An Auslander-Reiten principle in derived categories

TL;DR

This paper introduces a new principle in derived categories that underpins classical and generalized Auslander-Reiten dualities, and applies it to prove the Auslander-Reiten conjecture for certain Gorenstein rings.

Contribution

It presents a novel principle in derived categories that explains and extends Auslander-Reiten dualities, and uses it to verify the conjecture in specific high-dimensional Gorenstein rings.

Findings

Established a new principle underlying Auslander-Reiten duality.

Proved the Auslander-Reiten conjecture for Gorenstein rings with dimension > 2 and limited singular locus.

Extended the applicability of Auslander-Reiten theory to new classes of rings.

Abstract

We give a principle in derived categories, which lies behind the classical Auslander-Reiten duality and its generalized version by Iyama and Wemyss. We apply the principle to show the validity of the Auslander-Reiten conjecture over a Gorenstein ring in the case where the ring has dimension larger than two and the singular locus has at most one dimension.