Gorenstein global dimensions relative to balanced pairs

Haiyu Liu; Rongmin Zhu; Yuxian Geng; Haiyu Liu; Rongmin Zhu; Yuxian Geng

doi:10.3934/era.2020082

Electronic Research Archive

2020, Volume 28, Issue 4: 1563-1571. doi: 10.3934/era.2020082

Previous Article Next Article

Special Issues

Gorenstein global dimensions relative to balanced pairs

Haiyu Liu ^{1
,},
Rongmin Zhu ^{1
,
,},
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

Received: 01 January 2020 Revised: 01 May 2020 Published: 31 July 2020
16E05, 16E30, 18E10, 18G20

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 $.
- Abelian category,
- balanced pair,
- Gorenstein subcategory,
- homological dimension,
- global dimension
Citation: Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs[J]. Electronic Research Archive, 2020, 28(4): 1563-1571. doi: 10.3934/era.2020082

Related Papers:

Abstract

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 $.

References

[1]	The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-)stablization. Comm. Algebra (2000) 28: 4547-4596.
[2]	On the stability question of Gorenstein categories. Appl. Categ. Structures (2017) 25: 907-915.
[3]	Homotopy equivalences induced by balanced pairs. J. Algebra (2010) 324: 2718-2731.
[4]	On the finiteness of Gorenstein homological dimensions. J. Algebra (2012) 372: 376-396.
[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
[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
[7]	Complete cohomological functors on groups. Topology Appl. (1987) 25: 203-223.
[8]	Model structures on moules over Ding-Chen rings. Homology Homotopy Appl. (2010) 12: 61-73.
[9]	On Ding injective, Ding projective and Ding flat modules and complexes. Rocky Mountain J. Math. (2017) 47: 2641-2673.
[10]	Gorenstein homological dimensions. J. Pure Appl. Algebra (2004) 189: 167-193.
[11]	Proper resolutions and Gorenstein categories. J. Algebra (2013) 393: 142-169.
[12]	Applications of balanced pairs. Sci. China Math. (2016) 59: 861-874.
[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
[14]	On pure global dimension of locally finitely presented Grothendieck categories. Fund. Math. (1977) 96: 91-116.
[15]	X. Tang, On $F$-Gorenstein dimensions, J. Algebra Appl., 13 (2014), 14pp. doi: 10.1142/S0219498814500224
[16]	On stability of Gorenstein categories. Comm. Algebra (2013) 41: 3793-3804.
[17]	Gorenstein categories $\mathcal{G(X, Y, Z)}$ and dimensions. Rocky Mountain J. Math. (2015) 45: 2043-2064.
[18]	Balanced pairs induce recollements. Comm. Algebra (2017) 45: 4238-4245.

Reader Comments

your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)