On a question of Auslander and Bridger on 2-reflexive modules

TL;DR

Problem: Auslander and Bridger asked whether every 2-reflexive module over a two-sided noetherian ring is reflexive. The paper provides the first explicit counterexamples in general: a radical-square-zero finite-dimensional quiver algebra $R=KQ/\operatorname{rad}^2(KQ)$ with an indecomposable module $N$ for which $\operatorname{Ext}_R^2(N,R)$ is a nonzero projective module, and a simple module $S_3$ that is 2-reflexive but not reflexive. The analysis relies on Auslander–Bridger transpose $\operatorname{Tr}(-)$, projective resolutions, and careful computations of $\operatorname{Tr}(S_3)$ and $\operatorname{Ext}_R^2(\operatorname{Tr}(S_3),R)$, showing $\operatorname{Ext}_R^2(\operatorname{Tr}(S_3),R)$ is nonzero and projective, hence $S_3$ is 2-reflexive. Consequently, the notions of reflexive and 2-reflexive modules diverge in general, addressing the AB question negatively; this also illuminates connections to Gorenstein characterizations within AB's framework. The authors also note that their explicit calculations were performed with GAP/QPA.

Abstract

We answer a question raised by Auslander and Bridger by showing that not every 2-reflexive module is reflexive.

Theorems & Definitions (3)

• Proposition 1
• Proposition 2
• proof