LAR's and PLAR's, during SARS

Abstract

The Weak Lefschetz Property (WLP) of polynomial quotient rings is studied in commutative algebra for the implications it has for a ring's Hilbert function, which is a chief object of study in commutative algebra. As simplicial complexes are a bridge between spatial and algebraic objects of study, we focus on the WLP and Stanley-Reisner rings, the algebraic allegory to simplical complexes, specifically squarefree Stanley-Reisner rings. We give an introduction to the key ideas here by following the results of a paper which develops several results for the WLP and squarefree Stanley-Reisner rings, while giving a modest generalization of these results and examining other related propositions.

Polarization is an operation on quotient rings of monomial ideals which returns a quotient ring whose ideal is generated entirely by squarefree monomials and has the same Hilbert function and free resolution. We examine the question of whether there is an analogous and natural operation which returns an Artinian ring with a similar or predictable Hilbert function. One property which we hope to preserve between them is called levelness. We provide compelling evidence that such an operation does not exist and we examine the relation between the WLP and levelness.