Projective space - Imported Knowlpedia Content

Let $k$ be a field. Projective $n$ -space over $k$ is the scheme obtained from the Proj construction

\mathbb P_k^n:=\operatorname{Proj}k[x_0,\ldots,x_n].

Its $k$ -valued points may be written in homogeneous coordinates $[a_0:\cdots:a_n]$ , where not all $a_i$ vanish and

[a_0:\cdots:a_n]=[\lambda a_0:\cdots:\lambda a_n]

for every nonzero $\lambda\in k$ .

For each $i$ , the condition $a_i\ne0$ defines a standard open subset $U_i$ . Dividing the other coordinates by $a_i$ gives an isomorphism

U_i\cong\mathbb A_k^n.

Consequently, $\mathbb P_k^n=\bigcup_{i=0}^nU_i$ is covered by $n+1$ copies of affine $n$ -space, proving directly that it is a scheme.

For example, $\mathbb P_k^1$ is the affine line together with one point at infinity. Unlike affine space, projective space is generally not an affine scheme.