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

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.

arXiv comment40 pages, added helpful comments from a referee including a construction of the real trace map

References in corpus (6)

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• The big de Rham-Witt complex
• Equivariant Structure on Smash Powers
• Real topological cyclic homology of spherical group rings
• $K$-theory of Hermitian Mackey functors and a reformulation of the Novikov Conjecture
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