### Bi-relative algebraic K-theory and topological cyclic homology, by Thomas Geisser and Lars Hesselholt

[An updated version was received September 27, 2004, and posted
October 11, 2004. Another update was received January 12, 2005, and
posted January 15, 2005.]

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 : k^p] = p^r. 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.

Thomas Geisser <geisser@math.usc.edu>

Lars Hesselholt <larsh@math.mit.edu>