View source for Gluing schemes

{{short description|Mathematical concept}}
In algebraic geometry, a new [[scheme (mathematics)|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 <math>\{ X_i \}_{i \in I}</math> and for pairs <math>i, j</math>, there are open subsets <math>U_{ij}</math> and isomorphisms <math>\varphi_{ij} : U_{ij} \overset{\sim}\to U_{ji}</math>. Now, if the isomorphisms are compatible in the sense: for each <math>i, j, k</math>,
# <math>\varphi_{ij} = \varphi_{ji}^{-1}</math>,
# <math>\varphi_{ij}(U_{ij} \cap U_{ik}) = U_{ji} \cap U_{jk}</math>,
# <math>\varphi_{jk} \circ \varphi_{ij} = \varphi_{ik}</math> on <math>U_{ij} \cap U_{ik}</math>,<!-- cocycle condition? -->
then there exists a scheme ''X'', together with the morphisms <math>\psi_i : X_i \to X</math> such that<ref>{{harvnb|Hartshorne|1977|loc=Ch. II, Exercise 2.12.}}</ref>
# <math>\psi_i</math> is an isomorphism onto an open subset of ''X'',
# <math>X = \cup_i \psi_i(X_i),</math>
# <math>\psi_i(U_{ij}) = \psi_i(X_i) \cap \psi_j(X_j),</math>
# <math>\psi_i = \psi_j \circ \varphi_{ij}</math> on <math>U_{ij}</math>.
<!-- discuss uniqueness -->

== Examples ==
=== Projective line ===
[[File:Real projective line.svg|thumb|The projective line is obtained by gluing two affine lines so that the origin and illusionary <math>\infty</math> on one line corresponds to illusionary <math>\infty</math> and the origin on the other line, respectively.]]
Let <math>X = \operatorname{Spec}(k[t]) \simeq \mathbb{A}^1, Y = \operatorname{Spec}(k[u]) \simeq \mathbb{A}^1</math> be two copies of the affine line over a field ''k''. Let <math>X_t = \{ t \ne 0 \} = \operatorname{Spec}(k[t, t^{-1}])</math> be the complement of the origin and <math>Y_u = \{ u \ne 0 \}</math> defined similarly. Let ''Z'' denote the scheme obtained by gluing <math>X, Y</math> along the isomorphism <math>X_t \simeq Y_u</math> given by <math>t^{-1} \leftrightarrow u</math>; we identify <math>X, Y</math> with the open subsets of ''Z''.<ref>{{harvnb|Vakil|2017|loc=§ 4.4.6.}}</ref> Now, the affine rings <math>\Gamma(X, \mathcal{O}_Z), \Gamma(Y, \mathcal{O}_Z)</math> are both polynomial rings in one variable in such a way
:<math>\Gamma(X, \mathcal{O}_Z) = k[s]</math> and <math>\Gamma(Y, \mathcal{O}_Z) = k[s^{-1}]</math>
where the two rings are viewed as subrings of the function field <math>k(Z) = k(s)</math>. But this means that <math>Z = \mathbb{P}^1</math>; because, by definition, <math>\mathbb{P}^1</math> is covered by the two open affine charts whose affine rings are of the above form.

=== Affine line with doubled origin ===
Let <math>X, Y, X_t, Y_u</math> be as in the above example. But this time let <math>Z</math> denote the scheme obtained by gluing <math>X, Y</math> along the isomorphism <math>X_t \simeq Y_u</math> given by <math>t \leftrightarrow u</math>.<ref>{{harvnb|Vakil|2017|loc=§ 4.4.5.}}</ref> So, geometrically, <math>Z</math> 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 <math>t^{-1} \leftrightarrow u</math>, then the resulting scheme is, at least visually, the projective line <math>\mathbb{P}^1</math>.<!-- finish the following in the future
=== Line bundles ===
Given an invertible sheaf ''L'' on ''X'', the geometric line bundle corresponding to ''L'' is the ''X''-scheme
:<math>\operatorname{Spec}_X(\mathbb{Sym}(L^*))</math>.

(The discussion is more or less valid for geometric vector bundle.)

=== Relative Proj ===
(E.g., blow up)
Given a quasi-coherent sheaf ''R'' of graded algebras on a scheme ''X'', one puts:
:<math>\operatorname{Proj}_X(R) = \varinjlim \operatorname{Proj} (R(U))</math>.

'''Example''': <math>\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))</math> is a rational surface that is not a projective space.

=== Algebraic space ===-->

== Fiber products and pushouts of schemes ==
{{see also|Fiber product of schemes}}

The category of schemes admits finite pullbacks and in some cases finite pushouts;<ref>{{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}}</ref> 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 ==
{{reflist}}
*{{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}}

== Further reading ==
*[[Stacks Project]], [https://stacks.math.columbia.edu/tag/01JA 26.14 Glueing schemes]

{{algebraic-geometry-stub}}
[[Category:Scheme theory]]