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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 \(\binom{g}{i}\), 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.

Koszul Algebras Defined by Three Relations