# Totally Reflexive Modules and Poincar\'{e} Series

## TL;DR

This paper explores the structure of Cohen-Macaulay non-Gorenstein local rings by analyzing Poincaré series through totally reflexive modules, extending Yoshino's results to higher dimensions and constructing diverse modules.

## Contribution

It provides a new description of Poincaré series using totally reflexive modules and constructs large families of indecomposable modules in higher-dimensional rings.

## Key findings

- Generalizes Yoshino's result to higher dimensions
- Provides a method to compute Poincaré series via reflexive modules
- Constructs infinitely many indecomposable modules with large minimal generators

## Abstract

We study Cohen-Macaulay non-Gorenstein local rings $(R,\mathfrak{m},k)$ admitting certain totally reflexive modules. More precisely, we give a description of the Poincar\'{e} series of $k$ by using the Poincar\'{e} series of a non-zero totally reflexive module with minimal multiplicity. Our results generalize a result of Yoshino to higher-dimensional Cohen-Macaulay local rings. Moreover, from a quasi-Gorenstein ideal satisfying some conditions, we construct a family of non-isomorphic indecomposable totally reflexive modules having an arbitrarily large minimal number of generators.