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Global weak solutions for the two-component Novikov equation
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 |
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 $.
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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. |
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D. Bennis, J. R. García Rozas and L. Oyanarte,
On the stability question of Gorenstein categories, Appl. Categ. Structures, 25 (2017), 907-915.
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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. |
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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. |
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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.
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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.
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Complete cohomological functors on groups, Topology Appl., 25 (1987), 203-223.
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J. Gillespie,
Model structures on moules over Ding-Chen rings, Homology Homotopy Appl., 12 (2010), 61-73.
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[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. |
[10] |
H. Holm,
Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004), 167-193.
doi: 10.1016/j.jpaa.2003.11.007. |
[11] |
Z. Huang,
Proper resolutions and Gorenstein categories, J. Algebra, 393 (2013), 142-169.
doi: 10.1016/j.jalgebra.2013.07.008. |
[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. |
[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] |
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. |
[15] |
X. Tang, On $F$-Gorenstein dimensions, J. Algebra Appl., 13 (2014), 14pp.
doi: 10.1142/S0219498814500224. |
[16] |
A. Xu and N. Ding,
On stability of Gorenstein categories, Comm. Algebra, 41 (2013), 3793-3804.
doi: 10.1080/00927872.2012.677892. |
[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. |
[18] |
Y. Zheng,
Balanced pairs induce recollements, Comm. Algebra, 45 (2017), 4238-4245.
doi: 10.1080/00927872.2016.1262384. |
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. |
[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. |
[3] |
X. Chen,
Homotopy equivalences induced by balanced pairs, J. Algebra, 324 (2010), 2718-2731.
doi: 10.1016/j.jalgebra.2010.09.002. |
[4] |
I. Emmanouil,
On the finiteness of Gorenstein homological dimensions, J. Algebra, 372 (2012), 376-396.
doi: 10.1016/j.jalgebra.2012.09.018. |
[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] |
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. |
[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. |
[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. |
[10] |
H. Holm,
Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004), 167-193.
doi: 10.1016/j.jpaa.2003.11.007. |
[11] |
Z. Huang,
Proper resolutions and Gorenstein categories, J. Algebra, 393 (2013), 142-169.
doi: 10.1016/j.jalgebra.2013.07.008. |
[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. |
[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] |
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. |
[15] |
X. Tang, On $F$-Gorenstein dimensions, J. Algebra Appl., 13 (2014), 14pp.
doi: 10.1142/S0219498814500224. |
[16] |
A. Xu and N. Ding,
On stability of Gorenstein categories, Comm. Algebra, 41 (2013), 3793-3804.
doi: 10.1080/00927872.2012.677892. |
[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. |
[18] |
Y. Zheng,
Balanced pairs induce recollements, Comm. Algebra, 45 (2017), 4238-4245.
doi: 10.1080/00927872.2016.1262384. |
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