- Algebraic space
In

mathematics , an**algebraic space**is a generalization of the schemes ofalgebraic geometry introduced byMichael Artin for use indeformation theory .**Definition**An

**algebraic space**"X" comprises a schemeOne can always assume that "U" is anaffine 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 ofequivalence class es.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 function s 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 thelocal ring s 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**A**^{1}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 hascodimension ≥ 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.

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