I've been studying Weil divisors from Vakil's FOAG Chapter 14.2. I'm somewhat confused, as to why every Weil divisor on a Noetherian normal irreducible scheme $X$ needn't be principal. My reasoning is as follows.
Let $D = \sum_Y n_Y[Y]$ be a Weil divisor. Now, for each codimension 1 point in $\operatorname{Supp} D$, we have a uniformizer $f_Y$. Now, considering the element $f = \prod f_Y^{n_Y}$ of $K(X)^{\times}$, don't we have that $\operatorname{div} f = D$?
I would glad if someone could explain where I am going wrong.
Thanks.
Best Answer
The uniformizer $f_Y$ may have zeroes or poles somewhere else, and this scuppers your argument. For instance, if $X=\Bbb P^1_k$, then taking standard coordinates, a uniformizer for the point $[0:1]$ is $\frac{x_0}{x_1}\in k(\Bbb P^1)$ which has a pole at $[1:0]$ and therefore $\operatorname{div} \frac{x_0}{x_1}=[0:1]-[1:0]$.