The structure of Koszul algebras defined by four quadrics.

Avramov, Conca, and Iyengar ask whether β i S (R) ≤ ( g i ) 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. [ABSTRACT FROM AUTHOR]