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

TL;DR

This work extends Milnor's conjecture to the mod $2$ de Rham–Witt complex in characteristic $2$ by analyzing the $\mathbb{Z}/2$-geometric fixed points of real TRR and related invariants. It constructs a concrete bridge between the algebra of $2$-typical Witt vectors and the fixed-point data of $\mathrm{TRR}$ via explicit generators and a fundamental ideal, proving a strict isomorphism $W_{\langle 2^\bullet\rangle}\Omega^*_k/2 \cong J_{\langle 2^\bullet\rangle}^*/J_{\langle 2^\bullet\rangle}^{*+1}$. The paper also connects this Milnor-type description to real TC via the equaliser/coequaliser framework and a trace map from real K-theory, yielding a parallel statement for $\mathrm{TCR}$ and $\mathrm{TC}$ in characteristic $2$ in terms of Witt group data. Collectively, the results provide explicit, calculable, and structurally rich descriptions of the de Rham–Witt complex in characteristic $2$ that mirror Milnor–Kato-type phenomena and illuminate links to real K-theory and Witt groups.

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.

Theorems & Definitions (53)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Definition 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3