Restriction of Weil divisor at inflection point

Question

Let $X \subset \mathbb{P}^2$ be the nonsingular cubic curve $y^2 z = x^3 - x z^2$ (Hartshorne Ex. 6.10.2). I want to compute the restriction to $X$ of the divisor $H$ defined by the line $z = 0$. Since $H$ intersects $X$ in the inflection point $P_0 = (0:1:0)$, $H$ should restrict to $3 P_0$. In order to show this I think that I need to show that in $$ (k[x, z]/(z - x^3 + x z^2))_{(x,z)} $$ $z$ can be expressed as $z = t^3 u$, where $t$ is the generator of the maximal ideal and $u$ is a unit. However, when I try to compute it I find $z = t u$. Is the idea correct and how do I proceed?

Answer 1

Using the fact that quotient commutes with localization, we easily obtain that the quotient ring of $(k[x, z]/(z - x^3 + x z^2))_{(x,z)}$ by the principal ideal generated by $x$ is isomorphic to $k$, therefore $x$ is a local parameter in this ring. We have $$z=x^3-xz^2=x^3\left(1-\frac{z^2}{x^2}\right)=x^3(1-x^4+2x^2z^2-z^4).$$ In the factor ring $x^4-2x^2z^2+z^4=0$, therefore $1-x^4+2x^2z^2-z^4$ is a unit.