Homology of Centralizers

Abstract

Given a group Π, we study the group homology of centralizers Π _g , g ∊ Π, and of their central quotients Π _g /〈 g〉. This study is motivated by the structure of the Hochschild and the cyclic homology of group algebras, and is based on Quillen's approach to the cyclic homology of algebras via algebra extensions. A method of computing the de Rham complex of a group algebra by means of a Gruenberg resolution is also developed.

Key Words:

• Algebra extensions
• Connes periodicity
• Cyclic homology
• de Rham complex
• Exact sequence
• Group algebra
• Group homology
• Gruenberg resolution
• Hochschild homology

Mathematics Subject Classification:

• 16E40
• 20J05
• 20C07

ACKNOWLEDGMENTS

This article was completed when the authors were visiting Harish-Chandra Research Institute, Allahabad. They wish to express their gratitude to the Institute for its warm hospitality.

The research of the first author was supported by Russian Foundation for Basic Research, Grant No. 05-01-00993 and Russian Presidential Grant No. MK-3466.2007.1.

Notes

Communicated by A. Facchini.