Auslander‐reiten conjecture for gorenstein rings of krull dimension at least 2

Abstract

Auslander‐Reiten Conjecture is one of the most important and long‐standing conjectures in representation theory of finite dimensional algebras. It is known to be in close relationship with a string of homological conjectures on top of which lies the well‐known Finitistic Dimension Conjecture. Recently, a dual version of Auslander‐Reiten conjecture has been posed. This dual statement deserves further study from the point of view that its validity implies the validity of the conjecture itself. This paper is devoted to deal with this Dual Auslander‐Reiten Conjecture. We discuss it for Gorenstein rings of Krull dimension at least 2. To this end, we firstly show that it suffices to focus only on the case where the Krull dimension is exactly 2. Then, switching to such rings, we show that this conjecture holds for modules of finite length over Gorenstein rings of Krull dimension at least 2.

Keywords

• auslander-reiten conjecture (dual of)
• noetherian algebra
• gorenstein module