Algebraic space

In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Artin (1969, 1971) 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).

There are two common ways to define algebraic spaces: they can be defined as either quotients of schemes by etale equivalence relations, or as sheaves on a big etale site that are locally isomorphic to schemes. These two definitions are essentially equivalent.

An algebraic space X comprises a scheme U and a closed subscheme R ⊂ U × U satisfying the following two conditions:

Some authors, such as Knutson, add an extra condition that an algebraic space has to be quasi-separated, meaning that the diagonal map is quasi-compact.

One can always assume that R and U are affine schemes. Doing so means that the theory of algebraic spaces is not dependent on the full theory of schemes, and can indeed be used as a (more general) replacement of that theory.

If R is the trivial equivalence relation over each connected component of U (i.e. for all x, y belonging to the same connected component of U, we have xRy if and only if x=y), then the algebraic space will be a scheme in the usual sense. Since a general algebraic space X does not satisfy this requirement, it allows a single connected component of U to cover X with many "sheets". The point set underlying the algebraic space X is then given by |U| / |R| as a set of equivalence classes.

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