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

Jonathan Elmer, Peter Fleischmann

Research output: Chapter in Book/Report/Conference proceedingChapter (peer-reviewed)

4 Citations (Scopus)

Abstract

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.
Original languageEnglish
Title of host publicationSymmetry and Spaces
Subtitle of host publicationIn Honor of Gerry Schwarz
EditorsH E A Campbell, Aloysius G Helminck, Hanspeter Kraft, David Wehalu
PublisherSpringer
Pages45-61
Number of pages17
ISBN (Print)978-0-8176-4874-9, 978-0-8176-4875-6
DOIs
Publication statusPublished - 2009

Publication series

NameProgress in Mathematics
Volume278

Keywords

  • modular invariant theory
  • modular representation theory
  • depth, transfer
  • cohomology

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    Elmer, J., & Fleischmann, P. (2009). On the Depth of Modular Invariant Rings for the Groups C p × C p. In H. E. A. Campbell, A. G. Helminck, H. Kraft, & D. Wehalu (Eds.), Symmetry and Spaces: In Honor of Gerry Schwarz (pp. 45-61). (Progress in Mathematics; Vol. 278). Springer . https://doi.org/10.1007/978-0-8176-4875-6_4