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

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 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.

Keywords

Short local algebra / Reflexive module / Gorenstein-projective module / Bristle / Atom / Bar / Bristle-bar layout

References

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Funding

Universität Bielefeld (3146)