So algebraic geometry gives one a way of thinking about about rings geometrically. Like prime ideals correspond to points in the spectrum of a ring, maximal ideals are closed points and so on. This works well for rings like $\mathbb{C}[x_1,..,x_n]$ and the formalism of an affine scheme extends to other rings such as $\mathbb{Z}$ as well.

My question then is how should I think of fractional ideals geometrically?

If an ideal of some integral domain $R$ corresponds to some closed set in $\operatorname{Spec}R$ with smaller ideals corresponding to bigger closed sets, then what should a really big fractional ideal like $\frac{1}{p} \mathbb{Z} \subset \mathbb{Q}$ correspond to? Maybe there is a scheme here that one can define that is bigger than $\operatorname{Spec}Z$ in this case?

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    $\begingroup$ Fractional ideals of $\mathbb Z$ correspond to line bundles (or invertible sheaves, or locally free sheaves of rank $1$) on $\mathrm{Spec} \mathbb Z$. $\endgroup$ – Fredrik Meyer Aug 28 '14 at 17:20

Locally principal ideals turn into locally principal ideal sheaves, which are same thing as invertible sheaves equipped with an embedding $\mathcal L \hookrightarrow \mathcal O_X.$

Locally principal fractional ideals turn into locally principal fractional ideal sheaves, which are the same things as invertible sheaves equipped with an embedding $\mathcal L \hookrightarrow \mathcal K_X$, where $\mathcal K_X$ is the sheaf of rational functions on $X$ (an integral scheme, say).

General ideal sheaves (say on a Noetherian scheme $X$) are just coherent sheaves equipped with an embedding $\mathcal F \hookrightarrow \mathcal O_X$, and general fractional ideal sheaves are just coherent sheaves equipped with an embedding $\mathcal F \hookrightarrow \mathcal K_X$.

So when you work geometrically, the single piece of data (an ideal or fractional ideal) converts into two pieces of data: an abstract sheaf of $\mathcal O_X$-modules (invertible, or, more generally, coherent) together with an embedding into some standard sheaf, either $\mathcal O_X$ or $\mathcal K_X$. Forgetting the embedding corresponds to looking at equivalence classes of fractional ideals (in the sense of the class group).

This post is related (although it works with the dual concept of sections, rather than embeddings).


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