Let $X$ be a normal Noetherian algebraic space and $\mathscr{L}$ a line bundle on $X$. If $X$ is a scheme, then there is locally principal Weil divisor on $X$ that gives rise to $\mathscr{L}$. Is the same true in general? Something like this seems to be used without explanation on top of p. 263 of "Neron models," and I couldn't find a reference that would discuss such things (instead of assuming that they are known).
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Same proof as the scheme case works: Choose a meromorphic section and take the associated Weil divisor. This makes sense because algebraic spaces are schemes in codimension one, see Lemma Tag 0ADD.
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$\begingroup$ Thank you! Is separatedness important in the result you reference? In the reference I've mentioned this is applied to the relative identity component $(P, Y)^0$ of some rigidified Picard functor, and it doesn't seem to have been mentioned before that $(P, Y)^0$ is necessarily separated (part of the point of the proof I am reading is to prove the separatedness of the nonrigidified version $P^0$). $\endgroup$ Commented Jan 30, 2015 at 3:20
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1$\begingroup$ Sorry, I did not read your question carefully enough. For the general nonseparated case, in stead of using local ring, you could take the order of vanishing of the meromorphic section in the henselian local ring of the algebraic space at codimension 1 points (these henselian local rings have the correct residue field). $\endgroup$– user45795Commented Jan 30, 2015 at 3:41