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 etale equivalence relations, or as sheaves on a big etale 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.
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.
Let Y be an algebraic space defined by an equivalence relation S ⊂ V × V. The set Hom(Y, X) of morphisms of algebraic spaces is then defined by the condition that it makes the descent sequence
exact (this definition is motivated by a descent theorem of Grothendieck for surjective étale maps of affine schemes). With these definitions, the algebraic spaces form a category.
Let U be an affine scheme over a field k defined by a system of polynomials g(x), x = (x1, ..., xn), let
denote the ring of algebraic functions in x over k, and let X = { R ⊂ U × U} be an algebraic space.
The appropriate stalks ÕX, x on X are then defined to be the local rings of algebraic functions defined by ÕU, u, where u ∈ U is a point lying over x and ÕU, u is the local ring corresponding to u of the ring
k{ x1, ..., xn} / (g)
of algebraic functions on U.
A point on an algebraic space is said to be smooth if ÕX, x ≅ k{ z1, ..., zd} for some indeterminates z1, ..., zd. The dimension of X at x is then just defined to be d.
A morphism f: Y → X of algebraic spaces is said to be étale at y ∈ Y (where x = f(y)) if the induced map on stalks
ÕX, x → ÕY, y
is an isomorphism.
The structure sheaf OX on the algebraic space X is defined by associating the ring of functions O(V) on V (defined by étale maps from V to the affine line A1 in the sense just defined) to any algebraic space V which is étale over X.
Algebraic spaces as sheaves
An algebraic space can be defined as a sheaf of sets
such that
There is a surjective etale morphism
the diagonal morphism is representable. The second condition is equivalent to the property that given any schemes and morphisms , their fiber-product of sheaves
is representable by a scheme over . Note that 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.
Algebraic spaces and schemes
Algebraic spaces are similar to schemes, and much of the theory of schemes extends to algebraic spaces. For example, most properties of morphisms of schemes also apply to algebraic spaces, one can define cohomology of quasicoherent sheaves, this has the usual finiteness properties for proper morphisms, and so on.