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 $\_$ and for pairs $i, j$, there are open subsets $U_$ and isomorphisms $\varphi_ : U_ \overset\to U_$. Now, if the isomorphisms are compatible in the sense: for each $i, j, k$,
# $\varphi_ = \varphi_^$,
# $\varphi_(U_ \cap U_) = U_ \cap U_$,
# $\varphi_ \circ \varphi_ = \varphi_$ on $U_ \cap U_$,
then there exists a scheme ''X'', together with the morphisms $\psi_i : X_i \to X$ such that
# $\psi_i$ is an isomorphism onto an open subset of ''X'',
# $X = \cup_i \psi_i(X_i),$
# $\psi_i(U_) = \psi_i(X_i) \cap \psi_j(X_j),$
# $\psi_i = \psi_j \circ \varphi_$ on $U_$.

Examples

Projective line

Let X = \operatorname(k \simeq \mathbb^1, Y = \operatorname(k \simeq \mathbb^1 be two copies of the affine line over a field ''k''. Let X_t = \ = \operatorname(k , t^ be the complement of the origin and $Y_u = \$ defined similarly. Let ''Z'' denote the scheme obtained by gluing $X, Y$ along the isomorphism $X_t \simeq Y_u$ given by $t^ \leftrightarrow u$; we identify $X, Y$ with the open subsets of ''Z''. Now, the affine rings $\Gamma(X, \mathcal_Z), \Gamma(Y, \mathcal_Z)$ are both polynomial rings in one variable in such a way
:\Gamma(X, \mathcal_Z) = k /math> and \Gamma(Y, \mathcal_Z) = k ^/math>
where the two rings are viewed as subrings of the function field $k(Z) = k(s)$. But this means that $Z = \mathbb^1$; because, by definition, $\mathbb^1$ 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 $t^ \leftrightarrow u$, then the resulting scheme is, at least visually, the projective line $\mathbb^1$.

Fiber products and pushouts 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

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{{algebraic-geometry-stub
Scheme theory