The Structure of the Inverse System of Level $K$-Algebras

TL;DR

This paper extends the understanding of Macaulay's inverse system by establishing a correspondence between d-dimensional level K-algebras and specific submodules of the divided power ring, generalizing previous results for Gorenstein algebras.

Contribution

It introduces a new correspondence for level K-algebras, broadening the scope of inverse system applications beyond Gorenstein cases.

Findings

Established a one-to-one correspondence for level K-algebras and submodules.

Provided examples illustrating the new correspondence.

Extended the structure theory of inverse systems to level algebras.

Abstract

Macaulay's inverse system is an effective method to construct Artinian K-algebras with additional properties like, Gorenstein, level, more generally with any socle type. Recently, Elias and Rossi gave the structure of the inverse system of $d$-dimensional Gorenstein K-algebras for any $d>0$. In this paper we extend their result by establishing a one-to-one correspondence between $d$-dimensional level K-algebras and certain submodules of the divided power ring. We give several examples to illustrate our result.