Spec$(\mathbb{C}[x_1^2,x_2]) =$? I want to make sure that Spec($\mathbb{C}[x_1^2,x_2]) \cong \mathbb{C}^2/\mathbb{Z}_2(1,2)$. (Silly example while learning quotient singularities)
The variety $\mathbb{C}^2/\mathbb{Z}_2(1,2)$ is defined as the quotient of $\mathbb{C}^2$ where $-1 \in \mathbb{Z}_2$ acts by $(x_1, x_2) \to (-x_1, x_2)$. The ring of functions invariant under the action is $\mathbb{C}[x_1^2,x_2]$. How do I find at least all the maximal ideals? My guess is they are of form $(x_1^2-a, x_2-b)$.
How is the ring $\mathbb{C}[x_1^2,x_2]$ related to $\mathbb{C}[x_1, x_2]$?
This action gives us a morphism between varieties $\mathbb{C}^2 \to \mathbb{C}^2/\mathbb{Z}_2(1,2)$, so the induced homomorphism of coordinate rings should be $\mathbb{C}[x_1^2,x_2] \to \mathbb{C}[x_1, x_2]$ by sending $x_1^2 \to x_1$. But isn't that an isomorphism...?
 A: The canonical morphism of varieties $\mathbb{C}^2\to\mathbb{C}^2/\mathbb{Z}_2(1,2)$ does not correspond to the homomorphism $\mathbb{C}[x_1^2,x_2]\to\mathbb{C}[x_1,x_2]$ sending $x_1^2$ to $x_1$ (and $x_2$ to $x_2$), which you correctly observe is an isomorphism.  Instead, the corresponding homomorphism is just the inclusion map, which sends $x_1^2$ to $x_1^2$ and $x_2$ to $x_2$.  Indeed, when you identify the coordinate ring of $\mathbb{C}^2/\mathbb{Z}_2(1,2)$ with the invariant subring of the coordinate ring of $\mathbb{C}^2$, you are implicitly identifying the coordinate ring of $\mathbb{C}^2/\mathbb{Z}_2(1,2)$ with its image in $\mathbb{C}[x_1,x_2]$ under the canonical homomorphism.  Once you have made this identification the canonical homomorphism becomes just the inclusion map.
As for determining the maximal ideals of $\mathbb{C}[x_1^2,x_2]$, your guess is correct, and this follows by combining the Nullstellensatz with the isomorphism $\mathbb{C}[x_1^2,x_2]\to\mathbb{C}[y,z]$ sending $x_1^2$ to $y$ and $x_2$ to $z$ (I use different variable names to emphasize, as above, that this is not the canonical homomorphism given by the action).  In fact, this isomorphism gives an isomorphism of varieties $\mathbb{C}^2/\mathbb{Z}_2(1,2)\cong \mathbb{C}^2$.
