Let curves $A$ and $B$ be defined by $x^2-3x+y^2=0$ and $x^2-6x+10y^2=0$. Find the intersection multiplicities of all points of intersection of $A$ and $B$.
If we let $f=x^2-3x+y^2$ and $g=x^2-6x+10y^2$ we find that $<f,g>=<x,y^2>$. Then my book says that $\mathcal{O}_{\mathbb{A}_k^2,O}/<x,y^2>$ is isomorphic to $\mathbb C<1,y>$. How do they know this? Thanks
We have $<f,g>=<x^2-3x+y^2,x-3y^2>$ so that $\mathbb C[x,y]/<f,g>\cong \mathbb C[y]/<9y^4-8y^2>=\mathbb C[y]/<y^2(9y^2-8)>$ .
Localizing at the origin $<x,y>$, we get: $$\mathcal{O}_{\mathbb{A}_k^2,O}/<f,g>=(\mathbb C[x,y])_{<x,y>}/<f,g>\cong (\mathbb C[y])_{<y>}/<y^2(9y^2-8)> $$ $$=(\mathbb C[y])_{<y>}/<y^2>=(\mathbb C[y]/<y^2>)_{<y>}=\mathbb C[y]/<y^2>=\mathbb C\overline 1\oplus \mathbb C\overline y$$ just as your book says.
[I have used that that taking quotients and localizing commute, that $9y^2-8$ is invertible in the local ring $(\mathbb C[y])_{<y>}$ and at the end that localizing the already local ring $\mathbb C[y]/<y^2>$ at its maximal ideal $<y>$ does not change that ring]