Editing Scheme (mathematics) (section)

=== Non-affine schemes ===
* For any commutative ring ''R'' and natural number ''n'', '''projective space''' <math>\mathbb{P}^n_R</math> can be constructed as a scheme by gluing ''n'' + 1 copies of affine ''n''-space over ''R'' along open subsets. This is the fundamental example that motivates going beyond affine schemes. The key advantage of projective space over affine space is that <math>\mathbb{P}^n_R</math> is [[proper morphism|proper]] over ''R''; this is an algebro-geometric version of compactness. Indeed, [[complex projective space]] <math>\C\mathbb{P}^n</math> is a compact space in the classical topology, whereas <math>\C^n</math> is not.
* A [[homogeneous polynomial]] ''f'' of positive degree in the polynomial ring {{math|1=''R''[''x''<sub>0</sub>, ..., ''x''<sub>''n''</sub>]}} determines a closed subscheme {{math|1=''f'' = 0}} in projective space <math>\mathbb{P}^n_R</math>, called a [[projective hypersurface]]. In terms of the [[Proj construction]], this subscheme can be written as <math display="block"> \operatorname{Proj} R[x_0,\ldots,x_n]/(f).</math> For example, the closed subscheme {{math|1=''x''<sup>3</sup> + ''y''<sup>3</sup> = ''z''<sup>3</sup>}} of <math>\mathbb{P}^2_\Q</math> is an [[elliptic curve]] over the [[rational number]]s.
* The '''line with two origins''' (over a field ''k'') is the scheme defined by starting with two copies of the affine line over ''k'', and gluing together the two open subsets A<sup>1</sup> − 0 by the identity map. This is a simple example of a non-separated scheme. In particular, it is not affine.{{sfn|Hartshorne|1997|loc=Example II.4.0.1}}
* A simple reason to go beyond affine schemes is that an open subset of an affine scheme need not be affine. For example, let <math>X=\mathbb{A}^n\smallsetminus\{0\}</math>, say over the complex numbers <math>\C</math>; then ''X'' is not affine for ''n'' ≥ 2. (However, the affine line minus the origin is isomorphic to the affine scheme <math>\mathrm{Spec}\,\C[x,x^{-1}]</math>. To show ''X'' is not affine, one computes that every regular function on ''X'' extends to a regular function on <math>\mathbb{A}^n</math> when ''n'' ≥ 2: this is analogous to [[Hartogs's lemma]] in complex analysis, though easier to prove. That is, the inclusion <math>f:X\to\mathbb{A}^n</math> induces an isomorphism from <math>O(\mathbb{A}^n)=\C[x_1,\ldots,x_n] </math> to <math>O(X)</math>. If ''X'' were affine, it would follow that ''f'' is an isomorphism, but ''f'' is not surjective and hence not an isomorphism. Therefore, the scheme ''X'' is not affine.{{sfn|Hartshorne|1997|loc=Exercises I.3.6 and III.4.3}}
* Let ''k'' be a field. Then the scheme <math display="inline">\operatorname{Spec}\left(\prod_{n=1}^\infty k\right)</math> is an affine scheme whose underlying topological space is the [[Stone–Čech compactification]] of the positive integers (with the discrete topology). In fact, the prime ideals of this ring are in one-to-one correspondence with the [[ultrafilter]]s on the positive integers, with the ideal <math display="inline">\prod_{m \neq n} k</math> corresponding to the principal ultrafilter associated to the positive integer ''n''.{{sfn|Arapura|2011|loc=section 1}} This topological space is [[Krull dimension|zero-dimensional]], and in particular, each point is an [[irreducible component]]. Since affine schemes are [[quasi-compact]], this is an example of a non-Noetherian quasi-compact scheme with infinitely many irreducible components. (By contrast, a [[Noetherian scheme]] has only finitely many irreducible components.)