Thick subcategories of the stable module category

D. Benson; Jon Carlson; Jeremy Rickard

Fundamenta Mathematicae (1997)

  • Volume: 153, Issue: 1, page 59-80
  • ISSN: 0016-2736

Abstract

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We study the thick subcategories of the stable category of finitely generated modules for the principal block of the group algebra of a finite group G over a field of characteristic p. In case G is a p-group we obtain a complete classification of the thick subcategories. The same classification works whenever the nucleus of the cohomology variety is zero. In case the nucleus is nonzero, we describe some examples which lead us to believe that there are always infinitely many thick subcategories concentrated on each nonzero closed homogeneous subvariety of the nucleus.

How to cite

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Benson, D., Carlson, Jon, and Rickard, Jeremy. "Thick subcategories of the stable module category." Fundamenta Mathematicae 153.1 (1997): 59-80. <http://eudml.org/doc/212215>.

@article{Benson1997,
abstract = {We study the thick subcategories of the stable category of finitely generated modules for the principal block of the group algebra of a finite group G over a field of characteristic p. In case G is a p-group we obtain a complete classification of the thick subcategories. The same classification works whenever the nucleus of the cohomology variety is zero. In case the nucleus is nonzero, we describe some examples which lead us to believe that there are always infinitely many thick subcategories concentrated on each nonzero closed homogeneous subvariety of the nucleus.},
author = {Benson, D., Carlson, Jon, Rickard, Jeremy},
journal = {Fundamenta Mathematicae},
keywords = {triangulated categories; stable module categories; thick subcategories; varieties of modules; idempotent modules; nucleus; principal blocks; finite groups; finitely generated modules},
language = {eng},
number = {1},
pages = {59-80},
title = {Thick subcategories of the stable module category},
url = {http://eudml.org/doc/212215},
volume = {153},
year = {1997},
}

TY - JOUR
AU - Benson, D.
AU - Carlson, Jon
AU - Rickard, Jeremy
TI - Thick subcategories of the stable module category
JO - Fundamenta Mathematicae
PY - 1997
VL - 153
IS - 1
SP - 59
EP - 80
AB - We study the thick subcategories of the stable category of finitely generated modules for the principal block of the group algebra of a finite group G over a field of characteristic p. In case G is a p-group we obtain a complete classification of the thick subcategories. The same classification works whenever the nucleus of the cohomology variety is zero. In case the nucleus is nonzero, we describe some examples which lead us to believe that there are always infinitely many thick subcategories concentrated on each nonzero closed homogeneous subvariety of the nucleus.
LA - eng
KW - triangulated categories; stable module categories; thick subcategories; varieties of modules; idempotent modules; nucleus; principal blocks; finite groups; finitely generated modules
UR - http://eudml.org/doc/212215
ER -

References

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  1. [1] D. J. Benson, Representations and Cohomology I, Cambridge Stud. Adv. Math. 30, Cambridge Univ. Press, 1990. 
  2. [2] D. J. Benson, Representations and Cohomology II, Cambridge Stud. Adv. Math. 31, Cambridge Univ. Press, 1991. 
  3. [3] D. J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1995), 196-205. Zbl0879.20004
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  7. [7] L. Chouinard, Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra 7 (1976), 278-302. Zbl0327.20020
  8. [8] E. S. Devinatz, M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory, I, Ann. of Math. (2) 128 (1988), 207-241. Zbl0673.55008
  9. [9] K. Erdmann, Algebras and semidihedral defect groups I, Proc. London Math. Soc. 57 (1988), 109-150. Zbl0648.20007
  10. [10] K. Erdmann, Algebras and semidihedral defect groups II, Proc. London Math. Soc. 60 (1990), 123-165. Zbl0687.20006
  11. [11] M. J. Hopkins, Global methods in homotopy theory, in: Homotopy Theory (Durham, 1985), London Math. Soc. Lecture Note Ser. 117, Cambridge Univ. Press, 1987, 73-96. 
  12. [12] A. Neeman, Stable homotopy as a triangulated functor, Invent. Math. 109 (1992), 17-40. Zbl0793.55007
  13. [13] A. Neeman, The chromatic tower for D(R), Topology 31 (1992), 519-532. Zbl0793.18008
  14. [14] J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), 436-456. Zbl0642.16034
  15. [15] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303-317. Zbl0685.16016
  16. [16] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), 37-48. Zbl0683.16030
  17. [17] J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. (3) 72 (1996), 331-358. Zbl0862.20010
  18. [18] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc., to appear. Zbl0910.20034
  19. [19] G. Schneider, Die 2-modularen Darstellungen der Mathieu-Gruppe M 12 , doctoral dissertation, Essen, 1981. 

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