Totally acyclic complexes

JOURNAL OF ALGEBRA

卷:470

Article

Estrada, Sergio¹  Fu, Xianhui²  Iacob, Alina³ 

[1] Univ Murcia, E-30100 Murcia, Spain

[2] Northeast Normal Univ, Sch Math & Stat, Changchun, Peoples R China

[3] Georgia Southern Univ, Statesboro, GA 30460 USA

关键词: Totally acyclic complex;    Gorenstein injective module;    Gorenstein projective module;    Gorenstein flat module;   

DOI  :  10.1016/j.jalgebra.2016.09.009

来源: Elsevier

【 摘 要 】

It is known that over an Iwanaga-Gorenstein ring the Gorenstein injective (Gorenstein projective, Gorenstein flat) modules are simply the cycles of acyclic complexes of injective (projective, flat) modules. We consider the question: are these characterizations only working over Iwanaga-Gorenstein rings? We prove that if R is a commutative noetherian ring of finite Krull dimension then the following are equivalent: 1. R is an Iwanaga Gorenstein ring. 2. Every acyclic complex of injective modules is totally acyclic. 3. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 4. Every acyclic complex of projective modules is totally acyclic. 5. The cycles of every acyclic complex of Gorenstein projective modules are Gorenstein projective. 6. Every acyclic complex of flat modules is F-totally acyclic. 7. The cycles of every acyclic complex of Gorenstein fiat modules are Gorenstein fiat. Thus we improve slightly on a result of Iyengar and Krause; in [22] they proved that for a commutative noetherian ring R with a dualizing complex, the class of acyclic complexes of injectives coincides with that of totally acyclic complexes of injectives if and only if R is Gorenstein. We replace the dualising complex hypothesis by the finiteness of the Krull dimension, and add more equivalent conditions. In the second part of the paper we focus on the noncommutative case. We prove that for a two sided noetherian ring R of finite finitistic flat dimension that satisfies the Auslander condition the following are equivalent: 1. Every complex of injective (left and respectively right) R-modules is totally acyclic. 2. R is Iwanaga-Gorenstein. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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