Let $C$ be an irreducible curve over $\mathbb A^2$ and $P\in C$ a non-singular point. I want to prove that $\mathcal O_P(C)=\{f\in k(C)\mid f=a/b, b(P)\neq 0\}$ is a DVR.
I've already proved that $\mathfrak m_P(C)=\{f\in k(C)\mid f=a/b,a(P)=0, b(P)\neq 0\}$ is the only maximal ideal of $\mathcal O_P(C)$.
In order to prove this I'm using these equivalences. So, to prove what I want I only have to show $\mathcal O_P(C)$ is a PID. Any suggestion how can I do this?
Thanks
If I understand well we have to prove that the localization $(K[X,Y]/(F))_{(X-a,Y-b)}$ is a DVR provided $\frac{\partial F}{\partial X}(a,b)\ne 0$ or $\frac{\partial F}{\partial Y}(a,b)\ne 0$. We have $$(K[X,Y]/(F))_{(X-a,Y-b)}\simeq K[X,Y]_{(X-a,Y-b)}/(F).$$ Now use an automorphism of $K[X,Y]$ ($X\mapsto X+a$, $Y\mapsto Y+b$) and assume $(a,b)=(0,0)$. Suppose $\frac{\partial F}{\partial Y}(0,0)\ne 0$. Set $R=K[X,Y]_{(X,Y)}/(F)$. We claim that the maximal ideal of $R$ is generated by $x$, the residue class of $X$ modulo $(F)$. All we have to show is $y\in (x)$, that is, $Y\in (X,F)K[X,Y]_{(X,Y)}$. Now write $F(X,Y)=YF_0(Y)+XF_1(X,Y)$. Since $\frac{\partial F}{\partial Y}(0,0)\ne 0$ it follows $F_0(0)\ne 0$. Then $(X,F)K[X,Y]_{(X,Y)}=(X,Y)K[X,Y]_{(X,Y)}$ (since $F_0(Y)$ is invertible in $K[X,Y]_{(X,Y)}$) and we are done.