Algebraic space
generalization of a scheme; an étale equivalence relation on a scheme; a sheaf of sets on the big étale site with representable diagonal morphism and a surjective étale morphism from a scheme
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively,
schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology.
The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf. the Keel–Mori theorem).
Definition
There are two common ways to define algebraic spaces: they can be defined as either quotients of schemes by étale equivalence relations, or as sheaves on a big étale site that are locally isomorphic to schemes. These two definitions are essentially equivalent.
Algebraic spaces as quotients of schemes
An algebraic space X comprises a scheme U and a closed subscheme R ⊆ U × U satisfying the following two conditions:
1. R is an equivalence relation as a subset of U × U
2. The projections pi: R → U onto each factor are étale maps.
The public source identifies “Algebraic space” as generalization of a scheme; an étale equivalence relation on a scheme; a sheaf of sets on the big étale site with representable diagonal morphism and a surjective étale morphism from a scheme. This brief keeps that definition visible, then builds a research path around Algebraic, space and generalization.
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