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

In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H-vector (1, 13, 12, 13, 1). Migliore-Zanello show in (Migliore and Zanello, 2017) 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 shows in (Gondim, 2017) that WLP always holds for r < 4 and gives a family where WLP fails for any r >= 7, building on Ikeda's example (Ikeda, 1996) of failure 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 < 6.(c) 2023 Elsevier Ltd. All rights reserved.

Automated summary

This article studies an AG ring with a non-unimodal H-vector and its relation to Lefschetz properties and Jordan type. By analyzing Stanley's example and other research results, we discuss the Lefschetz properties of AG rings under restrictions of codimension and regularity.