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doi: 10.3934/era.2021042
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Balance of complete cohomology in extriangulated categories

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

* Corresponding author: tiweizhao@qfnu.edu.cn

Received  September 2020 Revised  May 2021 Early access June 2021

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.

Citation: Jiangsheng Hu, Dongdong Zhang, Tiwei Zhao, Panyue Zhou. Balance of complete cohomology in extriangulated categories. Electronic Research Archive, doi: 10.3934/era.2021042
References:
[1]

J. Asadollahi and S. Salarian, Tate cohomology and Gorensteinness for triangulated categories, J. Algebra, 299 (2006), 480-502.  doi: 10.1016/j.jalgebra.2005.10.024.  Google Scholar

[2]

M. Auslander and $\varnothing$. Solberg, Relative homology and representation theory I. Relative homology and homologically finite subcategories, Comm. Algebra, 21 (1993), 2995-3031.  doi: 10.1080/00927879308824717.  Google Scholar

[3]

L. L. Avramov, H.-B. Foxby and S. Halperin, Differential graded homological algebra, preprint, 2009. Google Scholar

[4]

L. L. Avramov and A. Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc., 85 (2002), 393-440.  doi: 10.1112/S0024611502013527.  Google Scholar

[5]

A. Beligiannis, Relative homological algebra and purity in triangulated categories, J. Algebra, 227 (2000), 268-361.  doi: 10.1006/jabr.1999.8237.  Google Scholar

[6]

D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra, 82 (1992), 107-129.  doi: 10.1016/0022-4049(92)90116-W.  Google Scholar

[7]

T. Bühler, Exact categories, Expo. Math., 28 (2010), 1-69.  doi: 10.1016/j.exmath.2009.04.004.  Google Scholar

[8]

L. W. Christensen, H.-B. Foxby and H. Holm, Derived Category methods in commutative Algebra, preprint, 2019. Google Scholar

[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.  Google Scholar

[10]

I. Emmanouil, Balance in complete cohomology, J. Pure Appl. Algebra, 218 (2014), 618-623. doi: 10.1016/j.jpaa.2013.08.001.  Google Scholar

[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.  Google Scholar

[12]

T. V. Gedrich and K. W. Gruenberg, Complete cohomological functors on groups, Topol. Appl., 25 (1987), 203-223.  doi: 10.1016/0166-8641(87)90015-0.  Google Scholar

[13]

F. Goichot, Homologie de Tate-Vogel $\acute{e}$quivariante, J. Pure Appl. Algebra, 82 (1992), 39-64.  doi: 10.1016/0022-4049(92)90009-5.  Google Scholar

[14]

J. Hu, D. Zhang, T. Zhao and P. Zhou, Complete cohomology for extriangulated categories, Algebra Colloq., (to appear), arXiv: 2003.11852v2. Google Scholar

[15]

J. HuD. Zhang and P. Zhou, Proper classes and Gorensteinness in extriangulated categories, J. Algebra, 551 (2020), 23-60.  doi: 10.1016/j.jalgebra.2019.12.028.  Google Scholar

[16]

J. HuD. Zhang and P. Zhou, Gorenstein homological dimensions for extriangulated categories, Bull. Malays. Math. Sci. Soc., 44 (2021), 2235-2252.  doi: 10.1007/s40840-020-01057-9.  Google Scholar

[17]

A. Iacob, Generalized Tate cohomology, Tsukuba J. Math., 29 (2005), 389-404.  doi: 10.21099/tkbjm/1496164963.  Google Scholar

[18]

G. Mislin, Tate cohomology for arbitrary groups via satellites, Topology Appl., 56 (1994), 293-300.  doi: 10.1016/0166-8641(94)90081-7.  Google Scholar

[19]

H. Nakaoka and Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. Géom. Différ. Catég., 60 (2019), 117-193.   Google Scholar

[20]

B. E. A. Nucinkis, Complete cohomology for arbitrary rings using injectives, J. Pure Appl. Algebra, 131 (1998), 297-318.  doi: 10.1016/S0022-4049(97)00082-0.  Google Scholar

[21]

W. Ren and Z. Liu, Balance of Tate cohomology in triangulated categories, Appl. Categ. Structures., 23 (2015), 819-828.  doi: 10.1007/s10485-014-9381-8.  Google Scholar

[22]

W. RenR. Zhao and Z. Liu, Cohomology theoreies in triangulated categories, Acta Math. Sin. (Engl. Ser.), 32 (2016), 1377-1390.  doi: 10.1007/s10114-016-5280-2.  Google Scholar

[23]

X. Tang, On $F$-Gorenstein dimensions, J. Algebra Appl., 13 (2014), 1450022, 14 pages. doi: 10.1142/S0219498814500224.  Google Scholar

[24]

X. Yang and W. Chen, Relative homological dimensions and Tate cohomology of complexes with respect to cotorsion pairs, Comm. Algebra, 45 (2017), 2875-2888.  doi: 10.1080/00927872.2016.1233226.  Google Scholar

show all references

References:
[1]

J. Asadollahi and S. Salarian, Tate cohomology and Gorensteinness for triangulated categories, J. Algebra, 299 (2006), 480-502.  doi: 10.1016/j.jalgebra.2005.10.024.  Google Scholar

[2]

M. Auslander and $\varnothing$. Solberg, Relative homology and representation theory I. Relative homology and homologically finite subcategories, Comm. Algebra, 21 (1993), 2995-3031.  doi: 10.1080/00927879308824717.  Google Scholar

[3]

L. L. Avramov, H.-B. Foxby and S. Halperin, Differential graded homological algebra, preprint, 2009. Google Scholar

[4]

L. L. Avramov and A. Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc., 85 (2002), 393-440.  doi: 10.1112/S0024611502013527.  Google Scholar

[5]

A. Beligiannis, Relative homological algebra and purity in triangulated categories, J. Algebra, 227 (2000), 268-361.  doi: 10.1006/jabr.1999.8237.  Google Scholar

[6]

D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra, 82 (1992), 107-129.  doi: 10.1016/0022-4049(92)90116-W.  Google Scholar

[7]

T. Bühler, Exact categories, Expo. Math., 28 (2010), 1-69.  doi: 10.1016/j.exmath.2009.04.004.  Google Scholar

[8]

L. W. Christensen, H.-B. Foxby and H. Holm, Derived Category methods in commutative Algebra, preprint, 2019. Google Scholar

[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.  Google Scholar

[10]

I. Emmanouil, Balance in complete cohomology, J. Pure Appl. Algebra, 218 (2014), 618-623. doi: 10.1016/j.jpaa.2013.08.001.  Google Scholar

[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.  Google Scholar

[12]

T. V. Gedrich and K. W. Gruenberg, Complete cohomological functors on groups, Topol. Appl., 25 (1987), 203-223.  doi: 10.1016/0166-8641(87)90015-0.  Google Scholar

[13]

F. Goichot, Homologie de Tate-Vogel $\acute{e}$quivariante, J. Pure Appl. Algebra, 82 (1992), 39-64.  doi: 10.1016/0022-4049(92)90009-5.  Google Scholar

[14]

J. Hu, D. Zhang, T. Zhao and P. Zhou, Complete cohomology for extriangulated categories, Algebra Colloq., (to appear), arXiv: 2003.11852v2. Google Scholar

[15]

J. HuD. Zhang and P. Zhou, Proper classes and Gorensteinness in extriangulated categories, J. Algebra, 551 (2020), 23-60.  doi: 10.1016/j.jalgebra.2019.12.028.  Google Scholar

[16]

J. HuD. Zhang and P. Zhou, Gorenstein homological dimensions for extriangulated categories, Bull. Malays. Math. Sci. Soc., 44 (2021), 2235-2252.  doi: 10.1007/s40840-020-01057-9.  Google Scholar

[17]

A. Iacob, Generalized Tate cohomology, Tsukuba J. Math., 29 (2005), 389-404.  doi: 10.21099/tkbjm/1496164963.  Google Scholar

[18]

G. Mislin, Tate cohomology for arbitrary groups via satellites, Topology Appl., 56 (1994), 293-300.  doi: 10.1016/0166-8641(94)90081-7.  Google Scholar

[19]

H. Nakaoka and Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. Géom. Différ. Catég., 60 (2019), 117-193.   Google Scholar

[20]

B. E. A. Nucinkis, Complete cohomology for arbitrary rings using injectives, J. Pure Appl. Algebra, 131 (1998), 297-318.  doi: 10.1016/S0022-4049(97)00082-0.  Google Scholar

[21]

W. Ren and Z. Liu, Balance of Tate cohomology in triangulated categories, Appl. Categ. Structures., 23 (2015), 819-828.  doi: 10.1007/s10485-014-9381-8.  Google Scholar

[22]

W. RenR. Zhao and Z. Liu, Cohomology theoreies in triangulated categories, Acta Math. Sin. (Engl. Ser.), 32 (2016), 1377-1390.  doi: 10.1007/s10114-016-5280-2.  Google Scholar

[23]

X. Tang, On $F$-Gorenstein dimensions, J. Algebra Appl., 13 (2014), 1450022, 14 pages. doi: 10.1142/S0219498814500224.  Google Scholar

[24]

X. Yang and W. Chen, Relative homological dimensions and Tate cohomology of complexes with respect to cotorsion pairs, Comm. Algebra, 45 (2017), 2875-2888.  doi: 10.1080/00927872.2016.1233226.  Google Scholar

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