\(K_ 2\) of p-adic group rings of abelian p-groups
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.
scientific article; zbMATH DE number 4014963