Reduced submodules of finite dimensional polynomial modules

TL;DR

This paper investigates the structure of reduced submodules in finite dimensional polynomial modules over a field of characteristic zero, revealing their connections to socles, Young diagrams, Macaulay inverse systems, and duality theories.

Contribution

It introduces a novel characterization of reduced submodules in polynomial modules, linking them to socles, Young diagrams, and Macaulay inverse systems, and explores their duality and torsion properties.

Findings

Largest reduced submodule equals the socle of the module.

Reduced submodules relate to outside corner elements of Young diagrams.

Modules satisfy the radical formula, exhibiting symmetry under duality.

Abstract

Let $k$ be a field with characteristic zero, $R$ be the ring $k[x_1, \cdots, x_n]$ and $I$ be a monomial ideal of $R$. We study the Artinian local algebra $R/I$ when considered as an $R$-module $M$. We show that the largest reduced submodule of $M$ coincides with both the socle of $M$ and the $k$-submodule of $M$ generated by all outside corner elements of the Young diagram associated with $M$. Interpretations of different reduced modules is given in terms of Macaulay inverse systems. It is further shown that these reduced submodules are examples of modules in a torsion-torsionfree class, together with their duals; coreduced modules, exhibit symmetries in regard to Matlis duality and torsion theories. Lastly, we show that any $R$-module $M$ of the kind described here satisfies the radical formula.