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.

References in corpus (1)

• Algebraic K-theory and trace invariants

Cited by in corpus (7)

• K-theory and topological cyclic homology of henselian pairs
• The $p$-completed cyclotomic trace in degree $2$
• Some recent advances in topological Hochschild homology
• Complex orientations for THH of some perfectoid fields
• The R(S^1)-graded equivariant homotopy of THH(F_p)
• On the p-typical de Rham-Witt complex over W(k)
• Local structure of the overconvergent de Rham-Witt complex