User:Leon2k2k2k/sandbox

Non-Examples
Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together.
 * The Projective $$n$$-Space $$\mathbb{P}^n_k = \operatorname{Proj}k[x_0,\ldots, x_n]$$ over a field $$k$$ .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 $$n$$-Space for $$ n \geq 1 $$ is not affine as the global section of $$\mathbb{P}^n_k$$ is $$k$$.
 * Affine plane minus the origin. Inside $$\mathbb{A}^2_k = \operatorname{Spec}\, k[x,y]$$ are distinguished open affine subschemes $$ D_x, D_y $$. Their union $$ D_x \cup D_y = U$$ is the affine plane with the origin taken out. The global sections of $$U$$ are pairs of polynomials on $$D_x,D_y $$ that restrict to the same polynomial on $$ D_{xy} $$, which can be shown to be $$ k[x,y] $$, the global section of $$\mathbb{A}^2_k $$. $$U$$ is not affine as $$ V_{(x)} \cap V_{(y)} = \varnothing $$ in $$ U$$.