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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 latershowed that for regularity $r=4$, Stanley's example has the smallest possiblecodimension $c$ for an AG ring with non-unimodal $H$-vector. The weak Lefschetzproperty (WLP) has been much studied for AG rings; it is easy to show that anAG ring with non-unimodal $H$-vector fails to have WLP. In codimension $c=3$ itis conjectured that all AG rings have WLP. For $c=4$, Gondim showed that WLPalways holds for $r \le 4$ and gives a family where WLP fails for any $r \ge7$, building on an earlier example of Ikeda of failure of WLP for $r=5$. Inthis note we study the minimal free resolution of $A$ and relation to Lefschetzproperties (both weak and strong) and Jordan type for $c=4$ and $r \le 6$.

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