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Statement

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 that¹

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

Projective line

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

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

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.

• 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

• Stacks Project, 26.14 Glueing schemes

References

1. Hartshorne 1977, Ch. II, Exercise 2.12. - Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 https://mathscinet.ams.org/mathscinet-getitem?mr=0463157 ↩

2. Vakil 2017, § 4.4.6. - Vakil, Ravi (November 18, 2017). "Math 216: Foundations of algebraic geometry". http://math.stanford.edu/~vakil/216blog/ ↩

3. Vakil 2017, § 4.4.5. - Vakil, Ravi (November 18, 2017). "Math 216: Foundations of algebraic geometry". http://math.stanford.edu/~vakil/216blog/ ↩

4. "Section 37.14 (07RS): Pushouts in the category of schemes, I—The Stacks project". https://stacks.math.columbia.edu/tag/07RS ↩