On a Projective Space Invariant of a Co-torsion Module of Rank Two over a Dedekind Domain

TL;DR

This paper introduces a new projective space invariant for rank two co-torsion modules over Dedekind domains, enabling complete classification and enumeration of such modules using ideal factorizations and relating their zeta functions to Dedekind zeta functions.

Contribution

It defines a novel projective space invariant for modules, linking module classification to ideal factorizations and projective spaces, and computes related zeta functions in number rings.

Findings

Invariant element in projective space determines module uniquely.

Zeta function of modules expressed via Dedekind zeta function.

Reinterpretation of Chinese remainder theorem through projective space intersection.

Abstract

For a Dedekind domain $\mathcal{O}$ and a rank two co-torsion module $M\subseteq \mathcal{O}^2$ with invariant factor ideals $\mathcal{L}\supseteq \mathcal{K}$ in $\mathcal{O}$, that is, $\frac{\mathcal{O}^2}{M}\cong \frac{\mathcal{O}}{\mathcal{L}}\oplus \frac{\mathcal{O}}{\mathcal{K}}$ we associate a new projective space invariant element in $\mathbb{PF}^1_{\mathcal{I}}$ where $\mathcal{I}$ is given by the ideal factorization $\mathcal{K} = \mathcal{L}\mathcal{I}$ in $\mathcal{O}$. This invariant element along with the invariant factor ideals determine the module $M$ completely as a subset of $\mathcal{O}^2$. As a consequence, projective spaces associated to ideals in $\mathcal{O}$ can be used to enumerate such modules. We compute the zeta function associated to such modules in terms of the zeta function of the one dimensional projective spaces for the ring $\mathcal{O}_K$ of integers in a number field $K/\mathbb{Q}$ and relate them to Dedekind zeta function. Using the projective spaces as parameter spaces, we re-interpret the Chinese remainder reduction isomorphism $\mathbb{PF}^1_{\mathcal{I}} \rightarrow \underset{i=1}{\overset{l}{\prod}} \mathbb{PF}^1_{\mathcal{I}_i}$ associated to a factorization of an ideal $\mathcal{I}=\underset{i=1}{\overset{l}{\prod}} \mathcal{I}_i$ into mutually co-maximal ideals $\mathcal{I}_i,1\leq i\leq l$ in terms of the intersection of associated modules arising from the projective space elements.