Is $(xy-1)$ a maximal ideal in $\mathbb C[x,y]$? I learnd that the maximal ideals in $\mathbb C[x,y]$ have the form $(x-z_1, y-z_2)$ by the Nullstellensatz. But if we set $I=(xy-1)$ then $\mathbb C[x,y]/I$ is isomorphic to $\mathbb C[x,1/x]$ which is in my opinion a field, thus $(xy-1)$ is maximal. Where did I make a mistake?

$\mathbb{C}[x,x^{-1}]$ is not a field, in fact its spectrum is the pointed affine line, in particular more than just a point. For example, $x+1$ is not invertible. In fact, the group of units of $\mathbb{C}[x,x^{-1}]$ is $\mathbb{C}^* \cdot \langle x \rangle$.
