Schemes 37: Comparison of Weil and Cartier divisors

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This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne.
In this lecture we compare Cartier and Weil divisors, showing that for Noethernian integral schems the map from Cartier to Weil divisors is injective if the scheme is normal ind an isomorphism if the scheme is locally factorial. We also give some examples where the the map is not injective or not surjective.
  1. Richard E Borcherds
  2. Weil divisor
  3. Cartier divisor
  4. locally factorial
  5. normal scheme
  6. algebraic geometry
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You only assume X is a Noetherian integral scheme, but in Hartshorne, it seems that we have to add "separated".
I wonder why you don't assume it. Thanks to great videos!

algeot
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10:30 p is the unique maximal ideal in the local ring R=k[[t^2, t^3]], so shouldn't R_p = R?
For the same reason, I am not convinced of the proof at 8:50 - for example 1+at has no zeros or poles, but 1/1+at is not in R_p = R.

rosieshen