meaning that for every scheme \(T \to X\) the base change \(U \times_X T \to T\), a morphism of schemes by condition 1, is étale and surjective.
A morphism \(U \to X\) as in condition 2 is an étale atlas.
If \(U \to X\) is an étale atlas of an algebraic space, then \(R = U \times_X U \to U \times_S U\) is an étale equivalence relation and \(X \cong U/R\). Conversely, the quotient \(U/R\) of any étale equivalence relation on a scheme \(U\) is an algebraic space, with étale atlas \(U \to U/R\).
Every scheme \(X\) is an algebraic space, with atlas \(\id \colon X \to X\) and \(R\) the diagonal. If a finite group \(G\) acts freely on an \(S\)-scheme \(U\), then \(R = U \times G \to U \times_S U\), \((u, g) \mapsto (u, ug)\), is an étale equivalence relation, and \(U/R\) is an algebraic space, the quotient \(U/G\); it is a scheme when, for instance, \(U\) is quasiprojective over a field.
Dependencies
D-ETFPPFThe étale and fppf topologies, and the small étale siteuses
Backlinks
D-ALGSTACKAlgebraic stacks, substacks, smooth stacks, and orbifoldsuses