Koszul algebras defined by three relations

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 ith Betti number of R over Q is at most (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 ≤ 3.