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$SL_2$ action on the cohomology of a rank two abelian group with arbitrary coefficient domain

Authors: Eric Jespers and Alexander Zimmermann
Journal: Proc. Amer. Math. Soc. 130 (2002), 315-325
MSC (2000): Primary 20J06, 20C05, 20F29
Published electronically: June 19, 2001
MathSciNet review: 1862108
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Abstract | References | Similar Articles | Additional Information


A rank two abelian group $C_n\times C_n$ is in a natural way an $SL_2({\mathbb Z})$-module. This induces an action of $SL_2({\mathbb Z})$ on its group cohomology $H^m(C_n\times C_n,R)$ for any trivial coefficient domain $R$. In the present note we determine this module, including the question of when the universal coefficient theorem sequence splits.

References [Enhancements On Off] (What's this?)

  • 1. D. Benson, Representations and Cohomology, Cambridge 1991.
  • 2. G. R. Chapman, The cohomology ring of a finite abelian group, Proc. London Math. Soc. 45 (1982) 564-576. MR 84g:20091

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Additional Information

Eric Jespers
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium

Alexander Zimmermann
Affiliation: LAMFA, Faculté de Mathématiques, Université de Picardie Jules Verne, 33 rue St Leu, 80039 Amiens Cedex, France

Received by editor(s): May 19, 2000
Received by editor(s) in revised form: June 12, 2000
Published electronically: June 19, 2001
Additional Notes: This research was done while the authors collaborated at the “Mathematisches Forschungsinstitut Oberwolfach” financed by the “Research in Pairs” program of the “Volkswagen Stiftung”. The first-named author is also supported in part by Fonds voor Wetenschappelijk Ondezoek (Belgium) and Onderzoeksraad Vrije Universiteit Brussel.
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2001 American Mathematical Society

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