On the rate of generic Gorenstein $K$-algebras

Abstract

The rate of a standard graded $K$-algebra $A$ is a measure of the growth of the shifts in a minimal free resolution of $K$ as an $A$-module. In particular $A$ has rate one if and only if it is Koszul. It is known that a generic Artinian Gorenstein algebra of embedding dimension $n \geq 3$ and socle degree $s=3$ is Koszul. We prove that a generic Artinian Gorenstein algebra with $n\geq 4$ and $s \ge 3 $ has rate $ \lfloor \frac{s}{2} \rfloor. $ In the process we show that such an algebra is generated in degree $\lfloor \frac{s}{2} \rfloor +1. $ This gives a partial positive answer to a longstanding conjecture stated by the first author on the minimal free resolution of a generic Artinian Gorenstein ring of odd socle degree.