### 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 language | English |
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Title of host publication | Symmetry and Spaces |

Subtitle of host publication | In Honor of Gerry Schwarz |

Editors | H E A Campbell, Aloysius G Helminck, Hanspeter Kraft, David Wehalu |

Publisher | Springer |

Pages | 45-61 |

Number of pages | 17 |

ISBN (Print) | 978-0-8176-4874-9, 978-0-8176-4875-6 |

DOIs | |

Publication status | Published - 2009 |

### Publication series

Name | Progress in Mathematics |
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Volume | 278 |

### Keywords

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

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## Cite this

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