Remark

In a projective space $\mathbf{P}(V)$ over a field $K$, let $H = \mathbf{P}(W)$ be a hyperplane. There exists a bijective linear mapping $f$ of $V$ onto $K_s \times W$ such that $f(W) = W$; let $g$ be the projective mapping obtained from $f$ by passing to the quotients. It has been seen (no. 8) that the complement of $\mathbf{P}(W)$ in $\mathbf{P}(K_s \times W)$ can be identified with an affine space whose translation space is $W$. When $\mathbf{P}(V)$ is identified with $\mathbf{P}(K_s \times W)$ by means of $g$, it is said that $H$ has been taken as hyperplane at infinity in $\mathbf{P}(V)$; the complement of $H$ in $\mathbf{P}(V)$ is then identified with an affine space whose translation space is $W$.