Gorenstein Homological Dimension of Groups Through Flat-Cotorsion Modules

Document Type : Original Scientific Paper

Author

‎Department of Mathematics, ‎University of Hormozgan, ‎Bandar Abbas‎, ‎I‎. ‎R‎. ‎Iran

Abstract

‎The representation theory of groups is one of the most interesting examples of the interaction between physics and pure mathematics‎, ‎where group rings play the main role‎. ‎The group ring $\rga$ is actually an associative ring that inherits the properties of the group $\ga$ and the ring of coefficients $R$‎. ‎In addition to the fact that the theory of group rings is clearly the meeting point of group theory and ring theory‎, ‎it also has applications in algebraic topology‎, ‎homological algebra‎, ‎algebraic K-theory and algebraic coding theory‎.
‎In this article‎, ‎we provide a complete description of Gorenstein flat-cotorsion modules over the group ring $\rga$‎,
‎where $\ga$ is a group and $R$ is a commutative ring‎. ‎It will be shown that if $\ga'\leqslant \ga$ is a finite-index subgroup‎, ‎then the restriction of scalars along the ring homomorphism $\rga'\rt\rga$ as well as its right adjoint $\rga\otimes_{\rga'}-$‎, ‎preserve the class of Gorenstein flat-cotorsion modules‎. ‎Then‎, ‎as a result‎, ‎Serre's Theorem is proved for the invariant $\Ghcd_{R}\ga$‎, ‎which refines the Gorenstein homological dimension of $\ga$ over $R$‎, ‎$\Ghd_{R}\ga$‎, ‎and is defined using flat-cotorsion modules‎. ‎Moreover‎, ‎we show that the inequality $\GF (\rga)\leqslant \GF (R)+{\cd_{R}\ga}$ holds for the group ring $\rga$‎, ‎where $\GF (R)$ denotes the supremum of Gorenstein flat-cotorsion dimensions of all $R$-modules and $\cd_{R}\ga$ is the cohomological dimension of $\ga$‎ over $R$‎.
 

Keywords

Main Subjects

References

[1] C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, American Mathematical Soc. 1966.
[2] P. I. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob and E. Yudovina, Introduction to Representation Theory, American Mathematical Soc. 2011.
[3] P. J. Sally and D. A Vogan, Representation Theory and Harmonic Analysis on Semisimple Lie Groups, American Mathematical Soc. 1989.
[4] S. Sternberg, Group Theory and Physics, Cambridge University Press, 1995.
[5] I. M. Gelfand and Z. Ya. Šhapiro, Representations of the group of rotations in three-dimensional space and their applications, Uspekhi Mat. Nauk (N.S.) 7 (1952) 3 - 117.
[6] E. P. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959.
[7] L. G. Chouinard, Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra 7 (1976) 287 - 302, https://doi.org/10.1016/0022-4049(76)90055-4.
[8] D. J. Benson and K. R. Goodearl, Periodic flat modules, and flat modules for finite groups, Pacific J. Math. 196 (2000) 45 - 67.
[9] J. R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. 88 (1968) 312 - 334, https://doi.org/10.2307/1970577.
[10] R. G. Swan, Groups of cohomological dimension one, J. Algebra 12 (1969) 585 - 610, https://doi.org/10.1016/0021-8693(69)90030-1.
[11] R. Bieri, Homological Dimension of Discrete Groups, Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, 1976.
[12] M. Auslander and M. Bridger, Stable Module Theory, Mem. Amer. Math. Soc. vol. 94, (Amer. Math. Soc., Providence, RI, 1969).
[13] E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995) 611 - 633, https://doi.org/10.1007/BF02572634.
[14] E. E. Enochs, O. M. G. Jenda and B. Torrecillas, Gorenstein flat modules, J. Nanjing Univ. Math. Biquarterly 10 (1993) 1 - 9.
[15] D. Bennis and N. Mahdou, Global Gorenstein dimensions, Proc. Amer. Math. Soc. 138 (2010) 461 - 465.
[16] H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004) 167 - 193, https://doi.org/10.1016/j.jpaa.2003.11.007.
[17] A. Bahlekeh, (Strongly) Gorenstein flat modules over group rings, Bull. Aust. Math. Soc. 90 (2014) 57 - 64, https://doi.org/10.1017/S000497271300107X.
[18] A. Bahlekeh, F. Dembegioti and O. Talelli, Gorenstein dimension and proper actions, Bull. Lond. Math. Soc. 41 (2009) 859 - 871, https://doi.org/10.1112/blms/bdp063.
[19] R. Biswas, Benson’s cofibrants, Gorenstein projectives and a related conjecture, Proc. Edinb. Math. Soc. 64 (2021) 779 - 799, https://doi.org/10.1017/S0013091521000481.
[20] I. Emmanouil and O. Talelli, Finiteness criteria in Gorenstein homological algebra, Trans. Amer. Math. Soc. 366 (2014) 6329 - 6351.
[21] I. Emmanouil and O. Talelli, Gorenstein dimension and group cohomology with group ring coefficients, J. Lond. Math. Soc. 97 (2018) 306 - 324, https://doi.org/10.1112/jlms.12103.
[22] I. Emmanouil and O. Talelli, On the Gorenstein cohomological dimension of group extensions, J. Algebra 605 (2022) 403 - 428, https://doi.org/10.1016/j.jalgebra.2021.12.042.
[23] O. Talelli, On the Gorenstein and cohomological dimension of groups, Proc. Amer. Math. Soc 142 (2014) 1175 - 1180.
[24] J. Asadollahi, A. Bahlekeh and Sh. Salarian, On the hierarchy of cohomological dimensions of groups, J. Pure Appl. Algebra 213 (2009) 1795 - 1803, https://doi.org/10.1016/j.jpaa.2009.01.011.
[25] J. Asadollahi, A. Bahlekeh, A. Hajizamani and Sh. Salarian, On certain homological invariants of groups, J. Algebra 335 (2011) 18 - 35, https://doi.org/10.1016/j.jalgebra.2011.03.018.
[26] J. Gillespie, The flat stable module category of a coherent ring, J. Pure Appl. Algebra 221 (2017) 2025 - 2031, https://doi.org/10.1016/j.jpaa.2016.10.012.
[27] L. W. Christensen, S. Estrada and P. Thompson, Homotopy categories of totally acyclic complexes with applications to the flat–cotorsion theory, Categorical, Homological and Combinatorial Methods in Algebra, Contemp. Math.
751 Amer. Math. Soc., Providence, RI (2020) 99 - 118.
[28] L. W. Christensen, S. Estrada, L. Liang, P. Thompson, D. Wu and G. Yang, A refinement of Gorenstein flat dimension via the flat-cotorsion theory, J. Algebra 567 (2021) 346-370, https://doi.org/10.1016/j.jalgebra.2020.09.024.
[29] L. W. Christensen, S. Estrada and P. Thompson, Gorenstein weak global dimension is symmetric, Math. Nachr. 294 (2021) 2121 - 2128, https://doi.org/10.1002/mana.202100141.
[30] L. W. Christensen, S. Estrada and P. Thompson, Five theorems on Gorenstein global dimensions, Algebra and coding theory, Contemp. Math. 785 Amer. Math. Soc., Providence, RI (2023).
[31] L. L. Avramov and H.-B. Foxby, Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71 (1991) 129 - 155, https://doi.org/10.1016/0022-4049(91)90144-Q.
[32] E. E. Enochs and J. R. García Rozas, Flat covers of complexes, J. Algebra 210 (1998) 86 - 102, https://doi.org/10.1006/jabr.1998.7582.
[33] T. Nakamura and P. Thompson, Minimal semi-flat-cotorsion replacements and cosupport, J. Algebra 562 (2020) 587 - 620, https://doi.org/10.1016/j.jalgebra.2020.07.001.
[34] S. Bazzoni, M. Cortés-Izurdiaga and S. Estrada, Periodic modules and acyclic complexes, Algebr. Represent. Theory 23 (2020) 1861 - 1883, https://doi.org/10.1007/s10468-019-09918-z.
[35] K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Springer, Berlin-Heidelberg-New York, 1982.
[36] D. J. Benson, Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, 1998.
[37] J. J. Rotman, An Introduction to Homological Algebra, Springer-Verlag New York, 2009.
[38] J. Xu, Flat Covers of Modules, Springer-Verlag Berlin Heidelberg 1996.
[39] E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter & Co., Berlin, 2000.
[40] J. Gillespie, The flat model structure on Ch(R), Trans. Amer. Math. Soc. 356 (2004) 3369 - 3390.
Mathematics Interdisciplinary Research
Volume 9, Issue 1
March 2024
Pages 23-43

Files

Share

How to cite

Statistics