Gluing schemes explained

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

\{ X _{i} \} _{i} and for pairs i, j , there are open subsets U _{ij} and isomorphisms \varphi _{ij} : U _{ij} \overset{\sim}\to U _{ji} . Now, if the isomorphisms are compatible in the sense: for each i, j, k ,\varphi _{ij} = \varphi ^{-1} _{ji} ,\varphi _{ij} (U _{ij} \cap U _{ik} ) = U _{ji} \cap U _{jk} ,\varphi _{jk} \circ \varphi _{ij} = \varphi _{ik} on U _{ij} \cap U _{ik} ,then there exists a scheme X, together with the morphisms \psi _{i} : X _{i} \to X such that\psi _{i} is an isomorphism onto an open subset of X,X = \cup _{i} \psi _{i(X} _{i),} \psi _{i(U} _{ij} ) = \psi _{i(X} _{i)} \cap \psi _{j(X} _{j),} \psi _{i} = \psi _{j} \circ \varphi _{ij} on U _{ij} .

Examples

Projective line

Let

X = \operatorname{Spec}(k[t]) \simeq A ^{1,} Y = \operatorname{Spec}(k[u]) \simeq A ^{1} be two copies of the affine line over a field k. Let X _{t} = \{ t \ne 0 \} = \operatorname{Spec}(k[t, t ^{-1} ]) be the complement of the origin and Y _{u} = \{ u \ne 0 \} defined similarly. Let Z denote the scheme obtained by gluing X, Y along the isomorphism X _{t} \simeq Y _{u} given by t ^{-1} \leftrightarrow u ; we identify X, Y with the open subsets of Z. Now, the affine rings \Gamma(X, l{O} _{Z),} \Gamma(Y, l{O} _{Z)} are both polynomial rings in one variable in such a way\Gamma(X, l{O} _{Z)} = k[s] and \Gamma(Y, l{O} _{Z)} = k[s ^{-1} ] where the two rings are viewed as subrings of the function field k(Z) = k(s) . But this means that Z = P ^{1} ; because, by definition, 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} be as in the above example. But this time let Z denote the scheme obtained by gluing X, Y along the isomorphism X _{t} \simeq Y _{u} given by t \leftrightarrow u . 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 isomrophism t ^{-1} \leftrightarrow u , then the resulting scheme is, at least visually, the projective line 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;^[1] 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

• • Web site: Vakil . Ravi . Math 216: Foundations of algebraic geometry . November 18, 2017.

Further reading

• Stacks Project, 26.14 Glueing schemes

Notes and References

1. Web site: Section 37.14 (07RS): Pushouts in the category of schemes, I—The Stacks project.