An paragraph from Vakil’s book. Here $U$ is plane minus the origin.
I know this question has appeared on this site. But I really get stuck by an step which is essentially the same as the above one from Vakil’s book. The only thing I can’t see is the reason why the prime ideal $(x,y)$ of $k[x,y]$ should cut out a point of $U$ if $U \cong \mathbb{A}^2_k$, which is used to derive an contra diction.
Could you help me? Thanks in advance.
A little edit: It seems that in scheme $U$ we could talk about points cut by ideals, as in scheme $\mathbb{A}^2_k$, but why these two ways of cutting should cut the same number of points? I couldn’t see why...
Like he says: there is bijection between prime ideals of $k[x,y]$ and points of $U$. Since $(x,y)$ is a maximal ideal, it corresponds to a closed point—although you don't need to know that the point is closed.
Since $V(x,y)$ is empty, by the constructive nature that he mentions, we should be able to recover $(x,y)$ as $I(V(x,y)) = I(\varnothing) = k[x,y]$, which is a contradiction.