Let $f\in R=k[x,y,z]_{(x,y,z)}$ be a homogeneous polynomial of degree $d$, monic in $x$. Show that $(y,z)$ is an ideal of finite colength on $M=R/(f)$. Compute the corresponding Hilbert-Samuel function.
Maybe someone can do it as an example.
Let $f\in R=k[x,y,z]_{(x,y,z)}$ be a homogeneous polynomial of degree $d$, monic in $x$. Show that $(y,z)$ is an ideal of finite colength on $M=R/(f)$. Compute the corresponding Hilbert-Samuel function.
Maybe someone can do it as an example.