2021/01/24 by Mantero, Paolo, Mastroeni, Matthew
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2101.09803
Avramov, Conca, and Iyengar ask whether βiS(R) ≤ \binomgi for all i when R=S/I is a Koszul algebra minimally defined by g quadrics. In recent work, we give an affirmative answer to this question when g ≤ 4 by completely classifying the possible Betti tables of Koszul algebras defined by height-two ideals of four quadrics. Continuing this work, the current paper proves a structure theorem for Koszul algebras defined by four quadrics. We show that all these Koszul algebras are LG-quadratic, proving that an example of Conca of a Koszul algebra that is not LG-quadratic is minimal in terms of number of defining equations. We then characterize precisely when these rings are absolutely Koszul, and establish the equivalence of the absolutely Koszul and Backelin--Roos properties up to field extensions for such rings (in characteristic zero). The combination of the above paper with the current one provides a fairly complete picture of all Koszul algebras defined by g ≤ 4 quadrics.