I'm trying to solve the following problem in Dino Lorenzini's book on arithmetic geometry:
Let $f,g\in k[x,y]$ be coprime and assume that $P=(0,0)\in V(f)\cap V(g)$ is a nonsingular point of $f$. Show that $V=k[x,y]/(f,g)$ is a finite dimensional $k$-vector space and that the intersection multiplicity of $f$ and $g$ at $P$ equals $\dim_k V$. Here $k$ is assumed to be algebraically closed.
The finite dimensional part is trivial and I can solve it in more than one way. The shortest is probably noting that if $A=k[x]$ then computing the resultant of $f,g$ in $A$, we get
$\textrm{Res}_A(f,g)=uf+vg\in (f,g)$,
so the resultant, being an element of $k[x]$, gives a nontrivial relation of $x$ in $k[x,y]/(f,g)$. The same can be repeated for $y$ by setting $A=k[y]$.
To show that this dimension equals the intersection multiplicity, let $M=(x,y)$. Then assuming the multiplicity is $n$, we have that $g\in M^n\setminus M^{n+1}$ in $B=k[x,y]/(f)$. Furthermore, we know that $B$ is a Dedekind domain, since $P$ is a nonsingular point.
I can't seem to figure out what to do next. We can assume that $MB_M=(x)$, so that $ug=vx^n$ for some $u,v\not\in M$. This shows that $vx^n=0$ in $k[x,y]/(f,g)$. However, this doesn't seem to give the relation I want for $x$, since $v$ is not a constant.
Any ideas?