Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

John W. Macquarrie, Peter Symonds, Pavel A. Zalesskii

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Abstract

One of the most beautiful results in the integral representation theory of finite groups is a theorem of A. Weiss that detects a permutation R-lattice for the finite p-group G in terms of the restriction to a normal subgroup N and the N-fixed points of the lattice, where R is a finite extension of the p-adic integers. Using techniques from relative homological algebra, we generalize Weiss' Theorem to the class of infinitely generated pseudocompact lattices for a finite p-group, allowing R to be any complete discrete valuation ring in mixed characteristic. A related theorem of Cliff and Weiss is also generalized to this class of modules. The existence of the permutation cover of a pseudocompact module is proved as a special case of a more general result. The permutation cover is explicitly described.
Original languageEnglish
Pages (from-to)106925
JournalAdvances in Mathematics
Volume361
Early online date2 Dec 2019
DOIs
Publication statusPublished - 12 Feb 2020

Keywords

  • Permutation modules
  • Pseudocompact modules
  • Profinite modules
  • Covers and precovers
  • Weiss' Theorem

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