Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four

TL;DR

This paper investigates the structure and Lefschetz properties of certain Artin Gorenstein rings of codimension four, focusing on their free resolutions, Lefschetz properties, and Jordan types for specific regularities.

Contribution

It provides new insights into the minimal free resolutions and Lefschetz properties of Artin Gorenstein rings with codimension four and regularity up to six.

Findings

Analysis of minimal free resolutions for c=4, r ≤ 6

Conditions for weak and strong Lefschetz properties

Relationship between Jordan type and Lefschetz properties

Abstract

In 1978, Stanley constructed an example of an Artinian Gorenstein (AG) ring $A$ with non-unimodal $H$-vector $(1,13,12,13,1)$. Migliore-Zanello later showed that for regularity $r=4$, Stanley's example has the smallest possible codimension $c$ for an AG ring with non-unimodal $H$-vector. The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal $H$-vector fails to have WLP. In codimension $c=3$ it is conjectured that all AG rings have WLP. For $c=4$, Gondim showed that WLP always holds for $r \le 4$ and gives a family where WLP fails for any $r \ge 7$, building on an earlier example of Ikeda of failure of WLP for $r=5$. In this note we study the minimal free resolution of $A$ and relation to Lefschetz properties (both weak and strong) and Jordan type for $c=4$ and $r \le 6$.