On the Weak Lefschetz Property for Hilbert functions of almost complete intersections

TL;DR

This paper proves that all Hilbert functions of almost complete intersection Artinian graded algebras of codimension 3 are Weak Lefschetz sequences, extending known properties of complete intersections.

Contribution

It establishes that the Hilbert functions of almost complete intersection Artinian algebras of codimension 3 are Weak Lefschetz sequences, a property previously known only for complete intersections.

Findings

Hilbert functions are unimodal.

First difference forms an O-sequence.

All such Hilbert functions are Weak Lefschetz sequences.

Abstract

It is known that all complete intersection Artinian standard graded algebras of codimension 3 have the Weak Lefschetz Property. Unfortunately, this property does not continue to be true when you increase the number of minimal generators for the ideal defining the algebra. For instance, it is not more valid for almost complete intersection Artinian standard graded algebras of codimension 3. On the other hand, the Hilbert functions of all Weak Lefschetz Artinian graded algebras are unimodal and with the positive part of their first difference forming an $O$-sequence (i.e. are Weak Lefschetz sequences). In this paper we show that all the Hilbert functions of the almost complete intersection Artinian standard graded algebras of codimension 3 are Weak Lefschetz sequences.