I am trying unsuccessfully to solve exercise II-20 (page 65) from the book "The geometry of schemes" by Eisenbud and Harris. In this exercise it is stated that there are two non-isomorphic subschemes of degree three in $A^2_{\mathbb R}$ supported at $(0,0)$ that become isomorhic to $Spec\mathbb C[x,y]/(x^2,xy,y^2)$ after complexification.
Could you give me a hint on what is the solution?
Let $A$ be the ring. Its residue field $k$ is either $\mathbb R$ or $\mathbb C$. But then $\mathbb C \otimes_{\mathbb R} A$ admits a surjection onto $\mathbb C\otimes_{\mathbb R} k.$ Since this base-change is local by assumption, with residue field $\mathbb C$, we see that necessarily $k = \mathbb R$.
Since the maximal ideal of $\mathbb C\otimes_{\mathbb R} A$ has square zero, the maximal ideal of $A$ must also have square zero.
So we can write $A = \mathbb R \oplus \mathfrak m,$ where $\mathfrak m^2 = 0$.
This answer seems to contradict the claim in Eisenbud and Harris --- namely it suggests that $A$ is uniquely determined up to isomorphism as the direct sum $\mathbb R \oplus \mathfrak m$, where $\mathfrak m$ is a two-dimensional $\mathbb R$-v.s. which is declared to be of square zero.