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Hartshorne gives a correspondence between Cartier divisors of X and Weil Divisors of X, when X is integral separated locally factorial noetherian scheme.

I understand given a Cartier Divisor how to define a Weil Divisor. But I don't understand the the other way correspondence.

I have seen a proof of how a Weil Divisor D induces a Weil Divisor $D_x$ om the local scheme Spec$\mathcal O_x$ in Restricting a Weil divisor to a local scheme. And $D_x$ is a principal divisor. But I don't understand after that.

Can someone please help Thanks in advance.

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  • $\begingroup$ You can take a look at (EGA, IV_4, 21.6). Probably there is also a translation in Goertz-Wedhorn. $\endgroup$ – user314 Dec 17 '13 at 16:45
  • $\begingroup$ I couldn't find a proof $\endgroup$ – Babai Dec 17 '13 at 17:26
  • $\begingroup$ It's theorem 21.6.9 on page 274: numdam.org/item?id=PMIHES_1967__32__5_0 $\endgroup$ – user314 Dec 17 '13 at 17:50
  • $\begingroup$ Unfortunately I don't know French. $\endgroup$ – Babai Dec 17 '13 at 18:12
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    $\begingroup$ Then look at Goertz-Wedhorn... Theorem 11.38 on page 307. $\endgroup$ – user314 Dec 17 '13 at 19:35
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The point is that in a factorial domain, the height one prime ideals are principal.

By definition a Weil divisor gives a height one prime ideal in the local ring a each point (this is the ideal that cuts out the Weil divisor), and if this local ring is factorial, it is principal, so we get an equation that cuts out the Weil divisor in a n.h. of this point. And a divisor cut out by a single equation is precisely a Cartier divisor.

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