Gluing schemes

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

Suppose there is a (possibly infinite) family of schemes

, there are open subsets

{\displaystyle \varphi _{ij}:U_{ij}{\overset {\sim }{\to }}U_{ji}}

Now, if the isomorphisms are compatible in the sense: for each

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

= Spec ⁡ ( k [ t ] ) ≃

= Spec ⁡ ( k [ u ] ) ≃

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

with the open subsets of Z.

[2] 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

is covered by the two open affine charts whose affine rings are of the above form.

denote the scheme obtained by gluing

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

, then the resulting scheme is, at least visually, the projective line

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.

This algebraic geometry–related article is a stub.

You can help Wikipedia by expanding it.