Totally acyclic complexes
Totally acyclic complexes
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DOI：
10.1016/j.jalgebra.2016.09.009
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发表时间：
2016-03
期刊：
影响因子：
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通讯作者：
S. Estrada;X. Fu;A. Iacob
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文献类型：
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作者：
S. Estrada;X. Fu;A. Iacob
众所周知，在Iwanaga-Gorenstein环上，Gorenstein内射(Gorenstein投射，Gorenstein Flat)模就是内射(投射，平坦)模的非循环复形的圈。我们考虑这样一个问题：这些刻画在Iwanaga-Gorenstein环上起作用吗？证明了如果R是有限Krull维的交换Noether环，则下列条件等价：1.R是Iwanaga-Gorenstein环。2.内射模的每个非循环复形都是完全非循环的。3.Gorenstein内射模的每个非循环复形的圈都是Gorenstein内射的。4.投射模的每个非循环复形都是完全非循环的。5.Gorenstein投射模的每个非循环复形的圈都是Gorenstein投射模。6.平坦模的每个非循环复形都是F-完全非循环的。7.Gorenstein平坦模的每个非循环复形的圈都是Gorenstein平坦的。因此，我们略微改进了Iyengar和Krause的一个结果；在[22]中，他们证明了对于具有对偶复形的交换Noether环R，内射的非循环复形与完全内射的非循环复形重合的充要条件是Ris Gorenstein。我们用Krull维的有限性代替了对偶化复假设，并增加了更多的等价条件。我们证明了对于满足Auslander条件的有限平坦维度的双边Noether环，下列条件是等价的：1.每个内射(左和右)R-模的复形是全非循环的。2.伊万纳加-戈伦斯坦。
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 characterizationsonlyworking over Iwanaga–Gorenstein rings? We prove that ifRis a commutative noetherian ring of finite Krull dimension then the following are equivalent: 1.Ris 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 Iyengar and Krause; in [22] they proved that for a commutative noetherian ringRwith a dualizing complex, the class of acyclic complexes of injectives coincides with that of totally acyclic complexes of injectives if and only ifRis 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 ringRof 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.Ris Iwanaga–Gorenstein.