Let $X$ be a Noetherian, regular and integral scheme with function field $K$. Moreover let $L$ be an invertible sheaf on $X$. Take a nonzero rational section $s\in \Gamma(L\otimes_{\mathcal O_X} K)$, then at each point $x\in X$ we can write $s_x=f_xe_x$ for $f_x\in K$ and $e_x\in L$. In other words we have chosen a local basis $\{e_x\}_{x\in X}$ for $L$.-
If $x$ is a point of codimension $1$ we have a discrete valuation on $K$ associated to $x$, and we denote it simply by $v_x$. Now we can put:
$$\operatorname{ord}_x(s):=v_x(f_x)$$
It is easy to show that $\operatorname{ord}_x(s)$ doesn't depend on the local basis chosen, but
I don't understand why $\operatorname{ord}_x(s)=0$ for all but finitely many points of codimension $1$.
Can you please explain it?
Edit: Equivalently a rational section $s$ can be seen as a nonzero element of $L_{\eta}$ where $\eta$ is the generic point of $X$. Clearly for any $x\neq \eta$ we have $L_\eta=L_x\otimes_{\mathcal O_{X,x}} K$. The problem is still there. I don't understand why the order is zero almost everywhere.
I think my hint should have made you think what to do.
First, you can find an affine open set $U=\mathrm{Spec}\, A$ where $L|U\cong A$.Since there are only finitely many irreducible components in $X-U$, we can ignore these for your question and thus assume $X=\mathrm{Spec}\, A$ and $L\cong A$. Now, $s$ is a non-zero rational section of $L$ implies $s$ can be written as $a/b$, $a,b\in A$, both non-zero. There are only finitely many codimension one supports in $a=0$ and $b=0$ and it is clear that for any $x$ of codimension one other than these, $\nu_x(s)=0$.