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 <>
Lars Hesselholt <>