Gluing Schemes

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

Statement

Suppose there is a (possibly infinite) family of schemes [math]\displaystyle{ \{ X_i \}_{i \in I} }[/math] and for pairs [math]\displaystyle{ i, j }[/math], there are open subsets [math]\displaystyle{ U_{ij} }[/math] and isomorphisms [math]\displaystyle{ \varphi_{ij} : U_{ij} \overset{\sim}\to U_{ji} }[/math]. Now, if the isomorphisms are compatible in the sense: for each [math]\displaystyle{ i, j, k }[/math],

1. [math]\displaystyle{ \varphi_{ij} = \varphi_{ji}^{-1} }[/math],
2. [math]\displaystyle{ \varphi_{ij}(U_{ij} \cap U_{ik}) = U_{ji} \cap U_{jk} }[/math],
3. [math]\displaystyle{ \varphi_{jk} \circ \varphi_{ij} = \varphi_{ik} }[/math] on [math]\displaystyle{ U_{ij} \cap U_{ik} }[/math],

then there exists a scheme X, together with the morphisms [math]\displaystyle{ \psi_i : X_i \to X }[/math] such that^[1]

1. [math]\displaystyle{ \psi_i }[/math] is an isomorphism onto an open subset of X,
2. [math]\displaystyle{ X = \cup_i \psi_i(X_i), }[/math]
3. [math]\displaystyle{ \psi_i(U_{ij}) = \psi_i(X_i) \cap \psi_j(X_j), }[/math]
4. [math]\displaystyle{ \psi_i = \psi_j \circ \varphi_{ij} }[/math] on [math]\displaystyle{ U_{ij} }[/math].

Examples

Projective line

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

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

[math]\displaystyle{ \Gamma(X, \mathcal{O}_Z) = k[s] }[/math] and [math]\displaystyle{ \Gamma(Y, \mathcal{O}_Z) = k[s^{-1}] }[/math]

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

Affine line with doubled origin

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

Fiber products and pushouts 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

• 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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