2020/04/21 by McCullough, Jason, Seceleanu, Alexandra · 1 citation
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2004.10237
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 Am, 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, Am is not Koszul, (3) for m ≥ 7, the Hilbert function of Am is not unimodal, and thus (4) for m ≥ 7, Am 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.