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I've done courses in Algebraic Number Theory and Algebraic Geometry, where I learned the theory of Dedekind domains (in ANT) and Divisors (in AG). Now, the wikipedia article on divisors says the following:

"The name "divisor" goes back to the work of Dedekind and Weber, who showed the relevance of Dedekind domains to the study of algebraic curves. The group of divisors on a curve (the free abelian group on its set of points) is closely related to the group of fractional ideals for a Dedekind domain."

So my question is:

What is this relation between fractional ideals and the group of divisors on a curve?

I would also appreciate some references that explain this relation in detail. Thank you in advance!

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  • $\begingroup$ The group of fractional ideals in a Dedekind domain is precisely the set of divisors in a smooth affine curve. $\endgroup$ – Mohan Nov 24 '16 at 21:13
  • $\begingroup$ Chapter 14 of Ravi Vakil's Foundations Of Algebraic Geometry discusses the connection between fractional ideals, divisors, and invertible sheaves. $\endgroup$ – André 3000 Nov 25 '16 at 4:56
  • $\begingroup$ @Mohan, could you elaborate a bit more, or perhaps give a reference for that connection? I don't quite see how to construct an affine curve from the group of fractional ideals. $\endgroup$ – u1571372 Nov 25 '16 at 16:31
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    $\begingroup$ You can not construct a curve from the fractional ideals, since the ring comes first, to even talk of fractional ideals. So, given a smoorth affine curve, the ring of regular functions is a Dedekind domain. The closed points are defined by maximal ideals and the fractional ideals are of the form $P_1^{r_1}\cdots P_n^{r_n}$, $P_i$ maximal and $r_i\in\mathbb{Z}$. But, these correspond to $\sum r_ix_i$, a divisor where $x_i$ are the points corresponding to $P_i$ s and vice versa. $\endgroup$ – Mohan Nov 25 '16 at 17:34
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The relation is what Mohan mentioned in his comment. Since you asked for reference, I will recommend section II.6 in Hartshorne's Algebraic Geometry. Example II.6.3.2. tells you precisely about this relation. Proposition II.6.2. generalizes the fact that one usually proves in number theory after showing unique factorization ideals, namely, that a Dedekind domain is a UFD if and only if it is a PID.

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