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.
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.
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.
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".
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).
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 is an isomorphism.
can be defined as a sheaf of sets such that The second condition is equivalent to the property that given any schemes
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.
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.