My book is Hulek's Elementary algebraic geometry. He defines the intersection multiplicity of $C,C'$ (given by $f,g$ respectively) at $P \in \mathbb{P}_k^2$ by $I_P(C,C')=\dim_k \mathcal{O}_{\mathbb{P}_k^2,P}/(f,g)$
He gives an example: $C$ is given in projective coordinates $X^2Z-Y^3=0$ and $L$ is $X=0$. Let $P=(0:0:1)$, then in local coordinates $x^2-y^3=0$ and, $x=0$.
Then $I_P(C,L)=\dim_k \mathcal{O}_{\mathbb{P}_k^2,P}/(X^2Z-Y^3,X)=\dim_k \mathcal{O}_{\mathbb{A}_k^2,O}/(x^2-y^3,x)=\dim_k \mathcal{O}_{\mathbb{A}_k^2,O}/(x,y^3)=3$
My question:
- Is the second equality ok? just changing to affine coordinates does not change the dimension? (If it is too complicated, don't need to explain in detail.)
- Why the last eq is true? I understand it roughly, maybe it related to the dimension of vector space spanned by $1, x, x^2$. But I want to know some precise details.