Some results on gorenstein flat (gorenstein cotorsion) dimensions of modules over group rings

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Sciences, University of Hormozgan, Bandar Abbas, Iran

Abstract

Let $R$ be a commutative ring, Γ be a group, and Γ′ be a subgroup of finite index. In this paper, we study the class of Gorenstein flat (resp. Gorenstein cotorsion) modules over the group ring RΓ. In particular, we discuss the relationship between Gorenstein flat (resp. Gorenstein cotorsion) dimension of RΓ−modules and that of RΓ′−modules. We will prove that, for any RΓ−module $M$ of finite Gorenstein flat dimension (∞>GfdM), there is an inequality GfdM ⩽ GfdRM + hdRΓ, where hdRΓ denotes the homological dimension of Γ over $R$ . Analogously, we prove that if Γ is finite, there is an inequality GctdM ⩽ GctdRM + cdRΓ, where GctdM (GctdRM) is the Gorenstein cotorsion dimension of module $M$ over RΓ (resp. over $R$) and cdRΓ is the cohomological dimension of $Γ over $R$.

Keywords


[1] J. Asadollahi, A. Bahlekeh and S. Salarian, On the hierarchy of cohomological dimensions of groups, J. Pure Appl. Algebra, 213 (2009) 1795–1803.
[2] J. Asadollahi, A. Bahlekeh, A. Hajizamani and Sh. Salarian, On certain homological invariants of groups, J. Algebra, 335 (2011) 18–35.
[3] M. Auslander, Anneaux de Gorenstein et torsion en algèbre commutative, Secrétariat mathématique, Paris, 1967, Séminaire dálgèbre commutative dirigé par Pierre Samuel, 1966/67. Texte rédigé, dáprès des exposés de Maurice Auslander, par Marquerite Mangeney, Christian Peskine et Lucien Szpiro, École Normale Supérieure de Jeunes Filles.
[4] A. Bahlekeh, (Strongly) Gorenstein flat modules over group rings, Bull. Aust. Math. Soc., 90 (2014) 57–64.
[5] A. Bahlekeh, F. Dembegioti and O. Talelli, Gorenstein dimension and proper actions, Bull. London. Math. Soc., 41 (2009) 859–871.
[6] D. Bennis, Rings over which the class of Gorenstein flat modules is closed under extensions, Commun. Algebra, 37 (2009) 855–868.
[7] D. Bennis, Weak Gorenstein global dimension, Int. Electron. J. Algebra, 8 (2010) 140–152.
[8] D. J. Benson, Representations and Cohomology I: Basic representation theory of finite groups and as-sociative algebras, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, 1991, reprinted in paperback, 1998.
[9] D. J. Benson and K. R. Goodearl, Periodic flat modules, and flat modules for finite groups, Pacific J. Math., 196 (2000) 45–66.
[10] D. Benson, S. B. Iyengar and H. Krause, Module categories for group algebras over commutative rings, J. K-Theory, 11 (2013) 297–329.
[11] R. Bieri, Homological dimension of discrete groups, Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, 1977.
[12] R. Biswas, Benson’s cofibrants, Gorenstein projectives and a related conjecture, Proc. Edinburgh Math. Soc., 64(4) (2021) 779–799.
[13] R. Biswas, On some cohomological invariants for large families of infinite groups, New York J. Math., 27 (2021) 818–839.
[14] K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87, Springer, Berlin-Heidelberg-New York, 1982.
[15] L. G. Chouinard, Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra, 7 (1976) 287–302.
[16] L. W. Christensen, S. Estrada and P. Thompson, Gorenstein weak global dimension is symmetric, Math. Nachr., 294 (2021) 2121–2128.
[17] I. G. Connell, On the group ring, Can. J. Math., 15 (1963) 650–685. [18] I. Emmanouil, On certain cohomological invariants of groups, Adv. Math., 225 (2010) 3446–3462.
[19] I. Emmanouil, On the finiteness of Gorenstein homological dimensions, J. Algebra, 372 (2012) 376–396.
[20] I. Emmanouil and O. Talelli, Finiteness criteria in Gorenstein homological algebra, Trans. Amer. Math. Soc., 366 (2014) 6429–6351.
[21] I. Emmanouil and O. Talell, Gorenstein dimension and group cohomology with group ring coefficients, J. London Math. Soc., 97 (2018) 306–324.
[22] I. Emmanouil and O. Talelli, On the Gorenstein cohomological dimension of group extensions, J. Algebra, 605 (2022) 403–428.
[23] E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z., 220 (1995) 611–633.
[24] E. E. Enochs, O. M. G. Jenda and B. Torrecillas, Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan, 10 (1993) 1–9.
[25] Z. Gao, On Gorenstein cotorsion dimension over GF -closed rings, Bulletin of the Korean Mathematical Society, 51 (2014) 173–187.
[26] D. G. Higman, Modules with a group of operators, Duke Math. J., 21 (1954) 369–376.
[27] H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004) 167–193.
[28] R. Lei and F. Meng, Notes on Gorenstein cotorsion modules, Mathematical Notes, 96 (2014) 716–731.
[29] J. J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, 85, Academic Press,
New York-London, 1979.
[30] J. Saroch and J. Sťovíček, Singular compactness and definability for Σ-cotorsion and Gorenstein modules,
Sel. Math., New Ser. 26 (2020).
[31] J. Stallings, On torsion-free groups with infinitely many ends, Ann. Math., 88 (1968) 312–334.
[32] R. G. Swan, Groups of cohomological dimension one, J. Algebra, 12 (1969) 585–601.
[33] O. Talelli, On the Gorenstein and cohomological dimension of groups, Proc. Amer. Math. Soc., 142 (2014) 1175–1180.
Volume 7, Issue 3 - Serial Number 3
September 2022
Pages 69-87
  • Receive Date: 19 October 2022
  • Revise Date: 22 January 2023
  • Accept Date: 24 January 2023
  • Publish Date: 22 November 2022