I recently learned about intersection multiplicity and tried to calculate a simple example. Unfortunately, I am having difficulty.
Consider the two curves in $\mathbb{C}^2$ given by $y=0$ and $y-x^n=0$. If there is any justice in the world, the intersection multiplicity at $(0,0)$ should be $n$.
By definition, the intersection number is the dimension as a $k$-module of
$\left( \frac{k[x,y]}{(y,y-x^n)} \right)_{(x,y)}.$
This is clearly equivalent to
$\left( \frac{k[x,y]}{(y,x^n)} \right)_{(x,y)}.$
Before localization, the quotient ring is all elements of the form
$a_0x^{n-1}+\cdots+a_n$
(polynomials in $x$ of degree less than $n$). This already has dimension $n$. Wouldn't taking the localization make the ring much bigger, with all sorts of nasty fractions, and make the dimension larger as well?
My question: Why is the dimension of the localized ring $n$?