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 ${\mathrm {TRR}}$ . This is analogous to the conjecture of Milnor, proved in [Kat82] 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 ${\mathrm {TRR}}$ and of real topological cyclic homology, for all fields.