Thinking About Fractional Ideals Geometrically

Question

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?

Answer 1

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).

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