The socle module of a monomial ideal

TL;DR

This paper introduces the socle module of an ideal in Noetherian local rings and graded algebras, exploring its properties, finite generation, and structure in special cases like polynomial rings and graph edge ideals.

Contribution

It defines the socle module for ideals in various algebraic settings and investigates its structure, finite generation, and applications to graph and polymatroidal ideals.

Findings

Socle module is finitely generated over the fiber cone.

Defined the module ^*(I) over the Rees ring for ideals with linear resolutions.

Studied the structure of socle modules for graph edge ideals and polymatroidal ideals.

Abstract

For any ideal $I$ in a Noetherian local ring or any graded ideal $I$ in a standard graded $K$-algebra over a field $K$, we introduce the socle module $\mathrm{Soc}(I)$, whose graded components give us the socle of the powers of $I$. It is observed that $\mathrm{Soc}(I)$ is a finitely generated module over the fiber cone of $I$. In the case that $S$ is the polynomial ring and all powers of $I\subseteq S$ have linear resolution, we define the module $\mathrm{Soc}^*(I)$ which is a module over the Rees ring of $I$. For the edge ideal of a graph and for classes of polymatroidal ideals we study the module structure of their socle modules.