A Description of Totally Reflexive Modules for a Class of non-Gorenstein Rings

TL;DR

This paper characterizes totally reflexive modules over specific non-Gorenstein rings, providing a matrix presentation form and establishing a bijection with matrix tuples, advancing understanding of module classification in this context.

Contribution

It introduces a explicit matrix presentation for totally reflexive modules over certain non-Gorenstein rings and establishes a classification bijection with tuples of matrices.

Findings

Every totally reflexive module has a specific matrix form.

A bijection exists between modules and tuples of matrices.

Classification reduces to matrix tuple equivalence.

Abstract

We consider local non-Gorenstein rings of the form $(S_i,\mathfrak{n}_i)=k[X, Y_1, \ldots ,Y_i]/\left(X^2, (Y_1, \ldots, Y_i)^2\right), $ where $i\geq 2.$ We show that every totally reflexive $S_i$-module has a presentation matrix of the form $I x + \sum_{j=1}^i B_j y_j, $ where $I$ is the identity matrix and $B_j$ is an square matrix with entries in the residue field, $k$. From there, we prove that there exists a bijection between the set of isomorphism classes of totally reflexive modules (without projective summands) over $S_i$ which are minimal generated by $n$ elements and the set of $i$-tuples of $n \times n$ matrices with entries in $k$ modulo a certain equivalence relation.