Algebraic Geometry - Find the pole sets of $z$ anda $z^2$.

Question

Let $V = V(Y² - X²(X + 1)) \subset \mathbb{A}^2$, and $\overline{X}$, $\overline{Y}$ the residues of $X$, $Y$ $\in$ $\Gamma (V)$; let $z = \frac{\overline{Y}}{\overline{X}} \in k(V)$. Find the pole sets of $z$ anda $z^2$.

In this solution Pole set of rational function defined on a variety, why "Because of the relation $Y^2 = X^2(X + 1)$, we see that any element of $\Gamma (V)$ can be written in a unique way as $f(X) + G(X)Y$"? Is this some algebra result? Sorry, I'm still pretty new to Algebra.

Answer 1

The ring you're considering is $k[X,Y]/(Y^2-X^2(X+1))$, so if you take any representative $f \in k[X,Y]$, then you may replace every $Y^2$ term by $X^2(X+1)$ and hence get rid of any higher powers of $Y$. A quick example should make all of this clear:

Example. In $k[X,Y]/(Y^2-X^2(X+1))$ we have $Y^5 = (Y^2)^2 Y = (X^2(X+1))^2 Y$.

With this in mind, you should also be able to convince yourself that the representative on the form $f(X) + g(X)Y$ is unique.