A Generalization of the Auslander-Reiten Conjecture for the Bounded Derived Category

TL;DR

This paper extends the Auslander-Reiten Conjecture to the bounded derived category of a Noetherian ring, establishing an equivalent and derived equivalence-preserved version of the classical conjecture.

Contribution

It introduces a generalized form of the Auslander-Reiten Conjecture applicable to the bounded derived category, maintaining equivalence with the classical form and invariance under derived equivalences.

Findings

The generalized conjecture is equivalent to the classical one.

The conjecture is preserved under derived equivalences.

Provides a new perspective on the conjecture within derived categories.

Abstract

We study the bounded derived category $\D^b(\Rmod)$ of a left Noetherian ring $R$. We give a version of the Generalized Auslander-Reiten Conjecture for $\D^b(\Rmod)$ that is equivalent to the classical statement for the module category and is preserved under derived equivalence.