Flatness of a projection

Question

What is the best way to see that the natural map $\pi:\mathbb{A}^n\setminus\{0\}\rightarrow\mathbb{P}^{n-1}$ is flat?

I see that it follows at once from miracle flatness (the fibres are lines), but that seems like an overkill.

Answer 1

On local charts, the map looks like $k[x_1/x_0,\dots,x_n/x_0] \to k[x_1/x_0,\dots,x_n/x_0][x_0,1/x_0]$ which is of the form $R \to R[t,1/t]$ which is certainly flat.