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

Abstract

We consider local non-Gorenstein rings of the form $(S_i,{\mathfrak{n}}_i)=k[X,Y_1,\dots ,Y_i]/\left(X^2,(Y_1,\dots ,Y_i{)}^2\right),$ where $i\ge 2.$ We show that every totally reflexive $S_i$-module has a presentation matrix of the form $Ix+{\sum }_{j=1}^i{B}_j{y}_j,$ 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.

Cited by in corpus (2)

• Gorenstein projective bimodules via monomorphism categories and filtration categories
• Differential projective modules over algebras with radical square zero