Kyoto Journal of Mathematics

Virtual Gorensteinness over group algebras

Abdolnaser Bahlekeh and Shokrollah Salarian

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Let Γ be a finite group, and let Λ be any Artin algebra. It is shown that the group algebra ΛΓ is virtually Gorenstein if and only if ΛΓ' is virtually Gorenstein, for all elementary abelian subgroups Γ' of Γ. We also extend this result to cover the more general context. Precisely, assume that Γ is a group in Kropholler’s hierarchy HF, Γ' is a subgroup of Γ of finite index, and R is any ring with identity. It is proved that, in certain circumstances, that RΓ is virtually Gorenstein if and only if RΓ' is so.

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Kyoto J. Math., Volume 55, Number 1 (2015), 129-141.

First available in Project Euclid: 13 March 2015

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Primary: 20J05: Homological methods in group theory 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) 16G10: Representations of Artinian rings 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]

virtually Gorenstein algebra Moore’s condition Group algebras


Bahlekeh, Abdolnaser; Salarian, Shokrollah. Virtual Gorensteinness over group algebras. Kyoto J. Math. 55 (2015), no. 1, 129--141. doi:10.1215/21562261-2848133.

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  • [1] E. Aljadeff, Profinite groups, profinite completions and a conjecture of Moore, Adv. Math. 201 (2006), 63–76.
  • [2] E. Aljadeff and Y. Ginosar, Induction from elementary abelian groups, J. Algebra 179 (1996) 599–606.
  • [3] E. Aljadeff and E. Meir, Nilpotency of Bocksteins, Kropholler’s hierarchy and a conjecture of Moore, Adv. Math. 226 (2011), 4212–4224.
  • [4] L. Angeleri-Hügel and J. Trilifaj, Tilting theory and finitistic dimension conjectures, Trans. Amer. Math. Soc. 354 (2002), 4345–4358.
  • [5] M. Auslander and M. Bridger, Stable Module Theory, Mem. Amer. Math. Soc. 94, Amer. Math. Soc., Montrouge, 1969.
  • [6] M. Auslander and I. Reiten, Applications of contravariantly finite subcategories, Adv. in Math. 86 (1991), 111–152.
  • [7] M. Auslander, I. Reiten, and S. Smalø, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math. 36, Cambridge Univ. Press, Cambridge, 1995.
  • [8] A. Bahlekeh, F. Dembegioti, and O. Talelli, Gorenstein dimension and proper actions, Bull. London Math. Soc. 41 (2009), 859–871.
  • [9] A. Bahlekeh and Sh. Salarian, New results related to a conjecture of Moore, Arch. Math. (Basel) 100 (2013), 231–239.
  • [10] H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.
  • [11] A. Beligiannis, Cohen–Macaulay modules, (co)torsion pairs and virtually Gorenstein algebra, J. Algebra 288 (2005), 137–211.
  • [12] A. Beligiannis and H. Krause, Thick subcategories and virtually Gorenstein Algebras, Illinois J. Math. 52 (2008), 551–562.
  • [13] A. Beligiannis and I. Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (2007), no. 883.
  • [14] D. J. Benson, Representations and Homology, II: Cohomology of Groups and Modules, Cambridge Stud. Adv. Math. 31, Cambridge Univ. Press, Cambridge, 1991.
  • [15] K. S. Brown, Cohomology of Groups, Grad. Texts in Math. 37, Springer, Berlin, 1982.
  • [16] J. F. Carlson, Cohomology and induction from elementary abelian subgroups, Q. J. Math. 51 (2000), 169–181.
  • [17] L. G. Chouinard, Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra 7 (1976), 287–302.
  • [18] L. W. Christensen, Gorenstein Dimensions, Lecture Notes in Math. 1747, Springer, Berlin, 2000.
  • [19] J. Cornick and P. H. Kropholler, Homological finiteness conditions for modules over group algebras, J. London Math. Soc. (2) 58 (1998), 49–62.
  • [20] E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), 611–633.
  • [21] E. Enochs and O. M. G. Jenda, Relative Homological Algebra, de Gruyter Exp. Math. 30, de Gruyter, Berlin, 2000.
  • [22] E. Green and B. Zimmerman-Huisgen, Finitistic dimension of artin rings with vanishing radical cube, Math. Z. 206 (1991), 505–526.
  • [23] K. Igusa and G. Todorov, “On the finitistic global dimension conjecture for Artin algebras” in Representations of Algebras and Related Topics, Fields Inst. Commun. 45, Amer. Math. Soc., Providence, 2005, 201–204.
  • [24] P. H. Kropholler, On groups of type $(\mathrm{FP})_{\infty}$, J. Pure Appl. Algebra 90 (1993), 55–67.
  • [25] D. Quillen, The spectrum of an equivariant cohomology ring, I, Ann. of Math. (2) 94 (1971), 549–572.
  • [26] D. Quillen and B. B. Venkov, Cohomology of finite groups and elementary abelian subgroups, Topology 11 (1972), 317–318.
  • [27] D. S. Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700–712.
  • [28] J. J. Rotman, An Introduction to Homological Algebra, 2nd ed., Universitext, Springer, New York, 2009.
  • [29] B. Zimmermann-Huisgen, Homological domino effects and the first finitistic dimension conjecture, Invent. Math. 108 (1992), 369–383.