Auslander-Reiten Conjecture in a Dual Vein
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Auslander-Reiten Conjecture in a Dual Vein Hossein Eshraghi1
· Ali Mahin Fallah2,3
Received: 4 June 2019 / Accepted: 2 September 2020 / © Springer Nature B.V. 2020
Abstract We consider a dual notion of the famous Auslander-Reiten Conjecture in case of Noetherian algebras over commutative Noetherian rings. Firstly, in the introduction, we will examine its relevance by showing that in an standard situation, the validity of this dual implies the validity of the Auslander-Reiten Conjecture itself. Moreover, in two important cases these two notions coincide: Artin algebras, and Noetherian algebras over complete local Noetherian rings. In this regard we will prove the following theorem: Let (R, m) be d-Gorenstein, d ≥ 2, and let be a Noetherian R-algebra which is Gorenstein and (maximal) CohenMacaulay as R-module. If M is an Artinian self-orthogonal Gorenstein injective -module such that Hom (p , M) is an injective p -module for every nonmaximal prime ideal p of R, then M is injective. Some applications are discussed afterwards. Keywords Noetherian algebra · Auslander-Reiten conjecture · Gorenstein rings and modules · Auslander-Reiten theory Mathematics Subject Classification (2010) 13D07 · 13E05 · 13H10 · 16G30 · 18G25
1 Introduction Unless explicitly mentioned, all modules considered in this paper are supposed to be left modules. (R, m) always denotes a commutative Noetherian local ring with maximal ideal m and stands for a Noetherian R-algebra.
Presented by: Vyjayanthi Chari Hossein Eshraghi
[email protected] Ali Mahin Fallah [email protected]; [email protected] 1
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, P.O. Box: 87317-51167, Kashan, Iran
2
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran
3
Department of Mathematics, Alzahra University, Vanak, 19834, Tehran, Iran
H. Eshraghi, A. Mahin Fallah
One of the famous and tenacious problems in the representation theory of finite dimensional algebras (or, more generally, Artin algebras) is the so-called Auslander-Reiten Conjecture. It was posed by Auslander and Reiten in [4] and can be formulated as follows: (ARC) Over an Artin algebra , a finitely generated -module M is projective if Exti (M, M ⊕ ) = 0 for i ≥ 1. It is known that (ARC) is linked to many other homological conjectures that emerge in the representation theory of Artin algebras; namely, Tachikawa Conjecture [25] and the Finitistic Dimension Conjecture [18]. However, it is usually said that (ARC) has its origins in the so-called (Generalized) Nakayama Conjecture [21]. Hence, an affirmative answer to it could affect numerous long-standing problems. Though having appeared originally in the representation theory of finite dimensional algebras, (ARC) has also penetrated to other relevant contexts both in commutative and noncommutative algebra. In fact, it has been considered in the study of commutative Noetherian rings [1, 12, 13, 20, 23, 24] and, more recently
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