On the topological cyclic homology of the algebraic closure of a local field

Abstract

The cyclotomic trace provides a comparison of the algebraic K-theory spectrum and a pro-spectrum TR that is built from the cyclic fixed points of topological Hochschild homology. In a previous paper with Ib Madsen, we used this comparison and an approximate evaluation of the structure of the pro-spectrum TR to evaluate the p-adic K-groups of a local field K of mixed characteristic (0,p) with perfect residue field. In this paper we completly determine the structure of the pro-spectrum TR for an algebrac closure of the local field K. This leads us to formulate a conjecture for the structure of the pro-spectrum TR for the field K. We also determine the structure of the absolute de Rham-Witt complex of the valuation ring R in the algebraic closure of K. The group in degree one is a p-divisible group whose Tate module is free module of rank one over the ring of Witt vectors in R. We give an explicit generator of this Tate module.