Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on complete resolutions


Authors: Fotini Dembegioti and Olympia Talelli
Journal: Proc. Amer. Math. Soc. 138 (2010), 3815-3820
MSC (2010): Primary 20J99
DOI: https://doi.org/10.1090/S0002-9939-2010-10422-7
Published electronically: May 20, 2010
MathSciNet review: 2679604
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the Eckmann-Shapiro Lemma holds for complete cohomology if and only if complete cohomology can be calculated using complete resolutions. It is also shown that for an $ {\scriptstyle\bf {LH}}\mathfrak{F}$-group $ G$ the kernels in a complete resolution of a $ \mathbb{Z}G$-module coincide with Benson's class of cofibrant modules.


References [Enhancements On Off] (What's this?)

  • 1. M. Auslander, Anneaux de Gorenstein, et torsion en algèbre commutative, Secrétariat mathématique, Paris, 1967, Séminaire d'Algèbre Commutative dirigé par Pierre Samuel, 1966/67. Texte rédigé, d'après des exposés de Maurice Auslander, par Marquerite Mangeney, Christian Peskine et Lucien Szpiro. École Normale Supérieure de Jeunes Filles. MR 0225844 (37:1435)
  • 2. M. Auslander and M. Bridger, Stable Module Theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, RI, 1969. MR 0269685 (42:4580)
  • 3. A. Bahlekeh, F. Dembegioti, and O. Talelli, Gorenstein dimension and proper actions, Bull. London Math. Soc. 41 (2009), 859-871.
  • 4. D. J. Benson, Complexity and varieties for infinite groups I, J. Algebra 193 (1997), 260-287. MR 1456576 (99a:20054)
  • 5. D. J. Benson and J. Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82 (1992), 107-129. MR 1182934 (93i:20058)
  • 6. L. W. Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, Vol. 1747, Springer, Berlin, 2000. MR 1799866 (2002e:13032)
  • 7. J. Cornick and P. H. Kropholler, On complete resolutions, Topology and its Applications 78 (1997), 235-250. MR 1454602 (98k:20087)
  • 8. T. V. Gedrich and K. W. Gruenberg, Complete cohomological functors on groups, Topology and its Applications 25 (1987), 203-223. MR 884544 (89h:20073)
  • 9. F. Goichot, Homologie de Tate-Vogel équivariante, J. Pure Appl. Algebra 82 (1992), 39-64. MR 1181092 (94d:55014)
  • 10. B. M. Ikenaga, Homological dimension and Farrell cohomology, J. Algebra 87 (1984), 422-457. MR 739945 (85k:20152)
  • 11. P. H. Kropholler, On groups of type $ \mathrm{FP}_\infty$, J. Pure Appl. Algebra 90 (1993), 55-67. MR 1246274 (94j:20051b)
  • 12. P. H. Kropholler and O. Talelli, On a property of fundamental groups of graphs of finite groups, J. Pure Appl. Alg. 74 (1991), 57-59. MR 1129129 (92h:57003)
  • 13. G. Mislin, Tate cohomology for arbitrary groups via satellites, Topology and its Applications 56 (1994), 293-300. MR 1269317 (95c:20072)
  • 14. G. Mislin and O. Talelli, On groups which act freely and properly on finite dimensional homotopy spheres, in Computational and Geometric Aspects of Modern Algebra, M. Atkinson et al. (Eds.), London Math. Soc. Lecture Note Ser., 275, Cambridge Univ. Press (2000), 208-228. MR 1776776 (2001i:20110)
  • 15. O. Talelli, Periodicity in group cohomology and complete resolutions, Bull. London Math. Soc. 37 (2005), 547-554. MR 2143734 (2006d:20095)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20J99

Retrieve articles in all journals with MSC (2010): 20J99


Additional Information

Fotini Dembegioti
Affiliation: Department of Mathematics, University of Athens, 15784 Athens, Greece

Olympia Talelli
Affiliation: Department of Mathematics, University of Athens, 15784 Athens, Greece

DOI: https://doi.org/10.1090/S0002-9939-2010-10422-7
Received by editor(s): May 20, 2009
Received by editor(s) in revised form: January 28, 2010
Published electronically: May 20, 2010
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society