Balance of complete cohomology in extriangulated categories

Jiangsheng Hu; Dongdong Zhang; Tiwei Zhao; Panyue Zhou; Jiangsheng Hu; Dongdong Zhang; Tiwei Zhao; Panyue Zhou

doi:10.3934/era.2021042

Electronic Research Archive

2021, Volume 29, Issue 5: 3341-3359. doi: 10.3934/era.2021042

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Balance of complete cohomology in extriangulated categories

Jiangsheng Hu ^{1
,},
Dongdong Zhang ^{2
,},
Tiwei Zhao ^{3
,
,},
Panyue Zhou ^{4
,}

1.
School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China
2.
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
3.
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
4.
College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China

Received: 01 September 2020 Revised: 01 May 2021 Published: 24 June 2021
Primary: 16E30, 18G25; Secondary: 18G10

Let $ (\mathcal{C}, \mathbb{E}, \mathfrak{s}) $ be an extriangulated category with a proper class $ \xi $ of $ \mathbb{E} $-triangles. In this paper, we study the balance of complete cohomology in $ (\mathcal{C}, \mathbb{E}, \mathfrak{s}) $, which is motivated by a result of Nucinkis that complete cohomology of modules is not balanced in the way the absolute cohomology Ext is balanced. As an application, we give some criteria for identifying a triangulated catgory to be Gorenstein and an Artin algebra to be $ F $-Gorenstein.
- Complete cohomology,
- balance,
- extriangulated category,
- proper class
Citation: Jiangsheng Hu, Dongdong Zhang, Tiwei Zhao, Panyue Zhou. Balance of complete cohomology in extriangulated categories[J]. Electronic Research Archive, 2021, 29(5): 3341-3359. doi: 10.3934/era.2021042

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Abstract

Let $ (\mathcal{C}, \mathbb{E}, \mathfrak{s}) $ be an extriangulated category with a proper class $ \xi $ of $ \mathbb{E} $-triangles. In this paper, we study the balance of complete cohomology in $ (\mathcal{C}, \mathbb{E}, \mathfrak{s}) $, which is motivated by a result of Nucinkis that complete cohomology of modules is not balanced in the way the absolute cohomology Ext is balanced. As an application, we give some criteria for identifying a triangulated catgory to be Gorenstein and an Artin algebra to be $ F $-Gorenstein.

References

[1]	Tate cohomology and Gorensteinness for triangulated categories. J. Algebra (2006) 299: 480-502.
[2]	Relative homology and representation theory I. Relative homology and homologically finite subcategories. Comm. Algebra (1993) 21: 2995-3031.
[3]	L. L. Avramov, H.-B. Foxby and S. Halperin, Differential graded homological algebra, preprint, 2009.
[4]	Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension. Proc. London Math. Soc. (2002) 85: 393-440.
[5]	Relative homological algebra and purity in triangulated categories. J. Algebra (2000) 227: 268-361.
[6]	Products in negative cohomology. J. Pure Appl. Algebra (1992) 82: 107-129.
[7]	Exact categories. Expo. Math. (2010) 28: 1-69.
[8]	L. W. Christensen, H.-B. Foxby and H. Holm, Derived Category methods in commutative Algebra, preprint, 2019.
[9]	L. W. Christensen and D. A. Jorgensen, Tate (co)homology via pinched complexes, Trans. Amer. Math. Soc., 366 (2014), 667-689. doi: 10.1090/S0002-9947-2013-05746-7
[10]	I. Emmanouil, Balance in complete cohomology, J. Pure Appl. Algebra, 218 (2014), 618-623. doi: 10.1016/j.jpaa.2013.08.001
[11]	E. E. Enochs, S. Estrada and A. C. Iacob, Balance with unbounded complexes, Bull. Lond. Math. Soc., 44 (2012), 439-442. doi: 10.1112/blms/bdr101
[12]	Complete cohomological functors on groups. Topol. Appl. (1987) 25: 203-223.
[13]	Homologie de Tate-Vogel $\acute{e}$quivariante. J. Pure Appl. Algebra (1992) 82: 39-64.
[14]	J. Hu, D. Zhang, T. Zhao and P. Zhou, Complete cohomology for extriangulated categories, Algebra Colloq., (to appear), arXiv: 2003.11852v2.
[15]	Proper classes and Gorensteinness in extriangulated categories. J. Algebra (2020) 551: 23-60.
[16]	Gorenstein homological dimensions for extriangulated categories. Bull. Malays. Math. Sci. Soc. (2021) 44: 2235-2252.
[17]	Generalized Tate cohomology. Tsukuba J. Math. (2005) 29: 389-404.
[18]	Tate cohomology for arbitrary groups via satellites. Topology Appl. (1994) 56: 293-300.
[19]	Extriangulated categories, Hovey twin cotorsion pairs and model structures. Cah. Topol. Géom. Différ. Catég. (2019) 60: 117-193.
[20]	Complete cohomology for arbitrary rings using injectives. J. Pure Appl. Algebra (1998) 131: 297-318.
[21]	Balance of Tate cohomology in triangulated categories. Appl. Categ. Structures. (2015) 23: 819-828.
[22]	Cohomology theoreies in triangulated categories. Acta Math. Sin. (Engl. Ser.) (2016) 32: 1377-1390.
[23]	X. Tang, On $F$-Gorenstein dimensions, J. Algebra Appl., 13 (2014), 1450022, 14 pages. doi: 10.1142/S0219498814500224
[24]	Relative homological dimensions and Tate cohomology of complexes with respect to cotorsion pairs. Comm. Algebra (2017) 45: 2875-2888.

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