Lower bounds for \(K_ 2^{top}({\hat {\mathbb{Z}}}_ p\pi)\) and \(K_ 2({\mathbb{Z}}\pi)\)
Let \(\pi\) be a finite group. In order to compute SK\({}_ 1\)(\({\mathbb{Z}}\pi)\) and find lower bounds for \(K_ 2({\mathbb{Z}}\pi)\) it is very useful to understand \(K_ 2({\hat {\mathbb{Z}}}_ p\pi)\), p a prime. In this paper the author detects elements of \(K_ 2({\hat {\mathbb{Z}}}_ p\pi)\) by describing \(K^*\!_ 2({\hat {\mathbb{Z}}}_ p\pi)=\lim_{\leftarrow}\) \(co\ker (K_ 2({\hat {\mathbb{Z}}}_ p{\tilde \pi})\to K_ 2({\hat {\mathbb{Z}}}_ p\pi))\), where the limit is taken over all surjections \({\tilde \pi}\to \pi\) of p-groups. More generally, \({\hat {\mathbb{Z}}}_ p\) can be replaced in these arguments by the valuation ring A in any finite unramified extension of \({\hat {\mathbb{Z}}}_ p.\) There are many applications. For example: (1) \(K^*\!_ 2({\hat {\mathbb{Z}}}_ p\pi)=0\) if and only if \(\pi\) is cyclic or quaternionic; (2) \(K_ 2\)(F\(\pi)\) can be described explicitly for any finite field F and abelian group \(\pi\) ; (3) For odd primes p, upper and lower bounds are obtained for \(Cl_ 1({\mathbb{Z}}\pi)\), \(\pi\) a p-group (an explicit combinatorial algorithm computing \(Cl_ 1({\mathbb{Z}}\pi)\) is conjectured here and proved in a subsequent paper). The methods used to derive these results involve the construction of several maps and exact sequences of great interest.
scientific article; zbMATH DE number 3957390
scientific article