# On the depth of modular invariant rings for the groups C_p x C_p

Elmer, Jonathan ORCID: https://orcid.org/0000-0001-5296-1987 and Fleischmann, Peter
(2009)
*On the depth of modular invariant rings for the groups C_p x C_p.*
In:
Symmetry and Spaces: in honour of Gerry Schwarz.
Progress in Mathematics
(278).
Birkhauser Boston, pp. 45-61.
9780817648749.
(doi:10.1007/978-0-8176-4875-6_4)

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## Abstract

Let G be a finite group, k a field of characteristic p and V a finite dimensional kG-module. Let R denote the symmetric algebra over the dual space V∗ 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(V^P)+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 C_p × C_p, generating many new examples of flat modules. We introduce the useful notion of “strongly flat” modules, classi-fying 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.

Item Type: | Book Section |
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Research Areas: | A. > School of Science and Technology > Design Engineering and Mathematics |

Item ID: | 19268 |

Notes on copyright: | Access to full text restricted pending copyright check |

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Depositing User: | Jonathan Elmer |

Date Deposited: | 14 Apr 2016 13:54 |

Last Modified: | 15 Oct 2019 05:14 |

ISBN: | 9780817648749 |

URI: | https://eprints.mdx.ac.uk/id/eprint/19268 |

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