How to understand the quotient of $K(\mathbb{P}^1_{x_0,x_1})$ by a local ring of some point

Question

Let $K=K(\mathbb{P}^1_{x_0,x_1})$ be the function field of $\mathbb{P}^1$ at let $P\in\mathbb{P}^1$. I'm interested in understanding the abelian group $K/\mathcal{O}_{\mathbb{P}^1,P}$, where $\mathcal{O}_{\mathbb{P}^1,P}$ is the local ring at $P$. Is it true that any class $C\in K/\mathcal{O}_{\mathbb{P}^1,P}$ has a representant $f\in K$ of the form $f=\frac{a}{(P_1x_0-P_0x_1)^n}$ for some $a\in k[x_0,x_1]$ not vanishing at $P$? If not, how else could we describe $K/\mathcal{O}_{\mathbb{P}^1,P}$?

Answer 1

If I'm understanding correctly, you want to understand the quotient of $K$ by the local ring considered as an additive subgroup? First, I would say that since $\mathbb P^1$ is homogeneous there is no harm in choosing $P = (0:1)$. In this case, elements of the local ring look like $\frac{f}{x_0^n}$. Now choosing an arbitrary rational function $g(x_0,x_1)$, we can "Laurent expand" about $x_0=0$: $$ g(x_0,x_1) = \frac{g_{-n}(x_1)}{x_0^n} + \frac{g_{-n+1}(x_1)}{x_0^{n-1}} + \cdots \frac{g_{-1}(x_1)}{x_0} + g_0(x_1) + g_1(x_1)x_0 + \cdots g_n(x_1)x_0^m \hspace{30pt} $$ $$ \hspace{34pt}= \frac{g_{-n}(x_1)+ g_{-n+1}(x_1)x_0 + \cdots g_{-1}(x_1)x_0^{n-1}}{x_0^n} + g_0(x_1) + g_1(x_1)x_0 + \cdots g_n(x_1)x_0^m, $$

where $g_i \in k(x_1)$. In the group you are interested in, we thus have $$ g(x_0,x_1) \equiv g_0(x_1) + g_1(x_1)x_0 + \cdots g_n(x_1)x_0^m $$

which, as far as I can tell, is an arbitrary element of $k[x_0](x_1)$ (at least considered as an additive group).