Nancy Abdallah: Free resolutions and Lefschetz properties of AG rings

Abstract

In 1978, Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H-vector (1,13,12,13,1). Migliore-Zanello 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 at most 4 and gives a family where WLP fails for any r >= 6, building on an earlier example of Ikeda of failure for r=5. 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 at most 6. We discuss some open problems related to free resolutions, Leschetz properties and doubling constructions.