Forcing the Strong Lefschetz and the Maximal Rank Properties

TL;DR

This paper characterizes the Hilbert functions that force all standard graded artinian algebras to have the Strong Lefschetz Property and the Maximal Rank Property, revealing their relationship and differences from the Weak Lefschetz Property.

Contribution

It provides the first characterizations of Hilbert functions that guarantee SLP and MRP, showing these classes are strictly smaller than those for WLP, and discusses characteristic zero assumptions.

Findings

Characterizations of Hilbert functions for SLP and MRP

SLP and MRP characterizations coincide

SLP can fail in positive characteristic for certain algebras

Abstract

Three basic properties that standard graded artinian $k$-algebras may or may not enjoy are the Weak and Strong Lefschetz Properties and the Maximal Rank Property (respectively WLP, SLP, and MRP). In this paper we will assume that the base field $k$ has characteristic zero. It is known that SLP implies MRP, which in turn implies WLP, but that both implications are strict. However, it surprisingly turned out that the set of Hilbert functions admitting any algebras with WLP coincides with the corresponding set for SLP (and therefore with that for MRP). J. Migliore and the first author, using Green's theorem and a result of Wiebe, characterized the Hilbert functions forcing all algebras to enjoy WLP. The purpose of this note is to prove the corresponding characterizations for both SLP and MRP. Unsurprisingly (or surprisingly??), the two characterizations coincide, but they define a class of Hilbert functions strictly smaller than that determined for WLP. Our methods include the Herzog-Popescu theorem on quotients of $k$-algebras modulo a general form, a result of Wiebe, and gins and stable ideals. At the end, we will also discuss the importance of assuming that the characteristic be zero, and we will exhibit a class of codimension 2 monomial complete intersections for which SLP (but not MRP) fails in positive characteristic.