Gluing schemes

Mathematical concept

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

## Statement

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Suppose there is a (possibly infinite) family of schemes  { X i } i ∈ I {\displaystyle \\{X_{i}\\}_{i\in I}} and for pairs  i , j {\displaystyle i,j} , there are open subsets  U i j {\displaystyle U_{ij}} and isomorphisms  φ i j : U i j -> ∼ U j i {\displaystyle \varphi _{ij}:U_{ij}{\overset {\sim }{\to }}U_{ji}} . Now, if the isomorphisms are compatible in the sense: for each  i , j , k {\displaystyle i,j,k} ,

1. φ i j = φ j i − 1 {\displaystyle \varphi _{ij}=\varphi _{ji}^{-1}} ,
2. φ i j ( U i j ∩ U i k ) = U j i ∩ U j k {\displaystyle \varphi _{ij}(U_{ij}\cap U_{ik})=U_{ji}\cap U_{jk}} ,
3. φ j k ∘ φ i j = φ i k {\displaystyle \varphi _{jk}\circ \varphi _{ij}=\varphi _{ik}} on  U i j ∩ U i k {\displaystyle U_{ij}\cap U_{ik}} ,

then there exists a scheme X, together with the morphisms  ψ i : X i -> X {\displaystyle \psi _{i}:X_{i}\to X} such that[1]

1. ψ i {\displaystyle \psi _{i}} is an isomorphism onto an open subset of X,
2. X = ∪ i ψ i ( X i ) , {\displaystyle X=\cup _{i}\psi _{i}(X_{i}),}
3. ψ i ( U i j ) = ψ i ( X i ) ∩ ψ j ( X j ) , {\displaystyle \psi _{i}(U_{ij})=\psi _{i}(X_{i})\cap \psi _{j}(X_{j}),}
4. ψ i = ψ j ∘ φ i j {\displaystyle \psi _{i}=\psi _{j}\circ \varphi _{ij}} on  U i j {\displaystyle U_{ij}} .

## Examples

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### Projective line

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

Let  X = Spec ⁡ ( k [ t ] ) ≃ A 1 , Y = Spec ⁡ ( k [ u ] ) ≃ A 1 {\displaystyle X=\operatorname {Spec} (k[t])\simeq \mathbb {A} ^{1},Y=\operatorname {Spec} (k[u])\simeq \mathbb {A} ^{1}} be two copies of the affine line over a field k. Let  X t = { t ≠ 0 } = Spec ⁡ ( k [ t , t − 1 ] ) {\displaystyle X_{t}=\\{t\neq 0\\}=\operatorname {Spec} (k[t,t^{-1}])} be the complement of the origin and  Y u = { u ≠ 0 } {\displaystyle Y_{u}=\\{u\neq 0\\}} defined similarly. Let Z denote the scheme obtained by gluing  X , Y {\displaystyle X,Y} along the isomorphism  X t ≃ Y u {\displaystyle X_{t}\simeq Y_{u}} given by  t − 1 ↔ u {\displaystyle t^{-1}\leftrightarrow u} ; we identify  X , Y {\displaystyle X,Y} with the open subsets of Z.[2] Now, the affine rings  Γ ( X , O Z ) , Γ ( Y , O Z ) {\displaystyle \Gamma (X,{\mathcal {O}}_{Z}),\Gamma (Y,{\mathcal {O}}_{Z})} are both polynomial rings in one variable in such a way

Γ ( X , O Z ) = k [ s ] {\displaystyle \Gamma (X,{\mathcal {O}}_{Z})=k[s]} and  Γ ( Y , O Z ) = k [ s − 1 ] {\displaystyle \Gamma (Y,{\mathcal {O}}_{Z})=k[s^{-1}]}

where the two rings are viewed as subrings of the function field  k ( Z ) = k ( s ) {\displaystyle k(Z)=k(s)} . But this means that  Z = P 1 {\displaystyle Z=\mathbb {P} ^{1}} ; because, by definition,  P 1 {\displaystyle \mathbb {P} ^{1}} is covered by the two open affine charts whose affine rings are of the above form.

### Affine line with doubled origin

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Let  X , Y , X t , Y u {\displaystyle X,Y,X_{t},Y_{u}} be as in the above example. But this time let  Z {\displaystyle Z} denote the scheme obtained by gluing  X , Y {\displaystyle X,Y} along the isomorphism  X t ≃ Y u {\displaystyle X_{t}\simeq Y_{u}} given by  t ↔ u {\displaystyle t\leftrightarrow u} .[3] So, geometrically,  Z {\displaystyle 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 {\displaystyle t^{-1}\leftrightarrow u} , then the resulting scheme is, at least visually, the projective line  P 1 {\displaystyle \mathbb {P} ^{1}} .

## Fiber products and pushouts of schemes

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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

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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".

* Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
* Vakil, Ravi (November 18, 2017). "Math 216: Foundations of algebraic geometry".

## Further reading

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* Stacks Project, 26.14 Glueing schemes