Gluing schemes

In algebraic geometry, a new scheme can be obtained by gluing existing schemes through gluing maps.

Statement

Suppose there is a family of schemes and for pairs, there are open subsets and isomorphisms. Now, if the isomorphisms are compatible in the sense: for each,

then there exists a scheme X, together with the morphisms such that

is an isomorphism onto an open subset of X,
on.

Examples

Projective line

Let be two copies of the affine line over a field k. Let be the complement of the origin and defined similarly. Let Z denote the scheme obtained by gluing along the isomorphism given by ; we identify with the open subsets of Z. Now, the affine rings are both polynomial rings in one variable in such a way
where the two rings are viewed as subrings of the function field. But this means that ; because, by definition, is covered by the two open affine charts whose affine rings are of the above form.

Affine line with doubled origin

Let be as in the above example. But this time let denote the scheme obtained by gluing along the isomorphism given by. So, geometrically, is obtained by identifying two parallel lines except the origin; i.e., it is an affine line with the doubled origin. In contrast, if two lines are glued so that origin on the one line corresponds to the point at infinity for the other line; i.e, use the isomorphism, then the resulting scheme is, at least visually, the projective line.

Fiber products and pushouts of schemes

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