Totally acyclic complexes. (English) Zbl 1405.16005
Summary: 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 flat modules are Gorenstein flat. Thus we improve slightly on a result of S. Iyengar and H. Krause; in [Doc. Math. 11, 207–240 (2006; Zbl 1119.13014)] 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 dualizing 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.
MSC:
| 16E10 | Homological dimension in associative algebras |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
Keywords:
totally acyclic complex; Gorenstein injective module; Gorenstein projective module; Gorenstein flat moduleCitations:
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