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Homotopy Formulas for Cyclic Groups Acting on Rings

Published online by Cambridge University Press:  20 November 2018

Christian Kassel
Christian Kassel*
Affiliation:
Institut de Recherche Mathématique Avancée, CNRS - Université Louis Pasteur, 7 rue René Descartes, 67084 Strasbourg Cedex, France e-mail: kassel@math.u-strasbg.fr
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Abstract

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The positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff.

Keywords

20J0620K0116W2218G35group cohomologynorm mapcyclic grouphomotopy
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 1 , 01 March 2008 , pp. 81 - 85
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Aljadeff, E. and Kassel, C., Explicit norm one elements for ring actions of finite abelian groups. Israel J. Math. 129(2002), 99–108.Google Scholar
[2] Aljadeff, E. and Kassel, C., Norm formulas for finite groups and induction from elementary abelian subgroups. J. Algebra 303(2006), no. 2, 677–706.Google Scholar
[3] Cartan, H. and Eilenberg, S., Homological Algebra. Princeton University Press, Princeton, 1956.Google Scholar
[4] Kassel, C., Homologie cyclique, caractère de Chern et lemme de perturbation. J. Reine Angew. Math. 408(1990), 159–180.Google Scholar