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 $i,j,k$,

1. $\varphi _{ij}=\varphi _{ji}^{-1}$,
2. $\varphi _{ij}(U_{ij}\cap U_{ik})=U_{ji}\cap U_{jk}$,
3. $\varphi _{jk}\circ \varphi _{ij}=\varphi _{ik}$ 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,
2. $X=\cup _{i}\psi _{i}(X_{i}),$
3. $\psi _{i}(U_{ij})=\psi _{i}(X_{i})\cap \psi _{j}(X_{j}),$
4. $\psi _{i}=\psi _{j}\circ \varphi _{ij}$ on $U_{ij}$.

Examples

Projective line

The projective line is obtained by gluing two affine lines so that the origin and illusionary $\infty$ on one line corresponds to illusionary $\infty$ and the origin on the other line, respectively.

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 $X_{t},Y_{u}$ with the open subsets of Z. Now, the affine rings $\Gamma (X_{t},{\mathcal {O}}_{Z}),\Gamma (Y_{u},{\mathcal {O}}_{Z})$ are both polynomial rings in one variable in such a way

$\Gamma (X_{t},{\mathcal {O}}_{Z})=k[s]$ and $\Gamma (Y_{u},{\mathcal {O}}_{Z})=k[s^{-1}]$

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 $t\leftrightarrow u$. 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 $\mathbb {P} ^{1}$.

Fiber products and pushouts of schemes

See also: Fiber product of schemes

The category of schemes admits both a finite fiber product and a finite pushout; 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, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Vakil, Math 216: Foundations of algebraic geometry Nov 2017 version.