I'm trying to use some Algebraic Geometry techniques to check my understanding on them. I'm using the most stupid of all the examples: trying to compute the multiplicities of the intersections of the curves $C_1$ given by $y=x^2$ and $C_2$ given by $y=-x^3$ in $\mathbb{C}^2$.
I know (though I might be wrong) that the points of intersections are $P_1=(0,0)$ and $P_2=(-1,1)$. Also, the multiplicities of these intersections (using the fact that both the curves are given, respectly, by the same equation around each intersection point) are $$ i(C_1,C_2,P_1)=\dim_{\mathbb{C}} \mathcal{O}_{\langle x,y\rangle}/(y-x^2,y-x^3) $$
where $\mathcal{O}_{\langle x,y\rangle}$ stands for the localization of the local ring on the ideal $\langle x,y\rangle$ that corresponds to the point $P_1$.
I'm not sure if this is correct at all. I've seen some people using that $\mathcal{O}_{\langle x,y\rangle} \cong \mathbb{C}[x,y]$, yet still I can't see how I would use this to calculate the multiplicity that I need.
I editted it. I believe there is some way to simplify the ideal $(y-x^2,y-x^3)$ so the calculation of the dimensions gets straightfoward?
If $\mathfrak m\subset A$ is a maximal ideal in a ring $A$ then for any ideal $I\subset \mathfrak m$ we have an isomorphism $A_\mathfrak m/I_\mathfrak m\cong (A/I)_\mathfrak m$.
So in your case:
$\mathcal{O}_{\langle x,y\rangle}/(y-x^2,y-x^3)=( \mathbb C[x,y]/(y-x^2,y-x^3))_{(x,y)}=( \mathbb C[x]/(x^2-x^3))_{(x)}=( \mathbb C[x]/(x^2(1-x)))_{(x)}=(\mathbb C[x]/(x^2))_{(x)}=\mathbb C[x]/(x^2)$
And thus the requested intersection number is $i(C_1,C_2;P_1)=2$.