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{{harvnb|Hartshorne|1977|loc=Ch. II, Exercise 2.12.}}

$\psi_i$ 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 =

File:Real projective line.svg

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 $X_t = \{ t \ne 0 \} = \operatorname{Spec}(k[t, t^{-1}])$ be the complement of the origin and $Y_u = \{ u \ne 0 \}$ 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.{{harvnb|Vakil|2017|loc=§ 4.4.6.}} 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 $t \leftrightarrow u$ .{{harvnb|Vakil|2017|loc=§ 4.4.5.}} 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;{{Cite web|url=https://stacks.math.columbia.edu/tag/07RS|title = Section 37.14 (07RS): Pushouts in the category of schemes, I—The Stacks project}} 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 AG}}
{{cite web |last=Vakil |first=Ravi |url=http://math.stanford.edu/~vakil/216blog/ |title=Math 216: Foundations of algebraic geometry |date=November 18, 2017}}