Betti numbers of Koszul algebras defined by four quadrics

TL;DR

This paper determines the Betti tables of height two ideals generated by four quadrics that define Koszul algebras, affirmatively answering a question about bounding Betti numbers by binomial coefficients.

Contribution

It provides a complete classification of Betti tables for four quadrics defining Koszul algebras, extending previous results to this specific case.

Findings

Betti tables are explicitly determined for the case of four quadrics.

The question of bounding Betti numbers by binomial coefficients is affirmatively answered for this case.

The results extend understanding of Koszul algebras defined by quadrics.

Abstract

Let $I$ be an ideal generated by quadrics in a standard graded polynomial ring $S$ over a field. A question of Avramov, Conca, and Iyengar asks whether the Betti numbers of $R = S/I$ over $S$ can be bounded above by binomial coefficients on the minimal number of generators of $I$ if $R$ is Koszul. This question has been answered affirmatively for Koszul algebras defined by three quadrics and Koszul almost complete intersections with any number of generators. We give a strong affirmative answer to the above question in the case of four quadrics by completely determining the Betti tables of height two ideals of four quadrics defining Koszul algebras.