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 $\{X_{i}\}_{i\in I}$ 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 $i,j,k$ ,

$\varphi _{ij}=\varphi _{ji}^{-1}$ ,
$\varphi _{ij}(U_{ij}\cap U_{ik})=U_{ji}\cap U_{jk}$ ,
$\varphi _{jk}\circ \varphi _{ij}=\varphi _{ik}$ on $U_{ij}\cap U_{ik}$ ,

then there exists a scheme X, together with the morphisms $\psi _{i}:X_{i}\to X$ such that^[1]

$\psi _{i}$ 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 $X=\operatorname {Spec} (k[t])\simeq \mathbb {A} ^{1},Y=\operatorname {Spec} (k[u])\simeq \mathbb {A} ^{1}$ be two copies of the affine line over a field k. Let $X_{t}=\{t\neq 0\}=\operatorname {Spec} (k[t,t^{-1}])$ be the complement of the origin and $Y_{u}=\{u\neq 0\}$ 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.^[2] Now, the affine rings $\Gamma (X,{\mathcal {O}}_{Z}),\Gamma (Y,{\mathcal {O}}_{Z})$ are both polynomial rings in one variable in such a way

\Gamma (X,{\mathcal {O}}_{Z})=k[s]

and

\Gamma (Y,{\mathcal {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 $Z=\mathbb {P} ^{1}$ ; because, by definition, $\mathbb {P} ^{1}$ 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 $t\leftrightarrow u$ .^[3] 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 isomorphism $t^{-1}\leftrightarrow u$ , then the resulting scheme is, at least visually, the projective line $\mathbb {P} ^{1}$ .

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 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".