Let \(B=A[[t;\sigma,\delta]]\) be a skew power series ring and assume that \(\sigma\) extends to an inner automorphism of \(B\). In this paper under review, the author exploits an observation first made by P. Schneider and O. Venjakob [J. Pure Appl. Algebra 204, No. 2, 349-367 (2006; Zbl 1143.16038)], namely the existence of a short exact sequence \[ 0\to B\widehat\otimes_AM\to B\widehat\otimes_AM\to M\to 0 \] for a pseudocompact \(B\)-module \(M\). This was used – first in the context of non-commutative Iwasawa theory by D. Burns [Publ. Res. Inst. Math. Sci. 45, No. 1, 75-87 (2009; Zbl 1225.11137)], then in general by P. Schneider and O. Venjakob [Am. J. Math. 132, No. 1, 1-36 (2010; Zbl 1191.16041)] – to construct a natural splitting of the boundary map \[ K_1(B_S)@>\partial_0>>K_0(B,B_S) \] in the K-theory localisation sequence of a regular Noetherian skew power series ring \(B\) and its localisation \(B_S\) with respect to a certain left denominator set \(S\).
In order to generalize this result, the author shows that by an application of Waldhausen’s additivity theorem [F. Waldhausen, Lect. Notes Math. 1126, 318-419 (1985; Zbl 0579.18006)] to the above exact sequence and by a well-known result in homotopy theory one obtains a natural splitting of the boundary map \(\partial_n\) for every \(n\). Using Waldhausen’s construction of K-theory for arbitrary Waldhausen categories also helps us to extend this result to non-regular skew power series and even to situations where no suitable left denominator set \(S\) is known to exist.
The author applies his result to the area of non-commutative Iwasawa theory. Let \(H\) be a \(p\)-adic Lie group and \(G\) a semi-direct product of \(H\) and \(\mathbb Z_p\). Fix a topological generator \(\gamma\) of \(\mathbb Z_p\). Then \(B=\mathbb Z_p[[G]]\) is a skew power series ring over \(A=\mathbb{Z}_p[[H]]\) with \(\sigma\) given by conjugation with \(\gamma\), \(t=\gamma-1\) and \(\delta=\sigma-\text{id}\). The set \(S\) is Venjakob’s canonical Ore set and the sequence \[ K_1(\mathbb Z_p[[G]])\to K_1(\mathbb Z_p[[G]]_S)@>\partial_0>>K_0(\mathbb Z_p[[G]],\mathbb Z_p[[G]]_S)\to 0 \] is used to formulate the main conjecture of non-commutative Iwasawa theory. By the author’s result, the leftmost map in this sequence is always injective and the splitting of \(\partial_0\) does exist even if \(G\) has \(p\)-torsion elements. It was part of the proof of the non-commutative main conjecture for totally real fields to verify this fact for Galois groups of admissible \(p\)-adic Lie extensions.