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.