Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T07:43:19.372Z Has data issue: false hasContentIssue false
  • English
  • Français

Complexity and varieties for infinitely generated modules, II

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

Extract

It has now been almost twenty years since Alperin introduced the idea of the complexity of a finitely generated kG-module, when G is a finite group and k is a field of characteristic p > 0. In proving one of the first major results in the area [1], Alperin and Evens demonstrated the connection of the study of complexity for modules to the group cohomology. That connection eventually led to the categorization of modules according to their associated varieties in the maximal ideal spectrum of the cohomology ring H*(G, k). In all of the work that has followed, two principles have proved to be extremely important. The first is that the associated variety of a module is directly related to the structure of the module through the rank variety which is defined by the matrix representation of the module. The second major result is the tensor product theorem which says that the variety associated to a tensor product M ⊗kN is the intersection of the varieties associated to the modules M and N. In this paper we generalize these results to infinitely generated kG-modules.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 4 , November 1996 , pp. 597 - 615
Copyright
Copyright © Cambridge Philosophical Society 1996

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]Alperin, J. L. and Evens, L.. Representations, resolutions, and Quillen's dimension theorem. J. Pure Appl. Algebra 22 (1981), 1–9.CrossRefGoogle Scholar
[2]Auslander, M.. Representation theory of Artin algebras I. Commun. in Alg. 1 (1974), 177–268.CrossRefGoogle Scholar
[3]Avrunin, G. S. and Scott, L. L.. Quillen stratification for modules. Invent. Math. 66 (1982), 277–286.CrossRefGoogle Scholar
[4]Benson, D. J., Representations and cohomology II: Cohomology of groups and modules. Cambridge studies in advanced mathematics, vol. 31 (Cambridge University Press, 1991).Google Scholar
[5]Benson, D. J., Carlson, J. F. and Rickard, J.. Complexity and varieties for infinitely generated modules. Math. Proc. Camb. Phil. Soc. 118 (1995), 223–243.CrossRefGoogle Scholar
[6]Benson, D. J., Carlson, J. F. and Robinson, G. R.. On the vanishing of group cohomology. J. Algebra 131 (1990), 40–73.CrossRefGoogle Scholar
[7]Carlson, J. F.. The varieties and the cohomology ring of a module. J. Algebra 85 (1983), 104–143.CrossRefGoogle Scholar
[8]Carlson, J. F.. The decompostion of the trivial module in the complexity quotient category. J. Pure Appl. Algebra 106 (1996), 23–44.CrossRefGoogle Scholar
[9]Carlson, J. F., Donovan, P. W. and Wheeler, W. W.. Complexity and quotient categories for group algebras. J. Pure Appl. Algebra 93 (1994), 147–167.CrossRefGoogle Scholar
[10]Carlson, J. F. and Peng, C.. Relative projectivity and ideals in cohomology rings, preprint.Google Scholar
[11]Carlson, J. F. and Wheeler, W. W.. Varieties and localizations of module categories. J. Pure Appl. Algebra 102 (1995), 137–153.CrossRefGoogle Scholar
[12]Chouinard, L.. Projectivity and relative projectivity over group rings. J. Pure Appl. Algebra 7 (1976), 287–302.CrossRefGoogle Scholar
[13]Dade, E. C.. Endo-permutation modules over p-groups, II. Ann. of Math. 108 (1978), 317–346.CrossRefGoogle Scholar
[14]Evens, L.. The cohomology of groups (Oxford University Press 1991).CrossRefGoogle Scholar
[15]Köthe, G.. Verallgemeinerte Abelsche Gruppen mit hyperkomplexen Operatorenring. Mat Z. 39 (1935), 31–44.CrossRefGoogle Scholar
[16]Quillen, D. G.. The spectrum of an equivariant cohomology ring, I, II. Ann. of Math. 94 (1971), 549–572, 573–602.CrossRefGoogle Scholar
[17]Quillen, D. G.. A cohomological criterion for p-nilpotence. J. Pure Appl. Algebra 1 (1971), 361–372.CrossRefGoogle Scholar
[18]Rickard, J.. Idempotent modules in the stable category, preprint.Google Scholar
[19]Zariski, O. and Samuel, P.. Commutative Algebra, II. Graduate Texts in Mathematics, vol. 29 (Springer-Verlag, 1975).Google Scholar