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

2022/11/30 by Ringel, Claus Michael · 1 citation
#Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2211.16885

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 J³ = 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: J² 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.

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• Claus Michael Ringel's main contributions to Gorenstein-projective modules

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