Hostname: page-component-7479d7b7d-fwgfc Total loading time: 0 Render date: 2024-07-13T18:18:58.786Z Has data issue: false hasContentIssue false
  • English
  • Français

Complexity and varieties for infinitely generated modules

Published online by Cambridge University Press:  24 October 2008

D. J. Benson ,
Jon F. Carlson  and
J. Rickard
D. J. Benson
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
Jon F. Carlson
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
J. Rickard
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW
Get access Rights & Permissions [Opens in a new window]

Extract

In the past fifteen years the theory of complexity and varieties of modules has become a standard tool in the modular representation theory of finite groups. Moreover the techniques have been used in the study of integral representations [8] and have been extended to the representation theories of objects such as groups of finite virtual cohomological dimension [1], infinitesimal subgroups of algebraic groups and restricted Lie algebras [14, 16]. In all cases some sort of finiteness condition on the module category has been required to make the theory work. Usually this comes in the form of stipulating that all modules under consideration be finitely generated. While the restrictions have been efficient for most applications to date, there are very good reasons for wanting to develop a theory that will accommodate infinitely generated modules. One reason might be the possibility of extending the techniques of representations to other classes of infinite groups. Another reason is that some recent work has revealed a few of the defects of the finiteness requirement. One such problem can be summarized as follows.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 2 , September 1995 , pp. 223 - 243
Copyright
Copyright © Cambridge Philosophical Society 1995

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

REFERENCES

[1]Adem, A.. On the exponent of cohomology of discrete groups. Bull. London Math. Soc. 21 (1989), 585–590.CrossRefGoogle Scholar
[2]Alperin, J. L. and Evens, L.. Representations, resolutions, and Quillen's dimension theorem. J. Pure Appl. Algebra 22 (1981), 1–9.CrossRefGoogle Scholar
[3]Arkin, M. and Mazur, B.. Etale homotopy. Springer Lecture Notes in Mathematics 100 (Springer-Verlag, 1969).Google Scholar
[4]Bass, H.. Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc. 95 (1960), 466–488.CrossRefGoogle Scholar
[5]Benson, D. J.. Representations and cohomology II: cohomology of groups and modules. Cambridge studies in advanced mathematics, vol. 31 (CUP, 1991).Google Scholar
[6]Bökstedt, M. and Neeman, A.. Homotopy limits in triangulated categories. Compositio Mathematica 86 (1993), 209–234.Google Scholar
[7]Carlson, J. F.. The variety of an indecomposable module is connected. Invent. Math. 77 (1984), 291–299.CrossRefGoogle Scholar
[8]Carlson, J. F.. Exponents of modules and maps. Invent. Math. 95 (1989), 13–24.CrossRefGoogle Scholar
[9]Carlson, J. F.. The decomposition of the trivial module in the complexity quotient category, preprint.Google Scholar
[10]Carlson, J. F., Donovan, P. W. and Wheeler, W. W.. Complexity and quotient categories for group algebras, preprint.Google Scholar
[11]Carlson, J. F. and Wheeler, W. W.. Varieties and localizations of module categories, preprint.Google Scholar
[12]Chouinard, L.. Projectivity and relative projectivity over group rings. J. Pure Appl. Algebra 7 (1976), 278–302.CrossRefGoogle Scholar
[13]Evens, L.. The cohomology 0f groups. Oxford Science Publications (OUP, 1991).CrossRefGoogle Scholar
[14]Friedlander, E. M. and Pakshall, B. J.. Support varieties for restricted Lie algebras. Invent. Math. 86 (1986), 553–562.CrossRefGoogle Scholar
[15]Happel, D.. Triangulated categories in the representation theory of finite dimensional algebras. London Math. Soc. Lecture Notes 119 (CUP, 1988).CrossRefGoogle Scholar
[16]Jantzen, J. C.. Kohomologie von p–Lie-Algebren und nilpotente Elemente. Abh. Mat. Sem. Univ. Hamburg 76 (1986), 191–219.CrossRefGoogle Scholar
[17]Lenzing, H.. Homological transfer from finitely presented to infinite modules. In Abelian group theory, Springer Lecture Notes in Math. 1006, 734–761.CrossRefGoogle Scholar
[18]Ringel, C. M.. The indecomposable representations of the dihedral 2-groups. Math. Ann. 214 (1975), 19–34.CrossRefGoogle Scholar
[19]Serre, J.-P.. Sur la dimension cohomologique des groupes profinis. Topology 3 (1965), 413–420.CrossRefGoogle Scholar