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HIGHER CONJUGATION COHOMOLOGY IN COMMUTATIVE HOPF ALGEBRAS

Published online by Cambridge University Press:  20 January 2009

M. D. Crossley  and
Sarah Whitehouse
M. D. Crossley
Affiliation:
Climatic Research Unit, University of East Anglia, Norwich NR4 7TJ, UK ([email protected])
Sarah Whitehouse
Affiliation:
Laboratoire de Géométrie-Algèbre, Université d’Artois—Pôle de Lens, Rue Jean Souvraz, S.P. 18-63207 Lens, FR ([email protected])
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Abstract

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Let $A$ be a graded, commutative Hopf algebra. We study an action of the symmetric group $\sSi_n$ on the tensor product of $n-1$ copies of $A$; this action was introduced by the second author in 1 and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory 2.

We show that for a certain class of Hopf algebras the cohomology ring $H^*(\sSi_n;A^{\otimes n-1})$ is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. With the simplest coproduct structure, the group action becomes particularly tractable and we discuss the implications this has for computations.

AMS 2000 Mathematics subject classification: Primary 16W30; 57T05; 20C30; 20J06; 55S25

Keywords

Hopf algebrascohomology operationscohomology of groupsrepresentations of symmetric groups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 44 , Issue 1 , February 2001 , pp. 19 - 26
Copyright
Copyright © Edinburgh Mathematical Society 2001

References

1 S. Whitehouse, Symmetric group actions on tensor products of Hopf algebroids, Université d’Artois preprint 99-5.

2 A. Robinson and S. Whitehouse, $\sGa$-homology of commutative rings and of $E_{\infty}$-ring spectra, Warwick preprint 76/1995.