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. 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 :\mathrm{Hom}(Y, X) \rightarrow \mathrm{Hom}(V, X) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \mathrm{Hom}(S, X) 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 :k\{x_1, \ldots, x_n\}\ 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 \mathfrak{X} can be defined as a sheaf of sets :\mathfrak{X} : (\text{Sch}/S)^{op}_{\text{et}} \to \text{Sets} such that • There is a surjective étale morphism h_X \to \mathfrak{X} • the diagonal morphism \Delta_{\mathfrak{X}/S}: \mathfrak{X} \to \mathfrak{X}\times \mathfrak{X} is representable. The second condition is equivalent to the property that given any schemes Y,Z and morphisms h_Y,h_Z\to \mathfrak{X}, their fiber-product of sheaves :h_Y\times_\mathfrak{X} h_Z is representable by a scheme over S. 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==