Quadratic Gorenstein algebras with many surprising properties

Abstract

Let $k$ be a field of characteristic $0$. Using the method of idealization, we show that there is a non-Koszul, quadratic, Artinian, Gorenstein, standard graded $k$-algebra of regularity $3$ and codimension $8$, answering a question of Mastroeni, Schenck, and Stillman. We also show that this example is minimal in the sense that no other idealization that is non-Koszul, quadratic, Artinian, Gorenstein algebra, with regularity $3$ has smaller codimension. We also construct an infinite family of graded, quadratic, Artinian, Gorenstein algebras $A_m$, indexed by an integer $m\ge 2$, with the following properties: (1) there are minimal first syzygies of the defining ideal in degree $m+2$, (2) for $m\ge 3$, $A_m$ is not Koszul, (3) for $m\ge 7$, the Hilbert function of $A_m$ is not unimodal, and thus (4) for $m\ge 7$, $A_m$ does not satisfy the weak or strong Lefschetz properties. In particular, the subadditivity property fails for quadratic Gorenstein ideals. Finally, we show that the idealization of a construction of Roos yields non-Koszul quadratic Gorenstein algebras such that the residue field $k$ has a linear resolution for precisely $\alpha$ steps for any integer $\alpha \ge 2$. Thus there is no finite test for the Koszul property even for quadratic Gorenstein algebras.

Added new section showing there is no finite test for the Koszul property for quadratic Gorenstein algebras