Editing Spectrum of a ring (section)

== Non-affine examples ==
Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together.
* The [[projective space|projective <math>n</math>-space]] <math>\mathbb{P}^n_k = \operatorname{Proj}k[x_0,\ldots, x_n]</math> over a field <math>k</math>. This can be easily generalized to any base ring, see [[Proj construction]] (in fact, we can define projective space for any base scheme). The projective <math>n</math>-space for <math> n \geq 1 </math> is not affine as the ring of global sections of <math>\mathbb{P}^n_k</math> is <math>k</math>.
* Affine plane minus the origin.{{sfnp|Vakil|n.d.|loc=Chapter 4, example 4.4.1|ps=}} Inside <math>\mathbb{A}^2_k = \operatorname{Spec} k[x,y]</math> are distinguished open affine subschemes <math> D_x , D_y </math>. Their union <math> D_x \cup D_y = U</math> is the affine plane with the origin taken out. The global sections of <math>U</math> are pairs of polynomials on <math>D_x,D_y </math> that restrict to the same polynomial on <math> D_{xy} </math>, which can be shown to be <math> k[x,y] </math>, the global sections of <math>\mathbb{A}^2_k </math>. <math>U</math> is not affine as <math> V_{(x)} \cap V_{(y)} = \varnothing </math> in <math> U</math>.