This is an example in Fulton, Algebraic Curves.

Let $V= V(xw-yz) \subset \mathbb{A^{4}(k)}$, and $\Gamma(V)$, the co-ordinate ring of $V$ is $k[x,y,z,w]/ (xw-yz)$.

and, $\bar{x}, \bar{y}, \bar{z}, \bar{w}$ denote residues of $x,y,z,w$ in $\Gamma(V)$.

Then $\frac{\bar{x}}{\bar{y}}= \frac{\bar{z}}{\bar{w}} =f \in k(V)$, (the field of rational functions on $V$) is defined at $P=(x,y,z,w) \in V$ if $y,w, \neq 0$

My question- I think points $P$ where f is defined should be where $\bar{z}, \bar{y} \neq \bar{0}$, the points $(x,y,z,w)$ where $y,z$ are not in $I(V)= (xw-yz)$. How come the book says otherwise? I'm confused, can someone help?



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