# The structure of the inverse system of Gorenstein k-algebras

## TL;DR

This paper extends Macaulay's inverse system to characterize Gorenstein k-algebras of any dimension, providing new methods for their construction and illustrating the results with examples.

## Contribution

It generalizes Macaulay's correspondence to higher-dimensional Gorenstein algebras and introduces effective construction techniques.

## Key findings

- Extended Macaulay's correspondence to all dimensions
- Provided methods for constructing Gorenstein graded rings
- Included illustrative examples of the new methods

## Abstract

Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given.