Gluing schemes
In algebraic geometry, a new scheme (e.g. an algebraic variety) can be obtained by gluing existing schemes through gluing maps.
Statement
[edit]Suppose there is a (possibly infinite) family of schemes and for pairs , there are open subsets and isomorphisms . Now, if the isomorphisms are compatible in the sense: for each ,
- ,
- ,
- on ,
then there exists a scheme X, together with the morphisms such that[1]
- is an isomorphism onto an open subset of X,
- on .
Examples
[edit]Projective line
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Real_projective_line.svg/220px-Real_projective_line.svg.png)
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.[2] Now, the affine rings are both polynomial rings in one variable in such a way
- and
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
[edit]Let be as in the above example. But this time let denote the scheme obtained by gluing along the isomorphism given by .[3] So, geometrically, 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 , then the resulting scheme is, at least visually, the projective line .
Fiber products and pushouts of schemes
[edit]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
[edit]- ^ Hartshorne 1977, Ch. II, Exercise 2.12.
- ^ Vakil 2017, § 4.4.6.
- ^ Vakil 2017, § 4.4.5.
- ^ "Section 37.14 (07RS): Pushouts in the category of schemes, I—The Stacks project".
- 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".