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 ${\displaystyle {\phi }_{ij}:U_{ij}\stackrel{\sim }{\to }{U}_{ji}}$. Now, if the isomorphisms are compatible in the sense: for each ${\displaystyle i,j,k}$,

1. ${\displaystyle {\phi }_{ij}={\phi }_{ji}^{-1}}$,
2. ${\displaystyle {\phi }_{ij}(U_{ij}\cap U_{ik})=U_{ji}\cap U_{jk}}$,
3. ${\displaystyle {\phi }_{jk}\circ {\phi }_{ij}={\phi }_{ik}}$ 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 that^[1]

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. ${\displaystyle {\psi }_i={\psi }_j\circ {\phi }_{ij}}$ on ${\displaystyle U_{ij}}$.

Let ${\displaystyle X=\mathrm{Spec}(k[t])\simeq {\mathbb{𝔸}}^1,Y=\mathrm{Spec}(k[u])\simeq {\mathbb{𝔸}}^1}$ be two copies of the affine line over a field k. Let ${\displaystyle X_t=\{t\ne 0\}=\mathrm{Spec}(k[t,t^{-1}])}$ be the complement of the origin and ${\displaystyle Y_u=\{u\ne 0\}}$ 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 ${\displaystyle \mathrm{\Gamma }(X,{{𝒪}}_Z),\mathrm{\Gamma }(Y,{{𝒪}}_Z)}$ are both polynomial rings in one variable in such a way

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

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

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.

• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9 
• Vakil, Ravi (November 18, 2017). "Math 216: Foundations of algebraic geometry". http://math.stanford.edu/~vakil/216blog/. 

• Stacks Project, 26.14 Glueing schemes

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