Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T19:24:43.850Z Has data issue: false hasContentIssue false
  • English
  • Français

Benson's cofibrants, Gorenstein projectives and a related conjecture

Part of: Homological algebra Theory of modules and ideals Commutative algebra: Homological methods

Published online by Cambridge University Press:  23 September 2021

Rudradip Biswas
Rudradip Biswas*
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, ManchesterM13 9PL, UK (rudradipbiswas@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this short article, we will be principally investigating two classes of modules over any given group ring – the class of Gorenstein projectives and the class of Benson's cofibrants. We begin by studying various properties of these two classes and studying some of these properties comparatively against each other. There is a conjecture made by Fotini Dembegioti and Olympia Talelli that these two classes should coincide over the integral group ring for any group. We make this conjecture over group rings over commutative rings of finite global dimension and prove it for some classes of groups while also proving other related results involving the two classes of modules mentioned.

Keywords

Benson's cofibrantsGorenstein projectivesKropholler's hierarchycomplete resolutions

MSC classification

Primary: 18G05: Projectives and injectives 13C10: Projective and free modules and ideals 13C60: Module categories
Secondary: 13D02: Syzygies, resolutions, complexes
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 4 , November 2021 , pp. 779 - 799
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asadollahi, J., Bahlekeh, A. and Salarian, S., On the hierarchy of cohomological dimensions of groups, J. Pure Appl. Algebra 213 (2009), 1795–1803.CrossRefGoogle Scholar
Bahlekeh, A., Dembegioti, F. and Talelli, O., Gorenstein dimension and proper actions, Bull. London Math. Soc. 41 (2009), 859–871.CrossRefGoogle Scholar
Benson, D., Complexity and varieties for infinite groups, I, J. Algebra 193 (1997), 260–287.CrossRefGoogle Scholar
Benson, D., Complexity and varieties for infinite groups, II, J. Algebra 193 (1997), 288–317.CrossRefGoogle Scholar
Biswas, R., Generation operators in the module category and their applications. Completed as a separate paper out of the author's PhD thesis, can be read https://www.dropbox.com/sh/li0y4v8mdfra4yr/AACn0ssafgL-r3l-_Pgg2jMMa?dl=0here. Submitted to the Int. J. Algebra Comput. in May 2020.Google Scholar
Biswas, R., On some cohomological invariants for large families of infinite groups, New York J. Math. 27 (2021), 817–838.Google Scholar
Chatterji, I., Collected by Guido's Book of Conjectures. L'Enseignement mathématique Monograph no. 40 (ISBN 2-940264-07-4). A gift to Guido Mislin on the occasion of his retirement from ETHZ (June 2006); 190 p., 2008Google Scholar
Cornick, J. and Kropholler, P. H., On complete resolutions, Topol. Appl. 78 (1997), 235–250.10.1016/S0166-8641(96)00126-5CrossRefGoogle Scholar
Cornick, J. and Kropholler, P. H., Homological finiteness conditions for modules over group algebras, J. London Math. Soc. (2) 58(1) (1998), 49–62.CrossRefGoogle Scholar
Dembegioti, F. and Talelli, O., A note on complete resolutions, Proc. Am. Math. Soc. 138(11) (2010), 3815–3820.CrossRefGoogle Scholar
Dicks, W., Kropholler, P., Leary, I. and Thomas, S., Classifying spaces for proper actions of locally finite groups, J. Group Theory 5(4) (2002), 453–480.CrossRefGoogle Scholar
Emmanouil, I., On certain cohomological invariants of groups, Adv. Math. 225(6) (2010), 3446–3462.10.1016/j.aim.2010.06.007CrossRefGoogle Scholar
Emmanouil, I., On the finiteness of Gorenstein homological dimensions, J. Algebra 372 (2012), 376–396.CrossRefGoogle Scholar
Emmanouil, I. and Talelli, O., Gorenstein dimension and group cohomology with group ring coefficients, J. Lond. Math. Soc. (2) 97(2) (2018), 306–324.CrossRefGoogle Scholar
Holm, H., Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), 167–193.CrossRefGoogle Scholar
Ikenaga, B., Homological dimension and Farrell cohomology, J. Algebra 87 (1984), 422–457.CrossRefGoogle Scholar
Januskiewicz, T., Kropholler, P. and Leary, I., Groups possessing extensive hierarchical decompositions, Bull. London Math. Soc. 42(5) (2010), 896–904.CrossRefGoogle Scholar
Kropholler, P. H., On groups of type $(FP)_{\infty }$, J. Pure Appl. Algebra 90 (1993), 55–67.CrossRefGoogle Scholar
Mazza, N. and Symonds, P., The stable category and invertible modules for infinite groups, Adv. Math. 358 (2019), 106853, 26 p.CrossRefGoogle Scholar
Mislin, G., Tate cohomology for arbitrary groups via satellites, Topol. Appl. 56(3) (1994), 293–300.CrossRefGoogle Scholar
Mislin, G. and Talelli, O., On groups which act freely and properly on finite dimensional homotopy spheres, in Computational and geometric aspects of modern algebra (ed. Atkinson, M. et al. ), London Math. Soc. Lecture Note Series, Volume 275, pp. 208–228 (Cambridge University Press, 2000)Google Scholar
Perez, C. M., A bound for the Bredon cohomological dimension, J. Group Theory 10(6) (2007), 731–747.Google Scholar
Symonds, P., Endotrivial modules for infinite groups, Notes from the PIMS Summer School on Geometric and Topological Aspects of the Representation Theory of Finite Groups in Vancouver, July 27–30, 2016.Google Scholar
Talelli, O., On groups of type $\Phi$, Arch. Math. (Basel) 89(1) (2007), 24–32.CrossRefGoogle Scholar