Simplicial complexes and Macaulay's inverse systems

TL;DR

This paper explores the structure of certain artinian algebras derived from simplicial complexes using Macaulay's inverse systems, identifying conditions under which these algebras are level, thus linking combinatorial topology with algebraic properties.

Contribution

It provides an explicit description of the socle of these algebras and characterizes simplicial complexes that admit a tuple making the algebra level.

Findings

Explicit socle description in terms of simplicial complexes

Identification of 'levelable' complexes for level algebra existence

Connection between combinatorial and algebraic properties

Abstract

Let $\Delta$ be a simplicial complex on $V = \{x_1,...,x_n\}$, with Stanley-Reisner ideal $I_{\Delta}\subseteq R = k[x_1,...,x_n]$. The goal of this paper is to investigate the class of artinian algebras $A=A(\Delta,a_1,...,a_n)= R/(I_{\Delta},x_1^{a_1},...,x_n^{a_n})$, where each $a_i \geq 2$. By utilizing the technique of Macaulay's inverse systems, we can explicitly describe the socle of $A$ in terms of $\Delta$. As a consequence, we determine the simplicial complexes, that we will call {\em levelable}, for which there exists a tuple $(a_1,...,a_n)$ such that $A(\Delta,a_1,...,a_n)$ is a level algebra.