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

Document Type : Research Paper

Author

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran

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

References

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History

  • Receive Date: 19 July 2022
  • Revise Date: 07 September 2022
  • Accept Date: 12 September 2022
  • Publish Date: 22 May 2022

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