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Nov 21, 2014 at 14:59 comment added Marten Thank you! Yes, I think you're right, it can't be deduced from 5.1.2. The equivalence of categories stated there should give the isomorphism for rank one locally free $mathcal{O}_X$-modules.
Nov 21, 2014 at 14:54 vote accept Marten
Nov 13, 2014 at 18:03 history answered ChrisLazda CC BY-SA 3.0