Gluing schemes

Short description: Mathematical concept

In algebraic geometry, a new scheme (e.g. an algebraic variety) can be obtained by gluing existing schemes through gluing maps.

## Contents

* 1 Statement
* 2 Examples
* 2.1 Projective line
* 2.2 Affine line with doubled origin
* 3 Fiber products and pushouts of schemes
* 4 References
* 5 Further reading

## Statement

Suppose there is a (possibly infinite) family of schemes {Xi}i∈I and for pairs i,j, there are open subsets Uij and isomorphisms φij:Uij->∼Uji. Now, if the isomorphisms are compatible in the sense: for each i,j,k,

1. φij=φji−1,
2. φij(Uij∩Uik)=Uji∩Ujk,
3. φjk∘φij=φik on Uij∩Uik,

then there exists a scheme X, together with the morphisms ψi:Xi->X such that[1]

1. ψi is an isomorphism onto an open subset of X,
2. X=∪iψi(Xi),
3. ψi(Uij)=ψi(Xi)∩ψj(Xj),
4. ψi=ψj∘φij on Uij.

## Examples

### Projective line

The projective line is obtained by gluing two affine lines so that the origin and illusionary ∞ on one line corresponds to illusionary ∞ and the origin on the other line, respectively.

Let X=Spec⁡(k[t])≃𝔸1,Y=Spec⁡(k[u])≃𝔸1 be two copies of the affine line over a field k. Let Xt={t≠0}=Spec⁡(k[t,t−1]) be the complement of the origin and Yu={u≠0} defined similarly. Let Z denote the scheme obtained by gluing X,Y along the isomorphism Xt≃Yu given by t−1↔u; we identify X,Y with the open subsets of Z.[2] Now, the affine rings Γ(X,𝒪Z),Γ(Y,𝒪Z) are both polynomial rings in one variable in such a way

Γ(X,𝒪Z)=k[s] and Γ(Y,𝒪Z)=k[s−1]

where the two rings are viewed as subrings of the function field k(Z)=k(s). But this means that Z=ℙ1; because, by definition, ℙ1 is covered by the two open affine charts whose affine rings are of the above form.

### Affine line with doubled origin

Let X,Y,Xt,Yu be as in the above example. But this time let Z denote the scheme obtained by gluing X,Y along the isomorphism Xt≃Yu given by t↔u.[3] So, geometrically, Z is obtained by identifying two parallel lines except the origin; i.e., it is an affine line with the doubled origin. (It can be shown that Z is not a separated scheme.) In contrast, if two lines are glued so that origin on the one line corresponds to the (illusionary) point at infinity for the other line; i.e, use the isomorphism t−1↔u, then the resulting scheme is, at least visually, the projective line ℙ1.

## Fiber products and pushouts of schemes

See also: Fiber product of schemes

The category of schemes admits finite pullbacks and in some cases finite pushouts;[4] they both are constructed by gluing affine schemes. For affine schemes, fiber products and pushouts correspond to tensor products and fiber squares of algebras.

## References

1. ↑ Hartshorne 1977, Ch. II, Exercise 2.12.
2. ↑ Vakil 2017, § 4.4.6.
3. ↑ Vakil 2017, § 4.4.5.
4. ↑ "Section 37.14 (07RS): Pushouts in the category of schemes, I—The Stacks project". https://stacks.math.columbia.edu/tag/07RS.

* Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9
* Vakil, Ravi (November 18, 2017). "Math 216: Foundations of algebraic geometry". http://math.stanford.edu/~vakil/216blog/.

## Further reading

* Stacks Project, 26.14 Glueing schemes

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