On the rate of generic Gorenstein $K$-algebras

TL;DR

The paper analyzes the Backelin rate of standard graded K-algebras, focusing on Artinian Gorenstein algebras. It derives an explicit graded Poincaré series for compressed Gorenstein algebras and obtains sharp bounds on the homological shifts, enabling a determination of the rate in the compressed case. For generic Artinian Gorenstein algebras with embedding dimension at least 4 and socle degree at least 3, it proves that the rate is $\left\lfloor\frac{s}{2}\right\rfloor$ and the defining ideal is generated in degree $\left\lfloor\frac{s}{2}\right\rfloor+1$, using a combination of Macaulay inverse systems, Golodness, and open dense subsets. The authors also construct explicit monomial level algebras to realize generation in the predicted degree for odd socle degrees, establishing a partial positive answer to Boij's conjecture at the start of resolutions and clarifying the structure of generic Artinian Gorenstein resolutions.

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.

Theorems & Definitions (24)

• Theorem A
• Theorem B
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Corollary 2.6