### Abstract

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 |

### Fingerprint

### Keywords

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

### Cite this

*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

**On the Depth of Modular Invariant Rings for the Groups C p × C p.** / Elmer, Jonathan; Fleischmann, Peter .

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

*Symmetry and Spaces: In Honor of Gerry Schwarz.*Progress in Mathematics, vol. 278, Springer , pp. 45-61. https://doi.org/10.1007/978-0-8176-4875-6_4

}

TY - CHAP

T1 - On the Depth of Modular Invariant Rings for the Groups C p × C p

AU - Elmer, Jonathan

AU - Fleischmann, Peter

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - modular invariant theory

KW - modular representation theory

KW - depth, transfer

KW - cohomology

U2 - 10.1007/978-0-8176-4875-6_4

DO - 10.1007/978-0-8176-4875-6_4

M3 - Chapter (peer-reviewed)

SN - 978-0-8176-4874-9

SN - 978-0-8176-4875-6

T3 - Progress in Mathematics

SP - 45

EP - 61

BT - Symmetry and Spaces

A2 - Campbell, H E A

A2 - Helminck, Aloysius G

A2 - Kraft, Hanspeter

A2 - Wehalu, David

PB - Springer

ER -