Schematic fiber of $\mathbb{C}^2/\mu_2$ at $0$

Question

I wanned to have a confirmation from you guys concerning a basic computation of the schematic fiber of $\mathbb{C}^2/\mu_2$ at $0$ where $\mu_2$ is the cyclic group in two elements so $\{1,-1\} \subset \mathbb{C}^*$.
I have found that it is the variety associated to the ring $\mathbb{C}[x,y]/(x^2,y^2,xy)$ is that correct ?
Thanks in advance for your answers,
Best rhylx

Answer 1

Yes, it is correct. Computing $$\mathbb{C}[x,y] \otimes_{\mathbb{C}[x^2,y^2,xy]}\mathbb{C}$$ with $\operatorname{Spec}(\mathbb{C})\hookrightarrow \operatorname{Spec}(\mathbb{C}[x^2,y^2,xy]) $ being the origin gives $\dfrac{\mathbb{C}[x,y]}{(x^2,y^2,xy)}$.