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 ${\displaystyle \{X_{i}\}_{i\in I}}$ and for pairs ${\displaystyle i,j}$, there are open subsets ${\displaystyle U_{ij}}$ and isomorphisms ${\displaystyle \varphi _{ij}:U_{ij}{\overset {\sim }{\to }}U_{ji}}$. Now, if the isomorphisms are compatible in the sense: for each ${\displaystyle i,j,k}$,

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

then there exists a scheme X, together with the morphisms ${\displaystyle \psi _{i}:X_{i}\to X}$ such that¹

1. ${\displaystyle \psi _{i}}$ is an isomorphism onto an open subset of X,
2. ${\displaystyle X=\cup _{i}\psi _{i}(X_{i}),}$
3. ${\displaystyle \psi _{i}(U_{ij})=\psi _{i}(X_{i})\cap \psi _{j}(X_{j}),}$
4. ${\displaystyle \psi _{i}=\psi _{j}\circ \varphi _{ij}}$ on ${\displaystyle U_{ij}}$.

Let ${\displaystyle X=\operatorname {Spec} (k[t])\simeq \mathbb {A} ^{1},Y=\operatorname {Spec} (k[u])\simeq \mathbb {A} ^{1}}$ be two copies of the affine line over a field k. Let ${\displaystyle X_{t}=\{t\neq 0\}=\operatorname {Spec} (k[t,t^{-1}])}$ be the complement of the origin and ${\displaystyle Y_{u}=\{u\neq 0\}}$ defined similarly. Let Z denote the scheme obtained by gluing ${\displaystyle X,Y}$ along the isomorphism ${\displaystyle X_{t}\simeq Y_{u}}$ given by ${\displaystyle t^{-1}\leftrightarrow u}$; we identify ${\displaystyle X,Y}$ with the open subsets of Z.² Now, the affine rings ${\displaystyle \Gamma (X,{\mathcal {O}}_{Z}),\Gamma (Y,{\mathcal {O}}_{Z})}$ are both polynomial rings in one variable in such a way

${\displaystyle \Gamma (X,{\mathcal {O}}_{Z})=k[s]}$ and ${\displaystyle \Gamma (Y,{\mathcal {O}}_{Z})=k[s^{-1}]}$

where the two rings are viewed as subrings of the function field ${\displaystyle k(Z)=k(s)}$. But this means that ${\displaystyle Z=\mathbb {P} ^{1}}$; because, by definition, ${\displaystyle \mathbb {P} ^{1}}$ is covered by the two open affine charts whose affine rings are of the above form.

Let ${\displaystyle X,Y,X_{t},Y_{u}}$ be as in the above example. But this time let ${\displaystyle Z}$ denote the scheme obtained by gluing ${\displaystyle X,Y}$ along the isomorphism ${\displaystyle X_{t}\simeq Y_{u}}$ given by ${\displaystyle t\leftrightarrow u}$.³ So, geometrically, ${\displaystyle Z}$ 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 ${\displaystyle t^{-1}\leftrightarrow u}$, then the resulting scheme is, at least visually, the projective line ${\displaystyle \mathbb {P} ^{1}}$.

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.