# What is the relation between fractional ideals and divisors on curves?

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!

• The group of fractional ideals in a Dedekind domain is precisely the set of divisors in a smooth affine curve. – Mohan Nov 24 '16 at 21:13
• Chapter 14 of Ravi Vakil's Foundations Of Algebraic Geometry discusses the connection between fractional ideals, divisors, and invertible sheaves. – André 3000 Nov 25 '16 at 4:56
• @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. – u1571372 Nov 25 '16 at 16:31
• 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. – Mohan Nov 25 '16 at 17:34