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.

  • $\begingroup$ 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. $\endgroup$ – 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.

  • $\begingroup$ This is a helpful concept, but why is it flasque? $\endgroup$ – Lorenzo Najt Sep 5 '15 at 18:03
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    $\begingroup$ @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. $\endgroup$ – hochs Sep 5 '15 at 18:05
  • $\begingroup$ Oh of course. Thank you! $\endgroup$ – Lorenzo Najt Sep 5 '15 at 18:06
  • $\begingroup$ So the same thing would be true if I tensored a line bundle with any flasque sheaf? $\endgroup$ – Lorenzo Najt Sep 5 '15 at 18:07
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    $\begingroup$ I was imagining pulling back inductively on some chosen finite cover of trivializations - intersecting the ith open with the union of the previous ones, and pulling back the restriction of the section construted on the union to the intersection. $\endgroup$ – Lorenzo Najt Sep 5 '15 at 18:59

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