Suppose that $ X $ is a locally factorial variety, so that $ \mathcal{O}_{X,x} $ is a UFD for all $ x \in X. $ I have been able to show that every prime divisor $ D $ on $ X $ is such that $ I(D) = (p_{i}) $ for some irreducible polynomial $ p_{i} \in \mathcal{O}_{X,x}. $ I have to show that from here, it follows that every effective Weil divisor on $ X $ is locally principal/Cartier, but I do not know how best to proceed.
Now, I have been looking at two primary sources for ideas on this, and they seem to be saying the same thing in different language. I would appreciate some clarification, or any general help with this. I feel that I am nearly there. I just need some clarity.
The following is taken from "Algebraic Geometry" by Masayoshi Miyanishi.
Let $ D $ be a Weil divisor on $ X $ and write $$ D = n_{1}Y_{1}+\dots + n_{r}Y_{r}, \text{ where } n_{i} \in \mathbb{Z}. $$ Each $ Y_{i} $ is defined by a principal ideal $ a_{i}\mathcal{O}_{X,x}, $ where $ a_{i} \in \mathcal{O}_{X,x}^{*} $ if $ x \notin Y_{i}. $ Put $ f_{x} = \prod_{i=1}^{r} a_{i} $ which is in general a rational function
So far so good. Here's the problematic part for me:
We can consider that $ f_{x} $ defines an open neighbourhood $ U_{x} $ of $ x $ the divisor $ D $ with the multiplicity $ n_{i} $ of each irreducible component $ Y_{i} $ counted in
I have four questions here.
1) I'm not sure what this means. Does this mean that $$ D = \text{div}(f_{x}) = \text{div}\Bigg(\prod_{i=1}^{r}a_{i}^{n_{i}} \Bigg) = \sum_{i} v_{Y_{i}} \Bigg(\prod_{i=1}^{r}a_{i}^{n_{i}}\Bigg)Y_{i}?$$ How would we compute that?
2) I was thinking that since $ a_{i} $ is a unit for all $ i, $ there must be some open neighbourhood $ U_{i} \ni x $ where $ a_{i} $ is invertible(and therefore also non-zero). I expected that from here, we can somehow construct a Cartier divisor from this since we need rational functions which do not vanish, and are invertible on some open sets. But I am not sure this picture is completely accurate.
3) On page 151 of "Basic Algebraic Geometry(Volume 1)", Shafarevich says that if $ D = \sum_{i}n_{i}Y_{i} $ is a divisor, and each $ Y_{i} $ is defined by a local equation $ \pi_{i}, $ then $ D = \text{div}(f) $ in some open $ U $ where $ f = \prod_{i} \pi_{i}^{n_{i}}. $ Why is this so?
4) Comparing the statement of Shafarevich with the previous statements, it looks like $ \pi = a_{i} p_{i}. $ Is this true?
Miyanishi continues:
$ \lbrace U_{x} : x \in X \rbrace $ is an open covering of $ X. $ We denote by $ \mathcal{U} = \lbrace U_{i} \rbrace_{i} $ this open covering and by $ f $ the defining equation of $ f $ corresponding to $ U_{i}. $
I'm not sure what this means.