Poincar\'e series and deformations of Gorenstein local algebras with low socle degree

TL;DR

This paper characterizes Artinian Gorenstein local algebras with low socle degree, proving rational Poincaré series and smoothability under certain conditions, and explores properties of generic algebras with socle degree three.

Contribution

It provides a structure theorem for Gorenstein algebras with $M^4=0$, demonstrating rational Poincaré series and smoothability criteria, and analyzes generic algebras with socle degree three.

Findings

Gorenstein algebras with $M^4=0$ have rational Poincaré series.

Such algebras are smoothable if $ ext{dim}_K M^2/M^3 extless= 4$.

Generic Gorenstein algebras with socle degree three have rational Poincaré series.

Abstract

Let $K$ be an algebraically closed field of characteristic $0$, and let $A$ be an Artinian Gorenstein local commutative and Noetherian $K$--algebra, with maximal ideal $M$. In the present paper we prove a structure theorem describing such kind of $K$--algebras satisfying $M^4=0$. We use this result in order to prove that such a $K$--algebra $A$ has rational Poincar\'e series and it is always smoothable in any embedding dimension, if $\dim_K M^2/M^3 \le 4$. We also prove that the generic Artinian Gorenstein local $K$--algebra with socle degree three has rational Poincar\'e series, in spite of the fact that such algebras are not necessarily smoothable.