Advertisement

Algebras and Representation Theory

, Volume 21, Issue 2, pp 399–417 | Cite as

Separable Commutative Rings in the Stable Module Category of Cyclic Groups

  • Paul Balmer
  • Jon F. Carlson
Article
  • 47 Downloads

Abstract

We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group.

Keywords

Separable Etale Ring-object Stable category 

Mathematics Subject Classification (2010)

20C20 14F20 18E30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

Both authors would like to thank Bielefeld University for its support and kind hospitality during a visit when this work was undertaken. We are also thankful to Danny Krashen, Akhil Mathew and Greg Stevenson for valuable discussions.

References

  1. 1.
    Almkvist, G., Fossum, R.: Decomposition of exterior and symmetric powers of indecomposable Z/pZ-modules in characteristic p and relations to invariants. In: Séminaire d’Algèbre Paul Dubreil, 30ème année (Paris, 1976–1977), volume 641 of Lecture Notes in Mathematics, pp. 1–111. Springer, Berlin (1978)Google Scholar
  2. 2.
    Auslander, M., Goldman, O.: The Brauer group of a commutative ring. Trans. Amer. Math. Soc. 97, 367–409 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balmer, P.: Separability and triangulated categories. Adv. Math. 226(5), 4352–4372 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balmer, P.: Stacks of group representations. J. Eur. Math. Soc. (JEMS) 17(1), 189–228 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Balmer, P.: The derived category of an étale extension and the separable Neeman-Thomason theorem. J. Inst. Math. Jussieu 15(3), 613–623 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Balmer, P., Dell’Ambrogio, I., Sanders, B.: Restriction to finite-index subgroups as étale extensions in topology, KK-theory and geometry. Algebr. Geom. Topol. 15(5), 3025–3047 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Benson, D.J., Carlson, J.F.: Nilpotent elements in the Green ring. J. Algebra 104(2), 329–350 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Böhm, G., Brzeziński, T., Wisbauer, R.: Monads and comonads on module categories. J. Algebra 322(5), 1719–1747 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bruguières, A., Virelizier, A.: Hopf monads. Adv. Math. 215(2), 679–733 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Carlson, J.F., Friedlander, E.M., Pevtsova, J.: Modules of constant Jordan type. J. Reine Angew. Math. 614, 191–234 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Crawley, P., Jónsson, B.: Refinements for infinite direct decompositions of algebraic systems. Pac. J. Math. 14, 797–855 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Curtis, C.W., Reiner, I.: Representations of Finite Groups and Associated Algebras. Interscience Publishers, New York (1966)Google Scholar
  13. 13.
    DeMeyer, F., Ingraham, E.: Separable Algebras Over Commutative Rings. Lecture Notes in Mathematics, vol. 181. Springer-Verlag, Berlin (1971)Google Scholar
  14. 14.
    Kelly, G.M.: On the radical of a category. J. Austral. Math. Soc. 4, 299–307 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lam, T.Y.: A First Course in Noncommutative Rings, volume 131 of Graduate Texts in Mathematics. Springer-Verlag, New York (1991)CrossRefGoogle Scholar
  16. 16.
    Mathew, A.: The Galois group of a stable homotopy theory. Adv. Math. 291, 403–541 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Neeman, A.: Triangulated Categories, volume 148 of Annals of Mathematics Studies. Princeton University Press (2001)Google Scholar
  18. 18.
    Neeman, A.: Separable monoids in D qc(X). Journal Reine for Angewandte Mathematik. Preprint, 44 pages, to appear, available at doi: 10.1515/crelle-2015-0039 (2015)
  19. 19.
    Premet, A: The green ring of a simple three-dimensional lie p-algebra (Russian). Izv. Vyssh. Uchebn. Zaved. Mat. 10, 56–67 (1991)Google Scholar
  20. 20.
    Rickard, J.: Idempotent modules in the stable category. J. London Math. Soc. (2) 56(1), 149–170 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Warfield, R.B., Jr.: A Krull-Schmidt theorem for infinite sums of modules. Proc. Amer. Math. Soc. 22, 460–465 (1969)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Mathematics DepartmentUCLALos AngelesUSA
  2. 2.Department of MathematicsUniversity of GeorgiaAthensUSA

Personalised recommendations