When is the maximal ideal of the localization principal? My friend asked me the following exercise in commutative algebra:
Let $F\in \mathbb{C}[X,Y]$, and $R:=\mathbb{C}[X,Y]/(F)$. Denote by $x,y$ the residue classes of $X,Y$ in $R$, and let $\mathfrak{m}:=xR+yR\subset R$. Now consider the localization of $R$ at $\mathfrak{m}$, denoted by $R_\mathfrak{m}$.
(1) If $F(X,Y)=X^a+Y^b~(a,b\geq 2)$, prove $\mathfrak{m}R_\mathfrak{m}$ is not a principal ideal.
(2) Suppose $F(0,0)=0$ and $((\frac{\partial F}{\partial X})(0,0),(\frac{\partial F}{\partial Y})(0,0))\neq (0,0)$. Prove: in this case $\mathfrak{m}R_\mathfrak{m}$ is a principal ideal.
I don't know very much commutative algebra (maybe roughly Atiyah-Macdonald and a little Eisenbud). I had no idea what theorem is related to this exercise: is there any theorem dealing with when the maximal ideal of a local ring is principal? It looks like there's something to do with the Zariski tangent space (related to implicit function theorem?), but I have no idea how to deal with the example.
Can anyone give me some hint or background? Thanks a lot in advance!!
 A: Let $F(X,Y)=X^a+Y^b$ for $a,b\geq 2$, and I will also assume $\text{gcd}(n,m)=1$ so that $X^a+Y^b$ is irreducible (hence prime since $\Bbb{C}[x,y]$ is a GCD domain), and hence $\Bbb{C}[x,y]/(F)$ is a one dimensional integral domain, and thus $(x,y)+(F)$ is height one, so that $R_\mathfrak{m}$ is one dimensional. Since you said you have read Atiyah-Macdonald, I'll use results from that freely.
Since the quotient of a Noetherian commutative ring, and the localisation of a Noetherian commutative ring, are both Noetherian, we have that $R_{\mathfrak{m}}$ is a Noetherian local ring. For contradiction say that $\mathfrak{m}R_\mathfrak{m}$ is principal, then recall from A&M 9.2. that it follows that $R_\mathfrak{m}$ is a DVR, and letting $M=\mathfrak{m}R_\mathfrak{m}$ and $k=R_\mathfrak{m}/M$ we have (from the same result) that $\text{dim}_k(M/M^2)=1$. Then from Theorem 11.22(ii and iii) and the paragraph above that theorem, you see that $R_\mathfrak{m}$ is a regular local ring of dimension $1$. Now look at exercise 11.1, and see that this contradicts the fact that by definition the origin is not a non-singular point of this variety (as defined in exercise 11.1).
Your second question also follows from 11.22 and exercise 11.1. I can include a solution of exercise 11.1 if you wish.
