The structure of Koszul algebras defined by four quadrics

Avramov, Conca, and Iyengar ask whether $\beta_i^S(R) \leq \binom{g}{i}$ 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 $g \leq 4$ 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 $g \leq 4$ quadrics.