This is from Exercise 2.4.P. June 2013 version of Ravi Vakil's Math 216 notes. The idea is to show $\mathscr{O}_X \xrightarrow{\text{exp}} \mathscr{O}^*_X$ is an epimorphism. It seems straightforward to show surjectivity of stalks by invoking the fact that the logarithm exists for simply connected components. I'm wondering if there is a more categorical approach I'm missing?

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    $\begingroup$ Why should there be a more categorical approach? This is a property of a very specific morphism. $\endgroup$ – Zhen Lin Jun 10 '14 at 5:36
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    $\begingroup$ I agree with Zhen Lin. The question doesn't make much sense ... $\endgroup$ – Martin Brandenburg Jun 10 '14 at 17:17
  • $\begingroup$ I don't even know what «more categorical» might mean in this context :-) $\endgroup$ – Mariano Suárez-Álvarez Jun 13 '14 at 0:55
  • $\begingroup$ I guess I should clarify... I was thinking of something along the lines of deducing a property of the $\mathscr{O}_X$ and $\mathscr{O}_X^*$ categories and then saying using something along the lines of "for any sheaf morphism from $\mathscr{O}_X$ to some object $A$ there is at most one commuting morphism because..." instead of using the stalk properties. Given the flow of the book and the comment immediately after the problem, it struck me as the way to start approaching the problem. It seems the consensus is that I didn't miss anything simple and am now just being dumb :( $\endgroup$ – arando Jun 14 '14 at 1:37

Don't use stalks, use the following characterization of epis of sheaves: Every section in the target admits a covering such that each restricted section has a preimage in the source. Here, you only need the trivial direction that such a morphism is an epi. Therefore, the existence of local logarithms is precisely what we need. You cannot get this for free by abstract nonsense, because somewhere we really have to use that we are looking at the exponential function.


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