Advertisement

On the depth of separating algebras for finite groups

  • Jonathan ElmerEmail author
Original Paper
  • 66 Downloads

Abstract

Consider a finite group G acting on a vector space V over a field \({{\Bbbk}}\) of characteristic p >  0. A separating algebra is a subalgebra A of the ring of invariants \({{{\Bbbk}[V]^G}}\) with the same point separation properties. In this article we compare the depth of an arbitrary separating algebra with that of the corresponding ring of invariants. We show that, in some special cases, the depth of A is bounded above by the depth of \({{{\Bbbk}[V]^G}}\) .

Keywords

Invariant theory Separating algebra Depth Modular representation theory Cohomology modules 

Mathematics Subject Classification (2000)

13A50 (Actions of groups on commutative rings; invariant theory) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bruns, W., Herzog, J.: Cohen–Macaulay rings. In: Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)Google Scholar
  2. Campbell H.E.A., Hughes I.P., Kemper G., Shank R.J., Wehlau D.L.: Depth of modular invariant rings. Transform. Groups 5(1), 21–34 (2000)MathSciNetCrossRefGoogle Scholar
  3. Derksen, H., Kemper, G.: Computational invariant theory. Invariant Theory and Algebraic Transformation Groups, I. Springer, Berlin (2002) (Encyclopaedia of Mathematical Sciences, 130)Google Scholar
  4. Dufresne E.: Separating invariants and finite reflection groups. Adv. Math. 221, 1979–1989 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. Dufresne E., Elmer J., Kohls M.: The Cohen–Macaulay property of separating invariants of finite groups. Transform. Groups 14(4), 771–785 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. Ellingsrud G., Skjelbred T.: Profondeur d’anneaux d’invariants en caractéristique p. Compositio Math. 41(2), 233–244 (1980)MathSciNetzbMATHGoogle Scholar
  7. Elmer J.: Associated primes for cohomology modules. Arch. Math. (Basel) 91(6), 481–485 (2008)MathSciNetzbMATHGoogle Scholar
  8. Elmer J.: Depth and detection in modular invariant theory. J. Algebra 322(5), 1653–1666 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  9. Elmer, J., Fleischmann, P.: On the depth of modular invariant rings for the groups C p ×  C p. In: Symmetry and Spaces, Progr. Math., vol. 278, pp. 45–61. Birkhäuser Boston Inc., Boston (2010)Google Scholar
  10. Evens L.: The cohomology of groups. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford Science Publications, New York (1991)Google Scholar
  11. Fleischmann P., KemperG. Shank R.J.: Depth and cohomological connectivity in modular invariant theory. Trans. Am. Math. Soc. 357(9), 3605–3621 (2005) (electronic)zbMATHCrossRefGoogle Scholar
  12. Kemper G.: On the Cohen–Macaulay property of modular invariant rings. J. Algebra 215(1), 330–351 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  13. Kemper, G.: The depth of invariant rings and cohomology. J. Algebra 245(2), 463–531 (2001) (with an appendix by Kay Magaard)Google Scholar

Copyright information

© The Managing Editors 2011

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK

Personalised recommendations