Bi-relative algebraic K-theory and topological cyclic homology
(with Thomas Geisser)
It was recently proved by G. Cortinas that rationally bi-relative
algebraic K-theory and bi-relative cyclic homology agree. In this
paper we show that with finite coefficients bi-relative algebraic
K-theory and bi-relative topological cyclic homology agree. As an
application, we show that for a (possibly singular) curve over a field
k of positive characteristic p, the cyclotomic trace induces
an isomorphism of the p-adic algebraic K-groups and the
p-adic topological cyclic homology groups in degrees greater
than or equal to r where [k : kp] = pr. As a
further application, we show that for every prime p, the
difference between the p-adic K-groups of the integral group
ring of a finite group and the p-adic K-groups of a maximal
Z-order in the rational group algebra can be expressed entirely
in terms of topological cyclic homology.
Lars Hesselholt <larsh@math.nagoya-u.ac.jp>