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Decomposing the complexity quotient category

Published online by Cambridge University Press:  24 October 2008

D. J. Benson
D. J. Benson
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
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Extract

In the modular representation theory of finite groups, much recent effort has gone into describing cohomological properties of the category of finitely generated modules. In recent joint work of the author with Jon Carlson and Jeremy Rickard[3], it has become clear that for some purposes the finiteness restriction is undesirable. In particular, in the quotient category of kG-modules by the subcategory of modules of less than maximal complexity, it turns out that finitely generated modules can have infinitely generated summands, and that including these summands in the category repairs the lack of Krull–Schmidt property.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 4 , November 1996 , pp. 589 - 595
Copyright
Copyright © Cambridge Philosophical Society 1996

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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]Benson, D. J.. Representations and cohomology II: cohomology of groups and modules. Cambridge studies in advantaged mathematics, vol. 31 (Cambridge University Press, 1991).Google Scholar
[3]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
[4]Carlson, J. F.. The cohomology ring of a module. J. Pure Appl. Algebra 36 (1985), 105–121.CrossRefGoogle Scholar
[5]Carlson, J. F.. Cohomology rings of induced modules. J. Pure Appl. Algebra 44 (1987), 85–97.CrossRefGoogle Scholar
[6]Carlson, J. F.. The decomposition of the trivial module in the complexity quotient category. J. Pure Appl. Algebra 106 (1996), 23–44.CrossRefGoogle Scholar
[7]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
[8]Carlson, J. F. and Wheeler, W. W.. Varieties and localizations of module categories. J. Pure Appl. Algebra 102 (1995), 137–153.CrossRefGoogle Scholar
[9]Evens, L.. The cohomology of groups (Oxford, 1991).CrossRefGoogle Scholar
[10]McConnell, J. C. and Robson, J. C.. Noncommutative Noetherian rings (J. Wiley, 1988).Google Scholar
[11]Quillen, D.. The spectrum of an equivariant cohomology ring, I, II. Ann. of Math. 94 (1971), 549–602.CrossRefGoogle Scholar