On the Rate of Generic Gorenstein K-Algebras

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 $A$-module. In particular $A$ has rate one if and only if $A$ is Koszul. It is known that a generic Artinian Gorenstein algebra of socle degree three is Koszul. We extend this result showing that if $A$ is a generic Artinian Gorenstein algebra of socle degree $s\ge 3$, then the rate of $A$ is $ \lfloor \frac{s}{2} \rfloor. $ In proving it, we get that a generic Artinian Gorenstein $K$-algebra of embedding dimension at least four and socle degree $s\ge 3$ is generated in degree $\lfloor \frac{s}{2} \rfloor +1.$ This gives a partial positive answer to a longstanding conjecture stated by M. Boij on the resolution of a generic Artinian Gorenstein ring of odd socle degree. This is a joint work with M.Boij, A. De Stefani and M.E. Rossi.

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