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Short version: Why is the normalization of a separated scheme again separated?

Long version: I am trying to understand the construction of the normalization of an irreducible variety (or scheme if you want). In the affine case one just takes the spectrum of the integral closure of the coordinate ring, this is okay for me. For arbitrary varieties one takes an affine cover normalizes the pieces and glues them together, this is also okay to me. However I don't see why the result is separated. We have an affine cover with affine intersections for free, but I don't see why the coordinate rings of the intersections are generated by coordinate rings of the pieces I intersect. Maybe I am missing a simple fact in commutative algebra.

Any explanation or a reference where separatedness ist proven?

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The normalization map is always affine, and an affine morphism is separated. Note that the composite of two separated morphisms is separated. –  Akhil Mathew Jun 30 '11 at 18:10
    
Thanks Akhil, this is very clear and elegant. However I would still be interested in an elementary argument, entirely in the language of varieties. –  Jan Jun 30 '11 at 19:37
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Btw the AG notes on your page were very useful to check the details in your argument! –  Jan Jun 30 '11 at 19:40
    
Thanks! I thought about proving this result via an "elementary" approach as you sketched but I am afraid it is not so clear to me. Sorry for not being more helpful here. –  Akhil Mathew Jul 2 '11 at 15:57
    
You were already very helpful! If you post your comment as an answer, I will happily accept it. –  Jan Jul 4 '11 at 8:36

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