Galois invariants of \(K_1\)-groups of Iwasawa algebras (Q2895465)

For an arbitrary ring \(S\) with unit, an arbitrary group \(\Delta\) acting on \(S\) by ring automorphisms, and an arbitrary finite group \(G,\) the authors address the general question whether the natural map \(i_\ast: K_1 (S^\Delta [G]) \to K_1 (S[G])^\Delta\) is an isomorphism. A particular case is when \(S = \displaystyle\widehat{{\mathbb Z}_p^{\mathrm{ur}}} =\) the ring of integers of the \(p\)-adic completion \(\displaystyle\widehat{{\mathbb Q}_p^{\mathrm{ur}}}\) of the maximal unramified extension of \({\mathbb Q}_p\) and \(\Delta\) is generated by the Frobenius automorphism \(\varphi\) (a question raised by Fukaya and Kato in their Iwasawa theory of \(\varepsilon\)-constants).NEWLINENEWLINEThe answer is negative in general: the authors show that when \(S\) is a totally ramified integral extension of the ring of Witt vectors of a \(p\)-closed algebraic extension of \({\mathbb F}_p,\) then \(S K_1 (S[G])\) is trivial for all finite groups \(G.\) This surprising property prevents \(S K_1\) from having good Galois descent. Concentrating then on the \(Det\)-part, the authors consider the case when \(S\) is the ring of integers of a Galois extension (possibly infinite) of \({\mathbb Q}_p\) with finite absolute ramification index or the \(p\)-adic completion of such a field. They conjecture that \(i_\ast: Det (S^\Delta [G])^\times \to \;Det(S[G]^\times)^\Delta\) is an isomorphism, and they prove this when \(S\) is the ring of Witt vectors of an algebraic extension of \({\mathbb F}_p.\) This Galois descent for the \(Det\)-part can also be generalized to compact \(p\)-adic Lie groups and their Iwasawa algebras, which turns out to be quite useful in non-commutative Iwasawa theory.NEWLINENEWLINEFinally the authors deal with the \(S K_1\)-part recalling and generalizing results of \textit{R. Oliver} on Whitehead groups of finite groups. This leads to descent theorems such as: there is an exact sequence \(1 \to S K_1 ({\mathbb Z}_p [G]) \to K_1 ({\mathbb Z}_p [G]) \buildrel i_\ast \over \to K_1 \Bigl(\widehat{{\mathbb Z}_p^{\mathrm{ur}}} [G] \Bigl)^{\varphi = \mathrm{id}} \to 1,\) which induces an isomorphism of the rational \(K\)-groups \(K_1 ({\mathbb Z}_p [G])_{\mathbb Q} \simeq K_1 \Bigl(\widehat{{\mathbb Z}_p^{\mathrm{ur}}} [G] \Bigl)^{\varphi = \mathrm{id}}_{\mathbb Q}\).NEWLINENEWLINEFor the entire collection see [Zbl 1232.16001].