The short local algebras of dimension 6 with non-projective reflexive modules

TL;DR

This paper investigates the structure of 6-dimensional local algebras with radical cube zero, focusing on the existence of non-projective reflexive modules and establishing conditions for their presence.

Contribution

It proves the converse of previous results, characterizing 6-dimensional short local algebras with non-projective reflexive modules.

Findings

6 is the minimal dimension for such algebras with non-projective reflexive modules

Conditions include $J^2$ being both the left and right socle of $A$

No uniform ideal of length 3 exists in these algebras

Abstract

Let $A$ be a finite-dimensional local algebra over an algebraically closed field, let $J$ be the radical of $A.$ The modules we are interested in are the finitely generated left $A$-modules. Projective modules are always reflexive, and an algebra is self-injective iff all modules are reflexive. We discuss the existence of non-projective reflexive module in case $A$ is not self-injective. We assume that $A$ is short (this means that $J^3 = 0$). In a joint paper with Zhang Pu, it has been shown that 6 is the smallest possible dimension of $A$ that can occur and that in this case the following conditions have to be satisfied: $J^2$ is both the left socle and the right socle of $A$ and there is no uniform ideal of length 3. The present paper is devoted to show the converse.