Valuation of a rational section of an invertible sheaf at a codim 1 point

Question

Let $X$ be an integral scheme. Let $\mathcal{L}$ be an invertible sheaf. By a rational section of $\mathcal{L}$ I mean a pair $(s, U)$ where $s$ is a section of $\mathcal{L}$ over $U$ defined up to the equivalence relation of agreeing on a smaller open subset. If $Y$ is an irreducible closed codimension $1$ subset of $X$ with generic point $\eta$, it is claimed that one can define the valuation of a rational section of $\mathcal{L}$ at $\eta$. The claim is that this can be done by considering an isomorphism $$ \psi_\eta : \mathcal{L}_\eta \longrightarrow \mathcal{O}_\eta $$ and then taking the valuation in $\mathcal{O}_\eta$. But I don't see how this is possible in general. It may be the case that there are no open subsets of $X$ containing $\eta$ on which $s$ is defined. Then it won't be represented in $\mathcal{L}_\eta$ at all. I know that in this situation it is supposed to have a negative valuation. But how do you actually get that valuation?

Answer 1

Instead of taking $\eta$ to be the generic point of $Y$, one should take $\eta$ to be the generic point of $X$. Then the composite $$\mathcal{L}(U)\to \mathcal{L}_\eta \to \mathcal{O}_\eta = K(X)$$ sends $s$ to an element of $K(X)$ and one applies the valuation corresponding to $Y$ inside $K(X)$.