The Short Local Algebras of Dimension 6 with Non-projective Reflexive Modules

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 modules 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 showing the converse.