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http://dx.doi.org/10.4134/JKMS.j220153

(𝓕, 𝓐)-GORENSTEIN FLAT HOMOLOGICAL DIMENSIONS  

Becerril, Victor (Centro de Ciencias Matematicas Universidad Nacional Autonoma de Mexico)
Publication Information
Journal of the Korean Mathematical Society / v.59, no.6, 2022 , pp. 1203-1227 More about this Journal
Abstract
In this paper we develop the homological properties of the Gorenstein (𝓛, 𝓐)-flat R-modules 𝓖𝓕(𝓕(R),𝓐) proposed by Gillespie, where the class 𝓐 ⊆ Mod(Rop) sometimes corresponds to a duality pair (𝓛, 𝓐). We study the weak global and finitistic dimensions that come with the class 𝓖𝓕(𝓕(R),𝓐) and show that over a (𝓛, 𝓐)-Gorenstein ring, the functor - ⊗R - is left balanced over Mod(Rop) × Mod(R) by the classes 𝓖𝓕(𝓕(Rop),𝓐) × 𝓖𝓕(𝓕(R),𝓐). When the duality pair is (𝓕(R), 𝓕𝓟Inj(Rop)) we recover the G. Yang's result over a Ding-Chen ring, and we see that is new for (Lev(R), AC(Rop)) among others.
Keywords
Bi-complete duality pair; ($\mathcal{L}$, $\mathcal{A}$)-Gorenstein ring; balanced pair; weak global dimension; finitistic dimension; relative Gorenstein flat;
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1 F. A. A. Almahdi, E. M. Bouba, and M. Tamekkante, On w-FI-flat and w-FI-injective modules, Ann. Univ. Ferrara 68 (2022), 91-102. https://doi.org/10.1007/s11565-022-00388-8   DOI
2 M. Auslander and M. Bridger, Stable Module Theory, American Mathematical Soc., 1969.
3 V. Becerril, Relative global Gorenstein dimensions, J. Alg. Appl. (2022) https://doi.org/10.1142/S0219498822502085   DOI
4 V. Becerril, O. Mendoza, and V. Santiago, Relative Gorenstein objects in abelian categories, Comm. Algebra 49 (2021), no. 1, 352-402. https://doi.org/10.1080/00927872.2020.1800023   DOI
5 D. J. Fieldhouse, Character modules, dimension and purity, Glasgow Math. J. 13 (1972), 144-146. https://doi.org/10.1017/S0017089500001567   DOI
6 J. Gillespie, Model structures on modules over Ding-Chen rings, Homology Homotopy Appl. 12 (2010), no. 1, 61-73. http://projecteuclid.org/euclid.hha/1296223822   DOI
7 J. Gillespie, AC-Gorenstein rings and their stable module categories, J. Aust. Math. Soc. 107 (2019), no. 2, 181-198. https://doi.org/10.1017/s1446788718000290   DOI
8 J. Gillespie, Duality pairs and stable module categories, J. Pure Appl. Algebra 223 (2019), no. 8, 3425-3435. https://doi.org/10.1016/j.jpaa.2018.11.010   DOI
9 H. Holm and P. Jorgensen, Cotorsion pairs induced by duality pairs, J. Commut. Algebra 1 (2009), no. 4, 621-633. https://doi.org/10.1216/JCA-2009-1-4-621   DOI
10 A. Iacob, Projectively coresolved Gorenstein flat and ding projective modules, Comm. Algebra 48 (2020), no. 7, 2883-2893. https://doi.org/10.1080/00927872.2020.1723612   DOI
11 E. E. Enochs and O. M. G. Jenda, Gorenstein balance of Hom and tensor, Tsukuba J. Math. 19 (1995), no. 1, 1-13. https://doi.org/10.21099/tkbjm/1496162796   DOI
12 J. J. Rotman, An Introduction to Homological Algebra, second edition, Universitext, Springer, New York, 2009. https://doi.org/10.1007/b98977   DOI
13 J. Wang and Z. Di, Relative Gorenstein rings and duality pairs, J. Algebra Appl. 19 (2020), no. 8, 2050147, 22 pp. https://doi.org/10.1142/S0219498820501479   DOI
14 G. Yang, Homological properties of modules over Ding-Chen rings, J. Korean Math. Soc. 49 (2012), no. 1, 31-47. https://doi.org/10.4134/JKMS.2012.49.1.031   DOI
15 E. E. Enochs and O. M. G. Jenda, Copure injective resolutions, flat resolvents and dimensions, Comment. Math. Univ. Carolin. 34 (1993), no. 2, 203-211.
16 D. Bennis, Rings over which the class of Gorenstein flat modules is closed under extensions, Comm. Algebra 37 (2009), no. 3, 855-868. https://doi.org/10.1080/ 00927870802271862   DOI
17 D. Bravo and M. A. Perez, Finiteness conditions and cotorsion pairs, J. Pure Appl. Algebra 221 (2017), no. 6, 1249-1267. https://doi.org/10.1016/j.jpaa.2016.09.008   DOI
18 X.-W. Chen, Homotopy equivalences induced by balanced pairs, J. Algebra 324 (2010), no. 10, 2718-2731. https://doi.org/10.1016/j.jalgebra.2010.09.002   DOI
19 E. E. Enochs and O. M. G. Jenda, Balanced functors applied to modules, J. Algebra 92 (1985), no. 2, 303-310. https://doi.org/10.1016/0021-8693(85)90122-X   DOI
20 X. Yang, Gorenstein categories G(X, Y, Z) and dimensions, Rocky Mountain J. Math. 45 (2015), no. 6, 2043-2064. https://doi.org/10.1216/RMJ-2015-45-6-2043   DOI
21 J. Gillespie, On Ding injective, Ding projective and Ding flat modules and complexes, Rocky Mountain J. Math. 47 (2017), no. 8, 2641-2673. https://doi.org/10.1216/RMJ2017-47-8-2641   DOI
22 J. Gillespie and A. Iacob, Duality pairs, generalized Gorenstein modules, and Ding injective envelopes, arXiv preprint arXiv:2105.01770, 2021.
23 C. U. Jensen, On the vanishing of $\lim_{\leftarrow}^{(i)}$, J. Algebra 15 (1970), 151-166. https://doi.org/10.1016/0021-8693(70)90071-2   DOI
24 Z. Wang, G. Yang, and R. Zhu, Gorenstein flat modules with respect to duality pairs, Comm. Algebra 47 (2019), no. 12, 4989-5006. https://doi.org/10.1080/00927872.2019.1609011   DOI
25 S. Estrada, A. Iacob, and M. A. Perez, Model structures and relative Gorenstein flat modules and chain complexes, in Categorical, homological and combinatorial methods in algebra, 135-175, Contemp. Math., 751, Amer. Math. Soc., RI, 2020. https://doi.org/10.1090/conm/751/15084   DOI
26 E. E. Enochs and O. M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter & Co., Berlin, 2000. https://doi.org/10.1515/ 9783110803662   DOI
27 D. Bravo, J. Gillespie, and M. Hovey, The stable module category of a general ring, Preprint, 2014, arXiv:1405.5768