Is $(xy-1)$ a maximal ideal in $\mathbb C[x,y]$?

Question

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?

Answer 1

$\mathbb{C}[x,x^{-1}]$ 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 $\mathbb{C}^* \cdot \langle x \rangle$.

Answer 2

The ideal $(\rm XY - 1)$ describes a curve if you draw it and over $\mathbb C$, maximal ideals describe points.

At a technical level, the answer of Martin is right.

Answer 3

Since $xy-1=x(y+1)-(x+1)$, we can see that $(xy-1)\subsetneq (x+1,y+1)$.

The containment is proper since in the left-hand side, you cannot have elements with degree of $y$ less than 1, but in the right hand side, you can get elements with no $y$'s.