Totally reflexive modules over rings that are close to Gorenstein

TL;DR

This paper proves that over certain Artinian Gorenstein local rings, all totally reflexive modules are free, and introduces a test module for freeness based on the second syzygy of the canonical module.

Contribution

It establishes conditions under which totally reflexive modules over non-Gorenstein quotients are necessarily free and introduces a new test module for freeness.

Findings

All totally reflexive modules over specified rings are free.

The second syzygy of the canonical module contains a test module for freeness.

Provides criteria for freeness in non-Gorenstein quotients of Gorenstein rings.

Abstract

Let $S$ be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if $R$ is a non-Gorenstein quotient of $S$ of small colength, then every totally reflexive $R$-module is free. Indeed, the second syzygy of the canonical module of $R$ has a direct summand $T$ which is a test module for freeness over $R$ in the sense that if $\mathrm{Tor}_+^R(T,N)=0$, for some finitely generated $R$-module $N$, then $N$ is free.