Notes on Milnor-Witt K-theory

Abstract

These notes develop the foundations of Milnor-Witt K-theory for fields of arbitrary characteristic, without any perfectness assumptions. Extending the work of Morel and Feld, we establish all functorial properties of Milnor-Witt K-theory with careful attention to twists. A main new contribution is the computation of transfers in the general, in particular inseparable, case using Grothendieck (differential) trace maps. The results provide a complete framework for the functoriality axioms of Milnor-Witt modules and form the basis for an upcoming work on Chow-Witt groups with Niels Feld and Fangzhou Jin.

arXiv commentFinal revised version, 94 pp. The local closedness lemma, 5.2.7, and the proof of the residue-transfer relation (R3b), Theorem 5.2.6, have been revised; the construction of differential traces §6.2 and the description of valuation-theoretic defects §5.3.12 have been corrected and clarified. To appear in the proceedings of the Summer School "Motives and Arithmetic Groups"