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et X be a noetherian scheme and let F and G be two locally free sheaves of the same rank over X . Is it true that for all x∈X we have fx:Fx→Gx is an isomorphism and if true, My question is the following:why couldnʻt say that F and G are isomorphic ?

Amin.H
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – CrSb0001 Sep 26 '23 at 18:54
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    Look up "invertible sheaves" and the relation to divisors and the Picard group of the scheme... – Daniel Schepler Sep 26 '23 at 19:02
  • Let X be a noetherian scheme and let F and G be two locally free sheaves of the same rank over X .

    Of course, for all x∈X we have , fx:Fx→Gx that is an isomorphism.

    My question is the following: can we say F and G are isomorphic ?

     – Amin.H Sep 26 '23 at 20:15
  • Please use MathJax in your questions and comments (see tutorial on meta, I think). As for your question, see Daniel Schepler’s comment, or, even more concretely, consider locally free modules of rank one over $\operatorname{Spec}{\mathbb{Z}[\sqrt{-5}]}$. – Aphelli Sep 26 '23 at 20:16
  • Welcome to MSE. This is a very natural question to come up with at this stage in your algebraic geometry study, and luckily it has already been answered on this website before. In the future, please remember to check to see if your question already exists before posting a new one. – KReiser Sep 26 '23 at 22:31

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