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The Quarterly Journal of Mathematics Advance Access published online on June 10, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/han008
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© 2008. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

ON MORITA THEORY FOR SELF-DUAL MODULES

Wolfgang Willems {dagger}

Institut für Algebra und Geometrie, Fakultät für Mathematik, Otto-von-Guericke-Universität, 39016 Magdeburg, Germany

Alexander Zimmermann{ddagger}

Université de Picardie, Faculté de Mathématiques et LAMFA (UMR 6140 du CNRS), 33 rue St Leu, F-80039 Amiens Cedex 1, France

{ddagger} Corresponding author. E-mail: alexander.zimmermann{at}u-picardie.fr

Received 15 June 2007; revised 25 March 2008
  Abstract

Let G be a finite group and k be a field of characteristic p. It is known that a kG-module V carries a non-degenerate G-invariant bilinear form b if and only if V is self-dual. We show that whenever a Morita bimodule M that induces an equivalence between two blocks such as B(kG) and B(kH) of group algebras kG and kH is self-dual, then the correspondence preserves self-duality. Even more, if the bilinear form on M is symmetric, then, for p odd, the correspondence preserves the geometric type of simple modules. In characteristic 2, this holds also true for projective modules.

{dagger} E-mail: wolfgang.willems{at}ovgu.de


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