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

Abstract

We construct a variant of Karoubi’s relative Chern character for smooth separated schemes over the ring of integers in a 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.