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December  2020, 28(4): 1563-1571. doi: 10.3934/era.2020082

Gorenstein global dimensions relative to balanced pairs

Haiyu Liu 1, , Rongmin Zhu 1,, and  Yuxian Geng 2,

1. 

Department of Mathematics, Nanjing University, Nanjing 210093, China
2. 

School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China

* Corresponding author: Rongmin Zhu

Received  January 2020 Revised  May 2020 Published  July 2020

Fund Project: The first author is supported by the NSF of China (Grants No. 11771202)

Let $ \mathcal{G}(\mathcal{X}) $ and $ \mathcal{G}(\mathcal{Y}) $ be Gorenstein subcategories induced by an admissible balanced pair $ (\mathcal{X}, \mathcal{Y}) $ in an abelian category $ \mathcal{A} $. In this paper, we establish Gorenstein homological dimensions in terms of these two subcategories and investigate the Gorenstein global dimensions of $ \mathcal{A} $ induced by the balanced pair $ (\mathcal{X}, \mathcal{Y}) $. As a consequence, we give some new characterizations of pure global dimensions and Gorenstein global dimensions of a ring $ R $.

Keywords: Abelian category, balanced pair, Gorenstein subcategory, homological dimension, global dimension.
Mathematics Subject Classification: 16E05, 16E30, 18E10, 18G20.
Citation: Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082
References:
[1]

A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-)stablization, Comm. Algebra, 28 (2000), 4547-4596.  doi: 10.1080/00927870008827105.  Google Scholar

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X. Chen, Homotopy equivalences induced by balanced pairs, J. Algebra, 324 (2010), 2718-2731.  doi: 10.1016/j.jalgebra.2010.09.002.  Google Scholar

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I. Emmanouil, On the finiteness of Gorenstein homological dimensions, J. Algebra, 372 (2012), 376-396.  doi: 10.1016/j.jalgebra.2012.09.018.  Google Scholar

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E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter GmbH & Co. KG, Berlin, 2011. doi: 10.1515/9783110215212.  Google Scholar

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S. Estrada, M. A. Pérez and H. Zhu, Balanced pairs, cotorsion triplets and quiver representations, Proc. Edinb. Math. Soc. (2), 63 (2020), 67–90. doi: 10.1017/S0013091519000270.  Google Scholar

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

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J. Gillespie, On Ding injective, Ding projective and Ding flat modules and complexes, Rocky Mountain J. Math., 47 (2017), 2641-2673.  doi: 10.1216/RMJ-2017-47-8-2641.  Google Scholar

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H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004), 167-193.  doi: 10.1016/j.jpaa.2003.11.007.  Google Scholar

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Z. Huang, Proper resolutions and Gorenstein categories, J. Algebra, 393 (2013), 142-169.  doi: 10.1016/j.jalgebra.2013.07.008.  Google Scholar

[12]

H. Li, J. Wang and Z. Huang, Applications of balanced pairs, Sci. China Math., 59 (2016), 861-874.  doi: 10.1007/s11425-015-5094-1.  Google Scholar

[13]

S. Sather-Wagstaff, T. Sharif, D. White, Stability of Gorenstein categories, J. Lond. Math. Soc. (2), 77 (2008), 481–502. doi: 10.1112/jlms/jdm124.  Google Scholar

[14]

D. Simson, On pure global dimension of locally finitely presented Grothendieck categories, Fund. Math., 96 (1977), 91-116.  doi: 10.4064/fm-96-2-91-116.  Google Scholar

[15]

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

[16]

A. Xu and N. Ding, On stability of Gorenstein categories, Comm. Algebra, 41 (2013), 3793-3804.  doi: 10.1080/00927872.2012.677892.  Google Scholar

[17]

X. Yang, Gorenstein categories $\mathcal{G(X, Y, Z)}$ and dimensions, Rocky Mountain J. Math., 45 (2015), 2043-2064.  doi: 10.1216/RMJ-2015-45-6-2043.  Google Scholar

[18]

Y. Zheng, Balanced pairs induce recollements, Comm. Algebra, 45 (2017), 4238-4245.  doi: 10.1080/00927872.2016.1262384.  Google Scholar

show all references

References:
[1]

A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-)stablization, Comm. Algebra, 28 (2000), 4547-4596.  doi: 10.1080/00927870008827105.  Google Scholar

[2]

D. Bennis, J. R. García Rozas and L. Oyanarte, On the stability question of Gorenstein categories, Appl. Categ. Structures, 25 (2017), 907-915.  doi: 10.1007/s10485-016-9478-3.  Google Scholar

[3]

X. Chen, Homotopy equivalences induced by balanced pairs, J. Algebra, 324 (2010), 2718-2731.  doi: 10.1016/j.jalgebra.2010.09.002.  Google Scholar

[4]

I. Emmanouil, On the finiteness of Gorenstein homological dimensions, J. Algebra, 372 (2012), 376-396.  doi: 10.1016/j.jalgebra.2012.09.018.  Google Scholar

[5]

E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter GmbH & Co. KG, Berlin, 2011. doi: 10.1515/9783110215212.  Google Scholar

[6]

S. Estrada, M. A. Pérez and H. Zhu, Balanced pairs, cotorsion triplets and quiver representations, Proc. Edinb. Math. Soc. (2), 63 (2020), 67–90. doi: 10.1017/S0013091519000270.  Google Scholar

[7]

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

[8]

J. Gillespie, Model structures on moules over Ding-Chen rings, Homology Homotopy Appl., 12 (2010), 61-73.  doi: 10.4310/HHA.2010.v12.n1.a6.  Google Scholar

[9]

J. Gillespie, On Ding injective, Ding projective and Ding flat modules and complexes, Rocky Mountain J. Math., 47 (2017), 2641-2673.  doi: 10.1216/RMJ-2017-47-8-2641.  Google Scholar

[10]

H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004), 167-193.  doi: 10.1016/j.jpaa.2003.11.007.  Google Scholar

[11]

Z. Huang, Proper resolutions and Gorenstein categories, J. Algebra, 393 (2013), 142-169.  doi: 10.1016/j.jalgebra.2013.07.008.  Google Scholar

[12]

H. Li, J. Wang and Z. Huang, Applications of balanced pairs, Sci. China Math., 59 (2016), 861-874.  doi: 10.1007/s11425-015-5094-1.  Google Scholar

[13]

S. Sather-Wagstaff, T. Sharif, D. White, Stability of Gorenstein categories, J. Lond. Math. Soc. (2), 77 (2008), 481–502. doi: 10.1112/jlms/jdm124.  Google Scholar

[14]

D. Simson, On pure global dimension of locally finitely presented Grothendieck categories, Fund. Math., 96 (1977), 91-116.  doi: 10.4064/fm-96-2-91-116.  Google Scholar

[15]

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

[16]

A. Xu and N. Ding, On stability of Gorenstein categories, Comm. Algebra, 41 (2013), 3793-3804.  doi: 10.1080/00927872.2012.677892.  Google Scholar

[17]

X. Yang, Gorenstein categories $\mathcal{G(X, Y, Z)}$ and dimensions, Rocky Mountain J. Math., 45 (2015), 2043-2064.  doi: 10.1216/RMJ-2015-45-6-2043.  Google Scholar

[18]

Y. Zheng, Balanced pairs induce recollements, Comm. Algebra, 45 (2017), 4238-4245.  doi: 10.1080/00927872.2016.1262384.  Google Scholar

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