Comparison of Karoubi's regulator and the \(p\)-adic Borel regulator (Q2905339)

Let \(K\) be a finite extension of the field of \(p\)-adic numbers \(\mathbb{Q}_p\). For \(n>1\), let \(K_{2n-1}(K)\) (resp. \(K^{\mathrm{rel}}_{2n-1}(K)\)) denote the Quillen \(K\)-theory group (resp. the Karoubi relative \(K\)-theory group). One can consider two different regulators on \(K_{2n-1}(K)\otimes \mathbb{Q}\). On the one hand, using Karoubi's relative Chern character, one can define a (Karoubi) regulator map \(r_p:K^{\mathrm{rel}}_{2n-1}(K)\to K\). This map has been analyzed by \textit{N. Hamida} [C. R., Math., Acad. Sci. Paris 342, No. 11, 807--812 (2006; Zbl 1104.19005)], who gave an explicit description for it in terms of Goodwillie's relative \(K\)-theory. On the other hand, if \(R\) is the ring of integers of \(K\), \textit{A. Huber} and \textit{G. Kings} [J. Inst. Math. Jussieu 10, No. 1, 149--190 (2011; Zbl 1243.11113)] defined a \(p\)-adic analogue of Borel's regulator map \(b_p:K_{2n-1}(R)\to K\). Now the canonical maps \(K_{2n-1}(R)\to K_{2n-1}(K)\) and \(K^{\mathrm{rel}}_{2n-1}(K)\to K_{2n-1}(K)\) become isomorphisms after tensoring with \(\mathbb{Q}\). Taking these isomorphisms as identifications one can consider two regulator maps \(r_p,b_p:K_{2n-1}(K)\otimes\mathbb{Q}\to K\) and it makes sense to compare them. Then the main result of this paper is that NEWLINE\[NEWLINEr_p=\frac{(-1)^{n-1}}{(n-1)!(2n-2)!}b_p.NEWLINE\]NEWLINE The proof uses above-mentioned Hamida's explicit description of Karoubi's regulator. The paper ends with an appendix on integration on the standard simplex.