Given $R:=\mathbb{C}[x,y,z]/(xy-z^2)$,
I want to show that this is not a local ring. So can I just apply the Nullstellensatz and create two different maximal ideals? Let's say $(\bar{x},\bar{y},\bar{z})$ and $(\bar{x}-1,\bar{y},\bar{z})$. How can I show that $R/ (\bar{x},\bar{y},\bar{z})$ and $R/ (\bar{x}-1,\bar{y},\bar{z})$ are both isomorphic to the field $\mathbb{C}$ in order to show that both ideals are maximal?
I don’t even think it takes the Nulstellensatz. If it were local, all of its quotients would be local, and would not have any nontrivial idempotents.
But look: the quotient by the ideal generated by $x-z$ and $y-1$ has $z$ as a nontrivial idempotent.