Remarks on Balance for Tate and Generalized Tate (Co)homology
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Remarks on Balance for Tate and Generalized Tate (Co)homology Chaoling Huang1
· Kaituo Liu2
Received: 3 September 2019 / Revised: 22 October 2019 / Accepted: 30 October 2019 © Iranian Mathematical Society 2019
Abstract Based on the analysis of Tate and generalized Tate cohomology in abelian categories, in this note, we consider the balance for Tate and generalized Tate cohomology. Some existing results are generalized and some applications in the category of modules are given. We also consider the balance for Tate and generalized Tate homology in broader sense. Keywords Tate (co)homology · Generalized Tate (co)homology · Balancedness · Complete Tate (co)resolution Mathematics Subject Classification 13D02 · 13D07 · 16E05 · 18G15 · 18G25
1 Introduction The idea of relative homological algebra that was first introduced by Eilenberg and Moore [11] was reinvigorated by Enochs, Jenda and Torrecillas [8–10]. Up to now, many authors studied the related subjects. For instance, In Avramov and Martsinkovsky’s paper [2], a complex T is called totally acyclic if its modules are
Communicated by Rahim Zaare-Nahandi. C. Huang is supported by the guidance project of scientific research plan of Educational Administration of Hubei Province, China (No. B2016162) and the plan of science and technology innovation team of the excellent young and middle-age of Hubei province, China (No. T201731).
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Chaoling Huang [email protected] Kaituo Liu [email protected]
1
School of Mathematics and Computer Science, Hanjiang Normal University, Shiyan 442000, China
2
School of Science, Hubei University of Automotive Technology, Shiyan 442002, China
123
Bulletin of the Iranian Mathematical Society
projective, it is exact and the complex Hom R (T , Q) is exact for every projective τ
π
module Q. For a finite module M, a diagram T P M , where π is a projective resolution of M, T is totally acyclic complex, τ is a morphism of complexes and τn is bijective for all n 0 is called a complete resolution of M. Based on the n complete resolution of M, the Tate cohomology functor E xt R (M, −) is defined for any n ∈ Z, and many properties are given. Especially there exists an exact sequence of functors about the absolute, relative and Tate cohomology functors R (M, −) → Ext 2G → · · · . 0 → Ext 1G → Ext 1R → Ext 1
The author in [23] investigated the relative and Tate cohomology theory with respect to Ding modules. Following the idea of [23], one can obtain the following concepts: A complex T is called totally F-acyclic if its modules are projective, it is exact and the complex Hom R (T , Q) is exact for every flat module Q. A diagram τ
π
T P M , where π is a projective resolution of M, T is totally Facyclic complex, τ is a morphism of complexes and τn is bijective for all n 0 is called a Tate F-resolution of M. For any R-module M, one can define Tate cohomology nR (M, N ) = H n (Hom R (T , N )). Also there is the Avramov–Martsinkovsky group Ext type exact sequence; see [23, Theorem 5.6]. Let R be a commutative coherent ring and C be a s
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