# Why do global rational sections of line bundles exist?

Let $L$ be a line bundle on a smooth curve X (reduced, irreducible, scheme over a field $k$, or whatever). Then consider the sheaf $L \otimes_{O_X} K_X$, the sheaf of rational sections of $L$. Here $K_X$ is the sheaf of rational functions, defined on each open set to be just the field of rational functions $K(X)$.

From the point of view of a rational section as a section that is just defined on an open set (+ an equivalence relation), it is obvious that there are global rational sections of this new line bundle, since $L$ is locally free. However, I don't see this from the point of view of sheaf theory.

The book I am reading insists that one obviously comes from the local trivializations. I believe it, but I do not see how this works formally sheaf theoretically - I don't see why they patch together correctly.

• Just a comment. A different point of view, also using flasqueness but in a different packaging, is given in Proposition 9.29 of these note of mine. It uses the "internal point of view", in which sheaf of modules look like ordinary plain old modules. – Ingo Blechschmidt Dec 8 '19 at 19:13

You can cook up a compatible nonzero global section of $L \otimes_{\mathscr{O}_X} K_X$ because $L \otimes K_X|_{U}$ is a flasque on each open subset $U \subseteq X$ where $L$ is trivialized. Choose any nonzero section over any trivialization, restrict to overlaps with other trivializations, pull it back up to an element that restricts to this restriction (under a different trivialization), then glue.
• @AreaMan because on $L \otimes_{\mathscr{O}_X} K_X$ is a constant sheaf $K_X$ when restricted to a trivialization. We are on an integral scheme. – hochs Sep 5 '15 at 18:05