群的Gorenstein同调维数
On Gorenstein Homological Dimension ofGroups
DOI: 10.12677/PM.2023.136180, PDF, HTML, 下载: 166  浏览: 347 
作者: 罗玉祥：重庆师范大学，数学科学学院，重庆
关键词: Gorenstein同调维数；群环；Gorenstein平坦；Erobenius扩张；Gorenstein Homological Dimension； Group Ring； Gorenstein Flat； Frobenius Extension
摘要: 设G是群，R是交换环。定义群G在系数环R上的Gorenstein同调维数GhdRG为平凡RG-模R的Gorenstein平坦维数。证明了对交换环的Erobenius扩张R→S，有GhdsG = GhdRG。此外,还研究群的Gorenstein同调维数与其子群的Gorenstein同调维数之间的关系。证明了对群G的一个升序过滤(Gλ)λ<µ，有GhdRG≤ supλ<µGhdRGλ。进而，如果[G:Gλ]λ<µ是有限的，那么GhdRG =supλ<µGhdRGλ。
Abstract: Let G be group and R commutative ring. The Gorenstein homological dimension GhdRG of the group G over the coefficient ring R is defined as the Gorenstein flat dimension of trivial RG-module R. It is proved that GhdsG = GhdRG for any Frobenius extension of commutative rings R→S. In addition, the relationship between the Gorenstein homological dimension of a group and the Gorenstein homological dimension of its subgroups is studied. It is proved that GhdRG≤ supλ<µGhdRGλ for an ascending filltering (Gλ)λ<µ of group G; furthermore, if [G:Gλ]λ<µ is finite, then GhdRG =supλ<µGhdRGλ.
文章引用：罗玉祥. 群的Gorenstein同调维数[J]. 理论数学, 2023, 13(6): 1758-1768. https://doi.org/10.12677/PM.2023.136180

参考文献

[1] Auslander, M. and Bridge, M. (1969) Stable Module Theory. American Mathematical Society, Providence.
[2] Enochs, E.E., Jenda, O.M.G. and Torrecillas, B. (1993) Gorenstein Flat Modules. Journal of Nanjing University (Mathematical Biquarterly), 10, 1-9.
[3] Enochs, E.E. and Jenda, O.M.G. (2011) Relative Homological Algebra: Volume 1. Walter de Gruyter, Berlin.
[4] Brown, K.S. (1982) Cohomology of Groups (Graduate Texts in Mathematics, No. 87). Springer, Berlin-Heidelberg-New York.
[5] Stallings, J. (1968) On Torsion-Free Groups with Infinitely Many Ends. Annals of Mathematics, 88, 312-334.
https://doi.org/10.2307/1970577
[6] Swan, R.G. (1969) Groups of Cohomological Dimension One. Journal of Algebra, 12, 585-601.
https://doi.org/10.1016/0021-8693(69)90030-1
[7] Asadollahi, J., Bahlekeh, A. and Salarian, S. (2009) On the Hierarchy of Cohomological Di- mensions of Groups. Journal of Pure and Applied Algebra, 213, 1795-1803.
https://doi.org/10.1016/j.jpaa.2009.01.011
[8] Bahlekeh, A., Dembegioti, F. and Talelli, O. (2009) Gorenstein Dimension and Proper Actions. Bulletin of the London Mathematical Society, 41, 859-871.
https://doi.org/10.1112/blms/bdp063
[9] Emmanouil, I. and Talelli, O. (2014) Finiteness Criteria in Gorenstein Homological Algebra. Transactions of the American Mathematical Society, 366, 6329-6351.
https://doi.org/10.1090/S0002-9947-2014-06007-8
[10] Emmanouil, I. and Talell, O. (2018) Gorenstein Dimension and Group Cohomology with Group Ring Coefficients. Journal of the London Mathematical Society, 97, 306-324.
https://doi.org/10.1112/jlms.12103
[11] Talelli, O. (2014) On the Gorenstein and Cohomological Dimension of Groups. Proceeding of the American Mathematical Society, 142, 1175-1180.
https://doi.org/10.1090/S0002-9939-2014-11883-1
[12] Asadollahi, J., Bahlekeh, A., Hajizamani, A. and Salarian, S. (2011) On Certain Homological Invariants of Groups. Journal of Algebra, 335, 18-35.
https://doi.org/10.1016/j.jalgebra.2011.03.018
[13] Luo, Y.X. and Ren, W. (2022) On Gorenstein Homological Dimension of Groups. arXiv.2205.15542
[14] Kasch, F. (1954) Grundlagen einer Theorie der Frobeniuserweiterungen. Mathematische An- nalen, 127, 453-474.
https://doi.org/10.1007/BF01361137
[15] Browna, K.A., Gordon, I.G. and Stroppel, C.H. (2008) Cherednik, Hecke and Quantum Alge- bras as Free Frobenius and Calabi-Yau Extensions. Journal of Algebra, 319, 1007-1034.
https://doi.org/10.1016/j.jalgebra.2007.10.026
[16] Kock, J. (2004) Frobenius Algebras and 2D Topological Quantum Field Theories. Cambridge University Press, Cambridge.
https://doi.org/10.1017/CBO9780511615443
[17] Xi, C. (2021) Frobenius Bimodules and Flat-Dominant Dimensions. Science China Mathematics, 64, 33-44.
https://doi.org/10.1007/s11425-018-9519-2
[18] 周芮,赵志兵.Frobenius扩张下模的无挠性和自反性[J].山东大学学报(理学版), 2022,57(10):52-58.
[19] Hu, J.S., Li, H.H., Geng, Y.X. and Zhang, D.D. (2020) Frobenius Functors and Gorenstein Flat Dimensions. Communications in Algebra, 48, 1257-1265.
https://doi.org/10.1080/00927872.2019.1677699
[20] Ren, W. (2018) Gorenstein Projective Modules and Frobenius Extensions. Science China Mathematics, 61, 1175-1186.
https://doi.org/10.1007/s11425-017-9138-y
[21] Holm, H. (2004) Gorenstein Homological Dimensions. Journal of Pure and Applied Algebra, 189, 167-193.
https://doi.org/10.1016/j.jpaa.2003.11.007
[22] Saroch, J. and Stovcek, J. (2020) Singular Compactness and Definability for Σ-Cotorsion and Gorenstein Modules. Selecta Mathematica, 26, Article No. 23.
https://doi.org/10.1007/s00029-020-0543-2
[23] Bennis, D. (2009) Rings over Which the Class of Gorenstein Flat Modules Is Closed under Extensions. Communications in Algebra, 37, 855-868.
https://doi.org/10.1080/00927870802271862
[24] Yang, G. and Liu, Z. (2012) Gorenstein Flat Covers over GF-Closed Rings. Communications in Algebra, 40, 1632-1640.
https://doi.org/10.1080/00927872.2011.553644
[25] Kadison, L. (1999) New Examples of Frobenius Extensions. American Mathematical Society, Providence.
https://doi.org/10.1090/ulect/014
[26] Zhao, Z. (2019) Gorenstein Homological Invariant Properties under Frobenius Extensions. Science China Mathematics, 62, 2487-2496.
https://doi.org/10.1007/s11425-018-9432-2
[27] Xi, C. and Yin, S. (2020) Cellularity of Centrosymmetric Matrix Algebras and Frobenius Extensions. Linear Algebra and Its Applications, 590, 317-329.
https://doi.org/10.1016/j.laa.2020.01.002
[28] Enochs, E.E. and Lopez-Ramos, J.A. (2002) Kaplansky Classes. Rendiconti del Seminario Matematico della Universita di Padova, 107, 67-79.

为你推荐