Consider the ring $R:= \mathbb{R}[x,y]/(y^2+x(x^2+1))$ and the maximal ideal $m := (x,y)$ in $R$. Then I am looking for a regular system of local parameters.
The (real) variety determined by $R$ is
Hence I expect $R_{m}$ to be regular, noetherian, local. However, I can only come up with a system of parameters of $mR_m$ which is not regular. It is
$$m = (\overline{x}, \overline{y}) = \sqrt{(\overline{x})}$$
This is a valid choice since $\overline{y}^2 = \overline{x} ( \overline{x}^2 + 1) \in (\overline{x})$. Furthermore, I cannot seem to get an intuition of what such a system should look like.
Initially, I expected it to be something like a good local coordinate system. Let me illustrate this via the following example.
Consider $R := \mathbb{R}[x,y]/(x^2 -y)$ and again $m := (x,y)$. Its (real) variety is given by
I expected $R_m$ to be regular, noetherian, local. And I expected its local system of parameters to be $(\overline{y})$, with the reasoning that the tangent to the variety at $(0,0)$ is the vanishing ideal of $(y)$ (with lots of hand waving).
However, $\overline{x}$ gives a regular system of parameters since $\overline{y} = \overline{x} \cdot \overline{x}$. However, $\overline{y}$ only gives a system of parameters since $\overline{x} \not\in (\overline{y})$ but $\overline{x} \in \sqrt{(\overline{y})}$.
Now, in the very first case I looked at $(\overline{x})$ is a system of parameters (which conincided with what I expected, since: "it's a good local coordinate system"). However, why then would it not be a regular system?
So to sum this up, I have the following questions:
- What is a good intuition for what a system of parameters of a noetherian local ring should look like?
- What is a good intuition for what a regular system of parameters of a regular local noetherian ring should look like?
- What is a regular system of parameters of $\mathbb{R}[x,y]/(y^2+x(x^2+1))$ localized at $(x,y)$?