An Extension of a Theorem of Frobenius and Stickelberger to Modules of Projective Dimension One over a Factorial Domain

TL;DR

This paper extends classical theorems of Frobenius and Stickelberger to modules of projective dimension one over factorial domains, introducing quasi-Gorenstein modules and characterizing certain diagonal matrices.

Contribution

It introduces a new class of quasi-Gorenstein modules and generalizes existing theorems to modules of projective dimension one over factorial domains.

Findings

Characterization of diagonal matrices of maximal rank over factorial domains

Extension of Frobenius and Stickelberger theorems to modules of projective dimension one

Properties of finitely generated quasi-Gorenstein modules

Abstract

Let $R$ be a commutative ring. A quasi-Gorenstein $R$-module is an $R$-module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Noetherian factorial domain $R$ extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative Noetherian factorial domain.