Consider the ring $A=\Bbb{C}[X,Y]/(Y^2-X^4+2X^2-1)$. Let $A_M$ denote the localization of $A$ with respect to maximal ideal $M$. My question is:
Is $A_N$ a DVR, where $N$ is the maximal ideal generated by $X+1$ and $Y$ in $A$? In general, I want to determine all the maximal ideals $M$ such that $A_M$ is a Discrete Valuation Ring. (DVR)
The maximal ideals of $A$ correspond to the points of the curve $C = V(Y^2 - X^4 + 2X^2 - 1)$, i.e., they are $(X - a, Y - b)$ for $a, b \in {\mathbb C}$ such that $b^2 - a^4 + 2a^2 - 1 = 0$.
The localization $A_{\frak m}$ at such a maximal ideal ${\frak m} = (X - a, Y - b)$ is a discrete valuation ring if and only if $(a,b)$ is a non-singular point of the curve $C$. (To clarify: it is regular if and only if $(a,b)$ is a non-singular point; discrete valuation rings are regular local rings of dimension 1; and the dimension of $C$ is indeed 1).
The singular points of $V$ are those points of $V$ for which the partial derivatives of the defining equation vanish, i.e., at which $2 Y = 0$ and $-4X^3 + 4X = 0$. That is, the singular points are $(1,0)$ and $(-1,0)$ and, hence, the maximal ideals of ${\frak m}$ of $A$ for which $A_{\frak m}$ is not a discrete valuation ring are $(X-1,Y)$ and $(X+1,Y)$.