Algebraic space

In mathematics, an algebraic space is a generalization of the schemes of algebraic geometry introduced by Artin (1969, 1971) for use in deformation theory. Intuitively, an algebraic space is a scheme modulo a "nice" equivalence relation; the resulting category of algebraic spaces extends the category of schemes and allows to carry out several natural constructions that are needed for example in deformation theory or in the construction of moduli spaces but are not possible in the smaller category of schemes.

Definition

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 p_i: R → U onto each factor are étale maps.

If a third condition

3. 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 iff x=y)

is satisfied, then the algebraic space will be a scheme in the usual sense. Since a general algebraic space 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, …, x_n), 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 Õ_{X, x} on X are then defined to be the local rings of algebraic functions defined by Õ_{U, u}, where u ∈ U is a point lying over x and Õ_{U, u} is the local ring corresponding to u of the ring

k{x₁, …, x_n} / (g)

of algebraic functions on U.

A point on an algebraic space is said to be smooth if Õ_{X, x} ≅ k{z₁, …, z_d} for some indeterminates z₁, …, z_d. 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

Õ_{X, x} → Õ_{Y, y}

is an isomorphism.

The structure sheaf O_X 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.

Facts about algebraic spaces

• Algebraic spaces of dimension one (curves) are schemes.
• Non-singular algebraic spaces of dimension two (smooth surfaces) are schemes.
• Group objects in the category of algebraic spaces over a field are schemes.
• Commutative-group objects in the category of algebraic spaces over an arbitrary scheme which are proper, locally finite presentation, flat, and cohomologically flat in dimension 0 are schemes.
• Not every singular algebraic surface is a scheme.
• Not every non-singular 3-dimensional algebraic space is a scheme.
• Every algebraic space contains a dense open affine subscheme, and the complement of such a subscheme always has codimension ≥ 1. Thus algebraic spaces are in a sense "close" to affine schemes.

See also

• Algebraic stack

References

• Artin, Michael (1969), "The implicit function theorem in algebraic geometry", in Abhyankar, Shreeram Shankar, Algebraic geometry: papers presented at the Bombay Colloquium, 1968, of Tata Institute of Fundamental Research studies in mathematics,, 4, Oxford University Press, pp. 13–34, MR0262237, http://books.google.com/books/?id=3evuAAAAMAAJ 
• Artin, Michael (1971), Algebraic spaces, Yale Mathematical Monographs, 3, Yale University Press, ISBN 978-0-300-01396-2, MR0407012, http://books.google.com/books/about/Algebraic_spaces.html?id=Gl7vAAAAMAAJ 
• Knutson, Donald (1971), Algebraic spaces, Lecture Notes in Mathematics, 203, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0059750, ISBN 978-3-540-05496-2, MR0302647 

External links

• Danilov, V.I. (2001), "Algebraic space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=a/a011630