\(K_ 2\) of p-adic group rings of abelian p-groups
type: Item
instance of: scholarly article
title: \(K_ 2\) of p-adic group rings of abelian p-groups
zbMATH Open document ID: 0625.18005
DOI: 10.1007/BF01166703
author: Bob Oliver
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183712
review text: Let A be the ring of integers in a finite unramified extension of \({\mathbb{Q}}_ p\). The author shows that for all abelian - and some specific non-abelian - p-groups G there exists a natural homomorphism \(\Gamma_ 2(G)\) from the continuous \(K_ 2\)-group \(K^ c_ 2(AG)\) to the cyclic homology group \(HC_ 1(AG)\), which factors through the Whitehead group \(Wh_ 2^{{\mathbb{Z}}}(AG)\). For abelian p-groups this yields a short exact sequence \[ 0\quad \to \quad Wh_ 2^{{\mathbb{Z}}}(AG)\quad \to \quad HC_ 1(AG)\quad \to \quad \tilde H_ 2(G)\quad \to \quad 0\quad, \] where \(\tilde H_ 2(G)\) is the extension of \(H_ 2(G)\) introduced by Dennis. As a consequence lower bounds for the order of \(K_ 2({\mathbb{Z}} G)\) are obtained.
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 11S70
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 4014963
zbMATH Keywords: algebraic K-theory
zbMATH Keywords: local group rings
zbMATH Keywords: continuous K-theory
zbMATH Keywords: p-adic field
zbMATH Keywords: ring of integers
zbMATH Keywords: finite unramified extension
zbMATH Keywords: p-groups
zbMATH Keywords: \(K_ 2\)
zbMATH Keywords: cyclic homology
zbMATH Keywords: Whitehead group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4014963