Koszul Algebras Defined by Three Relations

TL;DR

This paper proves that for certain commutative Koszul algebras generated by three relations, the Betti numbers are bounded by binomial coefficients, confirming a conjecture in this case.

Contribution

It establishes the conjecture that Betti numbers are bounded by binomial coefficients for Koszul algebras with up to three generators.

Findings

Betti number bounds confirmed for g ≤ 3

Projective dimension is at most g for these algebras

Supports conjecture on Betti number bounds in Koszul algebras

Abstract

This work concerns commutative algebras of the form $R=Q/I$, where $Q$ is a standard graded polynomial ring and $I$ is a homogenous ideal in $Q$. It has been proposed that when $R$ is Koszul the $i$th Betti number of $R$ over $Q$ is at most $\binom gi$, where $g$ is the number of generators of $I$; in particular, the projective dimension of $R$ over $Q$ is at most $g$. The main result of this work settles this question, in the affirmative, when $g\le 3$.