An analogue of the Milnor conjecture for the de Rham-Witt complex in characteristic 2

TL;DR

This paper describes the structure of the modulo 2 de Rham-Witt complex for fields of characteristic 2, linking it to real topological restriction homology and providing explicit calculations for all fields.

Contribution

It offers a new explicit description of the modulo 2 de Rham-Witt complex in characteristic 2, connecting it to TRR and real topological cyclic homology, extending Milnor conjecture analogies.

Findings

Explicit description of the modulo 2 de Rham-Witt complex

Calculation of homotopy groups of TRR fixed points

Connections established between de Rham-Witt complex and topological cyclic homology

Abstract

We describe the modulo $2$ de Rham-Witt complex of a field of characteristic $2$, in terms of the powers of the augmentation ideal of the $\mathbb{Z}/2$-geometric fixed points of real topological restriction homology TRR. This is analogous to the conjecture of Milnor, proved by Kato for fields of characteristic $2$, which describes the modulo $2$ Milnor K-theory in terms of the powers of the augmentation ideal of the Witt group of symmetric forms. Our proof provides a somewhat explicit description of these objects, as well as a calculation of the homotopy groups of the geometric fixed points of TRR and of real topological cyclic homology, for all fields.