On the existence of non-free totally reflexive modules

TL;DR

This paper investigates conditions under which standard graded Cohen-Macaulay rings and their Stanley-Reisner ring quotients admit non-free totally reflexive modules, revealing new insights into their module theory.

Contribution

It establishes that if a quotient of a Cohen-Macaulay ring admits such modules, then the ring itself does too, and applies this to Stanley-Reisner rings of graphs.

Findings

Non-free totally reflexive modules exist under certain conditions.

The property transfers from quotients to the original ring.

Application to Stanley-Reisner rings of graphs.

Abstract

For a standard graded Cohen-Macaulay ring $S$, if the quotient $S/(\underline{x})$ admits non-free totally reflexive modules, where $\underline{x}$ is a system of parameters consisting of elements of degree one, then so does the ring $S$. As an application, we consider the question of which Stanley-Reisner rings of graphs admit non-free totally reflexive modules.