Algebraic space

In mathematics, an algebraic space is a generalization of the schemes of algebraic geometry introduced by Michael Artin for use in deformation theory.

Definition

An algebraic space "X" comprises a schemeOne can always assume that "U" is an affine scheme. 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.] "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"

is satisfied, then the algebraic space will be a scheme in the usual sense. Thus, an algebraic space 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

:$mathrm\{Hom\}(Y,\; X)\; ightarrow\; mathrm\{Hom\}(V,\; X)${} 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" = ("x"₁, …, "x"_"n"), let

:"k"{"x"₁, …, "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"{"x"₁, …, "x"_"n"} / ("g")

of algebraic functions on "U".

A point on an algebraic space is said to be smooth if "Õ"_{"X", "x"} ≅ "k"{"z"₁, …, "z"_"d"} for some indeterminates "z"₁, …, "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 curves are schemes.
* Non-singular algebraic surfaces are schemes.
* Algebraic spaces with group structure 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.

Applications

"To be written"

ee also

* Algebraic stack

Notes

References

* Artin, Michael. "Algebraic Spaces". Yale University Press, 1971.
* Knutson, Donald. "Algebraic Spaces". Springer Lecture Notes in Mathematics, 203, 1971.