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

Gorenstein global dimensions relative to balanced pairs

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 $.

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

[2]

D. BennisJ. 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. LiJ. 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. BennisJ. 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. LiJ. 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

[1]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024

[2]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[3]

Lisa Hernandez Lucas. Properties of sets of Subspaces with Constant Intersection Dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052

[4]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[5]

Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054

[6]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

[7]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283

[8]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018

[9]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

[10]

Xianbo Sun, Zhanbo Chen, Pei Yu. Parameter identification on Abelian integrals to achieve Chebyshev property. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020375

[11]

Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126

[12]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127

[13]

Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164

[14]

Peter Frolkovič, Karol Mikula, Jooyoung Hahn, Dirk Martin, Branislav Basara. Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 865-879. doi: 10.3934/dcdss.2020350

[15]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[16]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[17]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

[18]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325

[19]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156

[20]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

 Impact Factor: 0.263

Metrics

  • PDF downloads (63)
  • HTML views (209)
  • Cited by (0)

Other articles
by authors

[Back to Top]