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On the Depth of Modular Invariant Rings for the Groups C p × C p

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Symmetry and Spaces

Part of the book series: Progress in Mathematics ((PM,volume 278))

Summary

Let G be a finite group, k a field of characteristic p and V a finite dimensional kG -module. Let R :=Sym(V* ), the symmetric algebra over the dual spaceV* , with G acting by graded algebra automorphisms. Then it is known that the depth of the invariant ring R G is at least min{ dim(V ), dim(VP )+cc G (R )+1} . A module V for which the depth of R G attains this lower bound was called flat by Fleischmann, Kemper and Shank [13]. In this paper some of the ideas in [13] are further developed and applied to certain representations of Cp ×Cp, generating many new examples of flat modules. We introduce the useful notion of “strongly flat” modules, classifying them for the group C 2 ×C 2, as well as determining the depth of R G for any indecomposable modular representation of C 2 ×C 2.

Mathematics Subject Classification (2000): 13A50

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Authors and Affiliations

  1. Department of Mathematics, University of Aberdeen, Meston Hall, Aberdeen, AB24 3UE, UK

    Jonathan Elmer

  2. Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent, CT2 7NF, UK

    Peter Fleischmann

Authors
  1. Jonathan Elmer
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  2. Peter Fleischmann
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Correspondence to Jonathan Elmer .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, University of New Brunswick, Fredericton, E3B 5A3, Canada

    H. E. A. Campbell

  2. Dept. Mathematics, North Carolina State University, Harrelson Hall 255, Raleigh, 27695, U.S.A.

    Aloysius G. Helminck

  3. Inst. Mathematik, Universität Basel, Rheinsprung 21, Basel, 4051, Switzerland

    Hanspeter Kraft

  4. Dept. Mathematics & Statistics, Queen's University, Kingston, University Ave. 99, Kingston, K7L 3N6, Canada

    David Wehlau

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Elmer, J., Fleischmann, P. (2010). On the Depth of Modular Invariant Rings for the Groups C p × C p . In: Campbell, H., Helminck, A., Kraft, H., Wehlau, D. (eds) Symmetry and Spaces. Progress in Mathematics, vol 278. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4875-6_4

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