Koszul almost complete intersections

TL;DR

This paper classifies Koszul almost complete intersections and confirms that their Betti numbers are bounded by binomial coefficients based on the number of generators, answering a question about their algebraic structure.

Contribution

It provides a complete classification of Koszul almost complete intersections and affirms the Betti number bounds for these rings.

Findings

Classified the structure of Koszul almost complete intersections.

Confirmed Betti number bounds for all such rings.

Answered a key question in the theory of Koszul algebras.

Abstract

Let $R = S/I$ be a quotient of a standard graded polynomial ring $S$ by an ideal $I$ generated by quadrics. If $R$ is Koszul, a question of Avramov, Conca, and Iyengar asks whether the Betti numbers of $R$ over $S$ can be bounded above by binomial coefficients on the minimal number of generators of $I$. Motivated by previous results for Koszul algebras defined by three quadrics, we give a complete classification of the structure of Koszul almost complete intersections and, in the process, give an affirmative answer to the above question for all such rings.