The local rings of $xy=0$ and $xy+x^3+y^3=0$ are not isomorphic, but have isomorphic completions?

Question

I know that if you have a commutative local ring $R$, and you take its completion $\widehat{R}$ the inverse limit of the $R/\mathfrak{m}^i$, you get another local ring. However, nonisomorphic local rings might have isomorphic completions. I'm a little mystified by an example.

Assume everything takes place over an algebraically closed field. Consider the two curves $xy=0$ and $xy+x^3+y^3=0$. I'll denote these two curves by $C_1$ and $C_2$, respectively, and let $O=(0,0)$. I suspect there is some bizarre behavior near the origin.

Is there a nice way to see that the corresponding local rings $\mathcal{O}_{O,C_1}$ and $\mathcal{O}_{O,C_2}$ are not isomorphic, but that their completions $\widehat{\mathcal{O}}_{O,C_1}$ and $\widehat{\mathcal{O}}_{O,C_2}$ are indeed isomorphic? Thank you.

Answer 1

It is well known in the algebraic geometry of curves that at a node the completion is isomorphic to $K[[X,Y]]/(XY)$ (sometimes taken as the definition). Your curve $x^3+xy+y^3=0$ has a node at $(0,0)$ and this can be easily verified; see this in order to learn what to do.

If you want a purely algebraic approach, then follow the suggestion of Hugh Thomas and try to write $X^3+XY+Y^3$ in $K[[X,Y]]$ as $(X+f_2+f_3+\cdots)(Y+g_2+g_3+\cdots)$ with $f_i,g_i$ homogeneous of degree $i$.