Mathematische Zeitschrift

, Volume 287, Issue 3–4, pp 947–965 | Cite as

Stratification and \(\pi \)-cosupport: finite groups

  • Dave Benson
  • Srikanth B. Iyengar
  • Henning Krause
  • Julia Pevtsova
Article

Abstract

We introduce the notion of \(\pi \)-cosupport as a new tool for the stable module category of a finite group scheme. In the case of a finite group, we use this to give a new proof of the classification of tensor ideal localising subcategories. In a sequel to this paper, we carry out the corresponding classification for finite group schemes.

Keywords

Cosupport Stable module category Finite group scheme Localising subcategory Support Thick subcategory 

Mathematics Subject Classification

Primary 16G10 Secondary 20C20 20G10 20J06 

Notes

Acknowledgements

Part of this article is based on work supported by the National Science Foundation under Grant No. 0932078000, while DB, SBI, and HK were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the 2012–2013 Special Year in Commutative Algebra. The authors thank the Centre de Recerca Matemàtica, Barcelona, for hospitality during a visit in April 2015 that turned out to be productive and pleasant. SBI and JP were partly supported by NSF Grants DMS-1503044 and DMS-0953011, respectively. We are grateful to Eric Friedlander for comments on an earlier version of this paper.

References

  1. 1.
    Alperin, J.L., Evens, L.: Varieties and elementary abelian groups. J. Pure Appl. Algebra 26, 221–227 (1982)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Avrunin, G.S., Scott, L.L.: Quillen stratification for modules. Invent. Math. 66, 277–286 (1982)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bendel, C.: Cohomology and projectivity of modules for finite group schemes. Math. Proc. Camb. Philos. Soc. 131, 405–425 (2001)MATHMathSciNetGoogle Scholar
  4. 4.
    Benson, D.J.: Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras, Cambridge Studies in Advanced Mathematics, vol. 30, 2nd edn. Cambridge University Press, Cambridge (1998)Google Scholar
  5. 5.
    Benson, D.J.: Representations and Cohomology II: Cohomology of Groups and Modules, Cambridge Studies in Advanced Mathematics, vol. 31, 2nd edn. Cambridge University Press, Cambridge (1998)Google Scholar
  6. 6.
    Benson, D.J., Carlson, J.F., Rickard, J.: Complexity and varieties for infinitely generated modules, II. Math. Proc. Camb. Philos. Soc. 120, 597–615 (1996)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Benson, D.J., Carlson, J.F., Rickard, J.: Thick subcategories of the stable module category. Fundam. Math. 153, 59–80 (1997)MATHMathSciNetGoogle Scholar
  8. 8.
    Benson, D.J., Iyengar, S.B., Krause, H.: Local cohomology and support for triangulated categories. Ann. Scient. Éc. Norm. Sup. (4) 41, 575–621 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Benson, D.J., Iyengar, S.B., Krause, H.: Stratifying triangulated categories. J. Topol. 4, 641–666 (2011)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Benson, D.J., Iyengar, S.B., Krause, H.: Stratifying modular representations of finite groups. Ann. Math. 174, 1643–1684 (2011)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Benson, D.J., Iyengar, S.B., Krause, H.: Colocalising subcategories and cosupport. J. Reine Angew. Math. 673, 161–207 (2012)MATHMathSciNetGoogle Scholar
  12. 12.
    Benson, D.J., Iyengar, S.B., Krause, H., Pevtsova, J.: Stratification for module categories of finite group schemes (2015). Preprint: arXiv:1510.06773
  13. 13.
    Benson, D.J., Krause, H.: Pure injectives and the spectrum of the cohomology ring of a finite group. J. Reine Angew. Math. 542, 23–51 (2002)MATHMathSciNetGoogle Scholar
  14. 14.
    Carlson, J.F.: The complexity and varieties of modules, Integral representations and their applications, Oberwolfach, 1980, Lecture Notes in Mathematics, vol. 882. Springer, Berlin, pp. 415–422 (1981)Google Scholar
  15. 15.
    Carlson, J.F.: The varieties and cohomology ring of a module. J. Algebra 85, 104–143 (1983)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Chouinard, L.: Projectivity and relative projectivity over group rings. J. Pure Appl. Algebra 7, 278–302 (1976)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Friedlander, E.M., Pevtsova, J.: Representation theoretic support spaces for finite group schemes. Am. J. Math. 127, 379–420 (2005). Correction: AJM 128, 1067–1068 (2006)Google Scholar
  18. 18.
    Friedlander, E.M., Pevtsova, J.: \(\Pi \)-supports for modules for finite groups schemes. Duke Math. J. 139, 317–368 (2007)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Friedlander, E.M., Suslin, A.: Cohomology of finite group schemes over a field. Invent. Math. 127, 209–270 (1997)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Jantzen, J.C.: Representations of Algebraic Groups, 2nd edn. American Mathematical Society, Providence (2003)MATHGoogle Scholar
  21. 21.
    Neeman, A.: The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Scient. Éc. Norm. Sup. (4) 25, 547–566 (1992)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Quillen, D.: The spectrum of an equivariant cohomology ring: I. Ann. Math. 94, 549–572 (1971)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Quillen, D.: The spectrum of an equivariant cohomology ring: II. Ann. Math. 94, 573–602 (1971)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Suslin, A., Friedlander, E., Bendel, C.: Support varieties for infinitesimal group schemes. J. Am. Math. Soc. 10, 729–759 (1997)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Waterhouse, W.C.: Introduction to Affine Group Schemes, Graduate Texts in Mathematics, vol. 66. Springer, Berlin (1979)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Dave Benson
    • 1
  • Srikanth B. Iyengar
    • 2
  • Henning Krause
    • 3
  • Julia Pevtsova
    • 4
  1. 1.Institute of MathematicsUniversity of AberdeenAberdeenScotland, UK
  2. 2.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  3. 3.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  4. 4.Department of MathematicsUniversity of WashingtonSeattleUSA

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