$SL_2$ action on the cohomology of a rank two abelian group with arbitrary coefficient domain
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- by Eric Jespers and Alexander Zimmermann PDF
- Proc. Amer. Math. Soc. 130 (2002), 315-325 Request permission
Abstract:
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
- D. Benson, Representations and Cohomology, Cambridge 1991.
- G. R. Chapman, The cohomology ring of a finite abelian group, Proc. London Math. Soc. (3) 45 (1982), no. 3, 564–576. MR 675422, DOI 10.1112/plms/s3-45.3.564
Additional Information
- Eric Jespers
- Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
- MR Author ID: 94560
- Email: efjesper@vub.ac.be
- Alexander Zimmermann
- Affiliation: LAMFA, Faculté de Mathématiques, Université de Picardie Jules Verne, 33 rue St Leu, 80039 Amiens Cedex, France
- MR Author ID: 326742
- Email: Alexander.Zimmermann@u-picardie.fr
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 315-325
- MSC (2000): Primary 20J06, 20C05, 20F29
- DOI: https://doi.org/10.1090/S0002-9939-01-06087-7
- MathSciNet review: 1862108