Geodesic
KAROUBI’S RELATIVE CHERN CHARACTER, THE RIGID SYNTOMIC REGULATOR, AND THE BLOCH–KATO EXPONENTIAL MAP
Forum of Mathematics, Sigma, Tome 2 (2014)

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We construct a variant of Karoubi’s relative Chern character for smooth separated schemes over the ring of integers in a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$ -adic field, and prove a comparison with the rigid syntomic regulator. For smooth projective schemes, we further relate the relative Chern character to the étale $p$ -adic regulator via the Bloch–Kato exponential map. This reproves a result of Huber and Kings for the spectrum of the ring of integers, and generalizes it to all smooth projective schemes as above. 
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GEORG TAMME. KAROUBI’S RELATIVE CHERN CHARACTER, THE RIGID SYNTOMIC REGULATOR, AND THE BLOCH–KATO EXPONENTIAL MAP. Forum of Mathematics, Sigma, Tome 2 (2014). doi: 10.1017/fms.2014.21

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