The cones of Hilbert functions of squarefree modules

TL;DR

This paper investigates the structure of Hilbert function cones for squarefree modules, providing their extremal rays, inequalities, and comparisons with classical bounds, thus extending combinatorial and algebraic understanding of these modules.

Contribution

It generalizes the concept of squarefreeness to modules and characterizes the cones of their Hilbert functions, including extremal rays and inequalities, with a focus on degree zero modules.

Findings

Describes the cones of Hilbert functions for squarefree modules

Identifies extremal rays and inequalities of these cones

Compares inequalities with Kruskal-Katona bounds

Abstract

In this paper, we study different generalizations of the notion of squarefreeness for ideals to the more general case of modules. We describe the cones of Hilbert functions for squarefree modules in general and those generated in degree zero. We give their extremal rays and defining inequalities. For squarefree modules generated in degree zero, we compare the defining inequalities of that cone with the classical Kruskal-Katona bound, also asymptotically.