Quadratic Gorenstein algebras with many surprising properties

• Published in: Archiv der Mathematik 2020-11, Vol.115 (5), p.509-521
• Main Authors: McCullough, Jason, Seceleanu, Alexandra
• Format: Article
• Language: English
• Subjects: Algebra; Integers; Mathematics; Mathematics and Statistics; Properties (attributes); Regularity
• ISSN: 0003-889X; 1420-8938
• DOI: 10.1007/s00013-020-01492-x

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 ≥ 2 , with the following properties: (1) there are minimal first syzygies of the defining ideal in degree m + 2 , (2) for m ≥ 3 , A m is not Koszul, (3) for m ≥ 7 , the Hilbert function of A m is not unimodal, and thus (4) for m ≥ 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 α steps for any integer α ≥ 2 . Thus there is no finite test for the Koszul property even for quadratic Gorenstein algebras.