Rational section line bundle

Let $X$ be an integral scheme (Integral schemes) with generic point $\eta$ (Generic points of irreducible closed subsets), let $K_X$ be its sheaf of meromorphic functions (Sheaf total quotient rings), and let $\mathcal{L}$ be an invertible ${\mathcal{O}}_X$-module (Invertible sheaves).

The sheaf of meromorphic sections of $\mathcal{L}$ is the ${\mathcal{O}}_X$-module $K_X(\mathcal{L})=\mathcal{L}{\otimes }_{{\mathcal{O}}_X}{K}_X,$ the tensor product of sheaves of modules (Tensor product of sheaves of modules). A meromorphic section of $\mathcal{L}$ is a global section of $K_X(\mathcal{L})$; it is regular, or a rational section, when it is nonzero.

On an integral scheme the sheaf $K_X$ is the constant sheaf with value the function field $K(X)={\mathcal{O}}_{X,\eta }$, so $K_X(\mathcal{L})$ is the constant sheaf with value the stalk ${\mathcal{L}}_{\eta }$. This stalk is a one-dimensional vector space over the function field: fixing a ${\mathcal{O}}_{X,\eta }$-basis of identity $1_{\eta }$ of the field $K(X)$, the vector space is $K(X){\otimes }_{{\mathcal{O}}_{X,\eta }}{\mathcal{L}}_{\eta }\cong K(X),$ and a rational section is a nonzero element of the one-dimensional $K(X)$-vector space ${\mathcal{L}}_{\eta }$. The stalk ${\mathcal{L}}_{\eta }$ is one-dimensional over $K(X)$ because $\mathcal{L}$ is locally free of rank one (Locally free sheaves of finite rank): on a neighbourhood of $\eta$ a generator identifies $\mathcal{L}$ with ${\mathcal{O}}_X$, and passing to stalks gives ${\mathcal{L}}_{\eta }\cong {\mathcal{O}}_{X,\eta }=K(X)$. A rational section is therefore the same thing as a $K(X)$-multiple of any chosen local generator of $\mathcal{L}$ near $\eta$, and two rational sections $s,s^{\mathrm{\prime }}$ satisfy $s^{\mathrm{\prime }}=gs$ for a unique $g\in K(X{)}^{\times }$ when both are nonzero.

On the empty scheme there is no generic point and no invertible module with a nonzero stalk, so the notation is not used there; on a nonempty integral scheme the generic point exists and the construction is never vacuous. The definition imposes no properness, finiteness or normality assumption on $X$; those enter only when one wants to associate divisors to the sections.