Artinian Gorenstein algebras that are free extensions over ${\sf k}[t]/(t^n)$, and Macaulay duality

TL;DR

This paper investigates conditions under which Artinian Gorenstein algebras, as free extensions over ${ m k}[t]/(t^n)$, maintain Macaulay duality, providing criteria, examples, and exploring special cases.

Contribution

It establishes sufficient conditions on Macaulay dual generators for free extensions over ${ m k}[t]/(t^n)$ to be Gorenstein, and analyzes related examples and special cases.

Findings

Provided criteria for Gorenstein free extensions using dual generators

Constructed examples demonstrating the sharpness of the criteria

Explored special coinvariant algebras that are free extensions but do not meet the criteria

Abstract

T. Harima and J. Watanabe studied the Lefschetz properties of free extension Artinian algebras $C$ over a base $A$ with fibre $B$. The free extensions are deformations of the usual tensor product, when $C$ is also Gorenstein, so are $A$ and $B$, and it is natural to ask for the relation among the Macaulay dual generators for the algebras. Writing a dual generator $F$ for $C$ as a homogeneous "polynomial" in $T$ and the dual variables for $B$, and given the dual generator for $B$, we give sufficient conditions on $F$ that ensure that $C$ is a free extension of $A={\sf k}[t]/(t^n)$ with fiber $B$. We give examples that explore the sharpness of the statements. We also consider a special set of coinvariant algebras $C$ which are free extensions of $A$, but which do not satisfy the sufficient conditions of our main result.