Karoubi's relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map (Q2879433)

For a Banach algebra \(A\) one can consider its algebraic \(K\)-theory (by viewing \(A\) as a ring) and its topological \(K\)-theory. The homotopy fibre of the map from the former to the latter \(K\)-theory is called the relative \(K\)-theory of \(A\). M. Karoubi constructed (cf. [\textit{M. Karoubi}, C. R. Acad. Sci., Paris, Sér. I 297, 557--560 (1983; Zbl 0532.18009); Homologie cyclique et K-théorie. Paris: Société Mathématique de France (1987; Zbl 0648.18008); \textit{A. Connes} and \textit{M. Karoubi}, \(K\)-Theory 2, No. 3, 431--463 (1988; Zbl 0671.46034)]) the relative Chern character as an homomorphism \(K_{i}^{\text{rel}}(A)\rightarrow HC_{i-1}(A)\) from relative \(K\)-theory to continuous cyclic homology. Following the idea of Karoubi, Hamida compared, for \(A={\mathbb C}\) the relative Chern character and the Borel regulator \(K_{2n-1}(\mathbb C).\) The generalization of this result to ultrametric Banach algebras was done by the author in [J. K-Theory 9, No. 3, 579--600 (2012; Zbl 1272.19003)]. One can also consider relative Chern character for smooth varieties over \({\mathbb C}.\) This can be compared with Beilinson's regulator which maps algebraic \(K\)-theory to Deligne-Beilinson cohomology. This was done by the author in [Ann. Sci. Éc. Norm. Supér. (4) 45, No. 4, 601--636 (2012; Zbl 1266.19004)]. In the present paper, the author considers a variant of the comparison result for the \(p\)-adic situation. Let \(R\) be a complete discrete valuation ring with field of fractions \(K\) of characteristic zero and perfect residue field \(k\) of characteristic \(p.\) Let \(X\) be a smooth \(R\)-scheme. The analogue of Beilinson regulator is the rigid syntomic regulator (cf. [\textit{M. Gros}, Invent. Math. 115, No. 1, 61--79 (1994; Zbl 0799.14010)], [\textit{A. Besser}, Isr. J. Math. 120, Part B, 291--334 (2000; Zbl 1001.19003)]). In the paper, the author introduces relative \(K\)-theory of \(X\), and relative cohomology groups \(H^{*}_{\text{rel}} (X,n)\) which map naturally to the rigid syntomic cohomology groups. The main result of the paper is commutativity, for \(i>0,\) of the following diagram: NEWLINE\[CARRIAGE_RETURNNEWLINE\begin{tikzcd} K_{i}^{\text{rel}}(X) \ar[r] \ar[d, "\mathrm{ch}^{\text{rel}}_{n,i}" '] & K_{i}(X)\ar[d, "\mathrm{ch}^{\text{syn}}_{n,i}"] \\ H^{2n-i}_{\text{rel}} (X,n) \ar[r] & H^{2n-i}_{\text{syn}} (X,n) \end{tikzcd}CARRIAGE_RETURNNEWLINE\]NEWLINE Using exponential map of Bloch-Kato the author is able to relate, for smooth projective schemes, the relative Chern charcter to the étale \(p\)-adic regulator. This reproves and generalizes a result of \textit{A. Huber} and \textit{G. Kings} [J. Inst. Math. Jussieu 10, No. 1, 149--190 (2011; Zbl 1243.11113)].